A002033 -id:A002033 - cp's OEIS frontend

A000040 The prime numbers.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271

Offset: 1

N. J. A. Sloane

See A065091 for comments, formulas etc. concerning only odd primes. For all information concerning prime powers, see A000961. For contributions concerning "almost primes" see A002808.
A number p is prime if (and only if) it is greater than 1 and has no positive divisors except 1 and p.
A natural number is prime if and only if it has exactly two (positive) divisors.
A prime has exactly one proper positive divisor, 1.
The paper by Kaoru Motose starts as follows: "Let q be a prime divisor of a Mersenne number 2^p-1 where p is prime. Then p is the order of 2 (mod q). Thus p is a divisor of q - 1 and q > p. This shows that there exist infinitely many prime numbers." - Pieter Moree, Oct 14 2004
1 is not a prime, for if the primes included 1, then the factorization of a natural number n into a product of primes would not be unique, since n = n*1.
Prime(n) and pi(n) are inverse functions: A000720(a(n)) = n and a(n) is the least number m such that a(A000720(m)) = a(n). a(A000720(n)) = n if (and only if) n is prime.
Second sequence ever computed by electronic computer, on EDSAC, May 09 1949 (see Renwick link). - Russ Cox, Apr 20 2006
Every prime p > 3 is a linear combination of previous primes prime(n) with nonzero coefficients c(n) and |c(n)| < prime(n). - Amarnath Murthy, Franklin T. Adams-Watters and Joshua Zucker, May 17 2006; clarified by Chayim Lowen, Jul 17 2015
The Greek transliteration of 'Prime Number' is 'Protos Arithmos'. - Daniel Forgues, May 08 2009 [Edited by Petros Hadjicostas, Nov 18 2019]
A number n is prime if and only if it is different from zero and different from a unit and each multiple of n decomposes into factors such that n divides at least one of the factors. This applies equally to the integers (where a prime has exactly four divisors (the definition of divisors is relaxed such that they can be negative)) and the positive integers (where a prime has exactly two distinct divisors). - Peter Luschny, Oct 09 2012
Motivated by his conjecture on representations of integers by alternating sums of consecutive primes, for any positive integer n, Zhi-Wei Sun conjectured that the polynomial P_n(x) = Sum_{k=0..n} a(k+1)*x^k is irreducible over the field of rational numbers with the Galois group S_n, and moreover P_n(x) is irreducible mod a(m) for some m <= n(n+1)/2. It seems that no known criterion on irreducibility of polynomials implies this conjecture. - Zhi-Wei Sun, Mar 23 2013
Questions on a(2n) and Ramanujan primes are in A233739. - Jonathan Sondow, Dec 16 2013
From Hieronymus Fischer, Apr 02 2014: (Start)
Natural numbers such that there is exactly one base b such that the base-b alternate digital sum is 0 (see A239707).
Equivalently: Numbers p > 1 such that b = p-1 is the only base >= 1 for which the base-b alternate digital sum is 0.
Equivalently: Numbers p > 1 such that the base-b alternate digital sum is <> 0 for all bases 1 <= b < p-1. (End)
An integer n > 1 is a prime if and only if it is not the sum of positive integers in arithmetic progression with common difference 2. - Jean-Christophe Hervé, Jun 01 2014
Conjecture: Numbers having prime factors <= prime(n+1) are {k|k^f(n) mod primorial(n)=1}, where f(n) = lcm(prime(i)-1, i=1..n) = A058254(n) and primorial(n) = A002110(n). For example, numbers with no prime divisor <= prime(7) = 17 are {k|k^60 mod 30030=1}. - Gary Detlefs, Jun 07 2014
Cramer conjecture prime(n+1) - prime(n) < C log^2 prime(n) is equivalent to the inequality (log prime(n+1)/log prime(n))^n < e^C, as n tend to infinity, where C is an absolute constant. - Thomas Ordowski, Oct 06 2014
I conjecture that for any positive rational number r there are finitely many primes q_1,...,q_k such that r = Sum_{j=1..k} 1/(q_j-1). For example, 2 = 1/(2-1) + 1/(3-1) + 1/(5-1) + 1/(7-1) + 1/(13-1) with 2, 3, 5, 7 and 13 all prime, 1/7 = 1/(13-1) + 1/(29-1) + 1/(43-1) with 13, 29 and 43 all prime, and 5/7 = 1/(3-1) + 1/(7-1) + 1/(31-1) + 1/(71-1) with 3, 7, 31 and 71 all prime. - Zhi-Wei Sun, Sep 09 2015
I also conjecture that for any positive rational number r there are finitely many primes p_1,...,p_k such that r = Sum_{j=1..k} 1/(p_j+1). For example, 1 = 1/(2+1) + 1/(3+1) + 1/(5+1) + 1/(7+1) + 1/(11+1) + 1/(23+1) with 2, 3, 5, 7, 11 and 23 all prime, and 10/11 = 1/(2+1) + 1/(3+1) + 1/(5+1) + 1/(7+1) + 1/(43+1) + 1/(131+1) + 1/(263+1) with 2, 3, 5, 7, 43, 131 and 263 all prime. - Zhi-Wei Sun, Sep 13 2015
Numbers k such that ((k-2)!!)^2 == +-1 (mod k). - Thomas Ordowski, Aug 27 2016
Does not satisfy Benford's law [Diaconis, 1977; Cohen-Katz, 1984; Berger-Hill, 2017]. - N. J. A. Sloane, Feb 07 2017
Prime numbers are the integer roots of 1 - sin(Pi*Gamma(s)/s)/sin(Pi/s). - Peter Luschny, Feb 23 2018
Conjecture: log log a(n+1) - log log a(n) < 1/n. - Thomas Ordowski, Feb 17 2023

			From _David A. Corneth_, Oct 22 2024: (Start)
7 is a prime number as it has exactly two divisors, 1 and 7.
8 is not a prime number as it does not have exactly two divisors (it has 1, 2, 4 and 8 as divisors though it is sufficient to find one other divisor than 1 and 8)
55 is not a prime number as it does not have exactly two divisors. One other divisor than 1 and 55 is 5.
59 is a prime number as it has exactly two divisors; 1 and 59. (End)

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M. Agrawal, A Short History of "PRIMES is in P"
P. Alfeld, Notes and Literature on Prime Numbers
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Anonymous, Prime Number Master Index (for primes up to 2*10^7)
Anonymous, prime number
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Christian Axler, New estimates for the n-th prime number, arXiv:1706.03651 [math.NT], 2017.
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Index entries for "core" sequences
Index entries for sequences related to Benford's law
Additional (less important) items -- comments, formulas, references, links, programs, etc. -- related to the prime numbers, A000040. [HTML version] - [Plain text (TXT) version].

For is_prime and next_prime, see A010051 and A151800.
Cf. A000720 ("pi"), A001223 (differences between primes), A002476, A002808, A003627, A006879, A006880, A008578, A080339, A233588.
Cf. primes in lexicographic order: A210757, A210758, A210759, A210760, A210761.
Cf. A003558, A179480 (relating to the Quasi-order theorem of Hilton and Pedersen).
Boustrophedon transforms: A000747, A000732, A230953.
a(2n) = A104272(n) - A233739(n).
Related sequences:
Primes (p) and composites (c): A002808, A000720, A065855.
Primes between p(n) and 2*p(n): A063124, A070046; between c(n) and 2*c(n): A376761; between n and 2*n: A035250, A060715, A077463, A108954.
Composites between p(n) and 2*p(n): A246514; between c(n) and 2*c(n): A376760; between n and 2*n: A075084, A307912, A307989, A376759.

A000040:=Filtered([1..10^5],IsPrime); # Muniru A Asiru, Sep 04 2017

-- See also Haskell Wiki Link.
import Data.List (genericIndex)
a000040 n = genericIndex a000040_list (n - 1)
a000040_list = base ++ larger where
base = [2,3,5,7,11,13,17]
larger = p : filter prime more
prime n = all ((> 0) . mod n) $ takeWhile (\x -> x*x <= n) larger
_ : p : more = roll $ makeWheels base
roll (Wheel n rs) = [n * k + r | k <- [0..], r <- rs]
makeWheels = foldl nextSize (Wheel 1 [1])
nextSize (Wheel size bs) p = Wheel (size * p)
[r | k <- [0..p-1], b <- bs, let r = size*k+b, mod r p > 0]
data Wheel = Wheel Integer [Integer]
-- Reinhard Zumkeller, Apr 07 2014

Magma
```
[n : n in [2..500] | IsPrime(n)];
```
Magma
```
a := func< n | NthPrime(n) >;
```

A000040 := n->ithprime(n); [ seq(ithprime(i),i=1..100) ];
# For illustration purposes only:
isPrime := s -> is(1 = sin(Pi*GAMMA(s)/s)/sin(Pi/s)):
select(isPrime, [$2..100]); # Peter Luschny, Feb 23 2018

Mathematica
```
Prime[Range[60]]
```

A000040(n) := block(
if n = 1 then return(2),
return( next_prime(A000040(n-1)))
)$ /* recursive, to be replaced if possible - R. J. Mathar, Feb 27 2012 */

PARI
```
{a(n) = if( n<1, 0, prime(n))};
```

/* The following functions provide asymptotic approximations, one based on the asymptotic formula cited above (slight overestimate for n > 10^8), the other one based on pi(x) ~ li(x) = Ei(log(x)) (slight underestimate): */
prime1(n)=n*(log(n)+log(log(n))-1+(log(log(n))-2)/log(n)-((log(log(n))-6)*log(log(n))+11)/log(n)^2/2)
prime2(n)=solve(X=n*log(n)/2,2*n*log(n),real(eint1(-log(X)))+n)
\\ M. F. Hasler, Oct 21 2013

forprime(p=2, 10^3, print1(p, ", ")) \\ Felix Fröhlich, Jun 30 2014

primes(10^5) \\ Altug Alkan, Mar 26 2018

from sympy import primerange
print(list(primerange(2, 272))) # Michael S. Branicky, Apr 30 2022

a = sloane.A000040
a.list(58)  # Jaap Spies, 2007

prime_range(1, 300)  # Zerinvary Lajos, May 27 2009

The prime number theorem is the statement that a(n) ~ n * log n as n -> infinity (Hardy and Wright, page 10).
For n >= 2, n*(log n + log log n - 3/2) < a(n); for n >= 20, a(n) < n*(log n + log log n - 1/2). [Rosser and Schoenfeld]
For all n, a(n) > n log n. [Rosser]
n log(n) + n (log log n - 1) < a(n) < n log n + n log log n for n >= 6. [Dusart, quoted in the Wikipedia article]
a(n) = n log n + n log log n + (n/log n)*(log log n - log n - 2) + O( n (log log n)^2/ (log n)^2). [Cipolla, see also Cesàro or the "Prime number theorem" Wikipedia article for more terms in the expansion]
a(n) = 2 + Sum_{k = 2..floor(2n*log(n)+2)} (1-floor(pi(k)/n)), for n > 1, where the formula for pi(k) is given in A000720 (Ruiz and Sondow 2002). - Jonathan Sondow, Mar 06 2004
I conjecture that Sum_{i>=1} (1/(prime(i)*log(prime(i)))) = Pi/2 = 1.570796327...; Sum_{i=1..100000} (1/(prime(i)*log(prime(i)))) = 1.565585514... It converges very slowly. - Miklos Kristof, Feb 12 2007
The last conjecture has been discussed by the math.research newsgroup recently. The sum, which is greater than Pi/2, is shown in sequence A137245. - T. D. Noe, Jan 13 2009
A000005(a(n)) = 2; A002033(a(n+1)) = 1. - Juri-Stepan Gerasimov, Oct 17 2009
A001222(a(n)) = 1. - Juri-Stepan Gerasimov, Nov 10 2009
From Gary Detlefs, Sep 10 2010: (Start)
Conjecture:
a(n) = {n| n! mod n^2 = n(n-1)}, n <> 4.
a(n) = {n| n!*h(n) mod n = n-1}, n <> 4, where h(n) = Sum_{k=1..n} 1/k. (End)
For n = 1..15, a(n) = p + abs(p-3/2) + 1/2, where p = m + int((m-3)/2), and m = n + int((n-2)/8) + int((n-4)/8). - Timothy Hopper, Oct 23 2010
a(2n) <= A104272(n) - 2 for n > 1, and a(2n) ~ A104272(n) as n -> infinity. - Jonathan Sondow, Dec 16 2013
Conjecture: Sequence = {5 and n <> 5| ( Fibonacci(n) mod n = 1 or Fibonacci(n) mod n = n - 1) and 2^(n-1) mod n = 1}. - Gary Detlefs, May 25 2014
Conjecture: Sequence = {5 and n <> 5| ( Fibonacci(n) mod n = 1 or Fibonacci(n) mod n = n - 1) and 2^(3*n) mod 3*n = 8}. - Gary Detlefs, May 28 2014
Satisfies a(n) = 2*n + Sum_{k=1..(a(n)-1)} cot(k*Pi/a(n))*sin(2*k*n^a(n)*Pi/a(n)). - Ilya Gutkovskiy, Jun 29 2016
Sum_{n>=1} 1/a(n)^s = P(s), where P(s) is the prime zeta function. - Eric W. Weisstein, Nov 08 2016
a(n) = floor(1 - log(-1/2 + Sum_{ d | A002110(n-1) } mu(d)/(2^d-1))/log(2)) where mu(d) = A008683(d) [Ghandi, 1971] (see Ribenboim). Golomb gave a proof in 1974: Give each positive integer a probability of W(n) = 1/2^n, then the probability M(d) of the integer multiple of number d equals 1/(2^d-1). Suppose Q = a(1)*a(2)*...*a(n-1) = A002110(n-1), then the probability of random integers that are mutually prime with Q is Sum_{ d | Q } mu(d)*M(d) = Sum_{ d | Q } mu(d)/(2^d-1) = Sum_{ gcd(m, Q) = 1 } W(m) = 1/2 + 1/2^a(n) + 1/2^a(n+1) + 1/2^a(n+2) + ... So ((Sum_{ d | Q } mu(d)/(2^d-1)) - 1/2)*2^a(n) = 1 + x(n), which means that a(n) is the only integer so that 1 < ((Sum_{ d | Q } mu(d)/(2^d-1)) - 1/2)*2^a(n) < 2. - Jinyuan Wang, Apr 08 2019
Conjecture: n * (log(n)+log(log(n))-1+((log(log(n))-A)/log(n))) is asymptotic to a(n) if and only if A=2. - Alain Rocchelli, Feb 12 2025
From Stefano Spezia, Apr 13 2025: (Start)
a(n) = 1 + Sum_{m=1..2^n} floor(floor(n/Sum_{j=1..m} A080339(j))^(1/n)) [Willans, 1964].
a(n) = 1 + Sum_{m=1..2^n} floor(floor(n/(1 + A000720(m)))^(1/n)) [Willans, 1964]. (End)

A001055 The multiplicative partition function: number of ways of factoring n with all factors greater than 1 (a(1) = 1 by convention).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 7, 2, 2, 3, 4, 1, 5, 1, 7, 2, 2, 2, 9, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 12, 2, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 11, 2, 5, 1, 4, 2, 5, 1, 16, 1, 2, 4, 4, 2, 5, 1, 12, 5, 2, 1, 11, 2, 2, 2, 7, 1, 11, 2, 4, 2, 2, 2, 19, 1, 4, 4, 9, 1, 5, 1

Offset: 1

N. J. A. Sloane

From David W. Wilson, Feb 28 2009: (Start)
By a factorization of n we mean a multiset of integers > 1 whose product is n.
For example, 6 is the product of 2 such multisets, {2, 3} and {6}, so a(6) = 2.
Similarly 8 is the product of 3 such multisets, {2, 2, 2}, {2, 4} and {8}, so a(8) = 3.
1 is the product of 1 such multiset, namely the empty multiset {}, whose product is by definition the multiplicative identity 1. Hence a(1) = 1. (End)
a(n) = # { k | A064553(k) = n }. - Reinhard Zumkeller, Sep 21 2001; Benoit Cloitre and N. J. A. Sloane, May 15 2002
Number of members of A025487 with n divisors. - Matthew Vandermast, Jul 12 2004
See sequence A162247 for a list of the factorizations of n and a program for generating the factorizations for any n. - T. D. Noe, Jun 28 2009
So a(n) gives the number of different prime signatures that can be found among the integers that have n divisors. - Michel Marcus, Nov 11 2015
For n > 0, also the number of integer partitions of n with product n, ranked by A301987. For example, the a(12) = 4 partitions are: (12), (6,2,1,1,1,1), (4,3,1,1,1,1,1), (3,2,2,1,1,1,1,1). See also A380218. In general, A379666(m,n) = a(n) for any m >= n. - Gus Wiseman, Feb 07 2025

			1: 1, a(1) = 1
2: 2, a(2) = 1
3: 3, a(3) = 1
4: 4 = 2*2, a(4) = 2
6: 6 = 2*3, a(6) = 2
8: 8 = 2*4 = 2*2*2, a(8) = 3
etc.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 292-295.
Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.4.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge University Press, 1995, p. 198, exercise 9 (in the third edition 2015, p. 296, exercise 211).

T. D. Noe, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
D. Beckwith, Problem 10669, Amer. Math. Monthly 105 (1998), p. 559.
R. E. Canfield, P. Erdős and C. Pomerance, On a Problem of Oppenheim concerning "Factorisatio Numerorum", J. Number Theory 17 (1983), 1-28.
R. E. Canfield, P. Erdős and C. Pomerance, On a Problem of Oppenheim concerning "Factorisatio Numerorum", J. Number Theory 17 (1983), 1-28. [A second link to the same paper.]
Marc Chamberland, Colin Johnson, Alice Nadeau, and Bingxi Wu, Multiplicative Partitions, Electronic Journal of Combinatorics, 20(2) (2013), #P57.
S. R. Finch, Kalmar's composition constant, Jun 05 2003. [Cached copy, with permission of the author]
Shamik Ghosh, Counting number of factorizations of a natural number, arXiv:0811.3479 [cs.DM], 2008.
R. K. Guy and R. J. Nowakowski, Monthly unsolved problems, 1969-1995, Amer. Math. Monthly, 102 (1995), 921-926.
John F. Hughes and J. O. Shallit, On the Number of Multiplicative Partitions, American Mathematical Monthly 90(7) (1983), 468-471.
Cao Hui-Zhong and Ku Tung-Hsin, On the Enumeration Function of Multiplicative Partitions, Math. Balkanica, Vol. 4 (1990), Fasc. 3-4.
Florian Luca, Anirban Mukhopadhyay and Kotyada Srinivas, On the Oppenheim's "factorisatio numerorum" function, arXiv:0807.0986 [math.NT], 2008.
Pankaj Jyoti Mahanta, On the number of partitions of n whose product of the summands is at most n, arXiv:2010.07353 [math.CO], 2020.
Amarnath Murthy, Generalization of Partition Function (Introducing the Smarandache Factor Partition) [Broken link]
Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.4.
Paul Pollack, On the parity of the number of multiplicative partitions and related problems, Proc. Amer. Math. Soc. 140 (2012), 3793-3803.
Marko Riedel, Calculating the multiplicative partition function in Maple with the Polya Enumeration Theorem
Eric Weisstein's World of Mathematics, Unordered Factorization
Wikipedia, Multiplicative Partition Function
Index entries for "core" sequences

A045782 gives the range of a(n).
For records see A033833, A033834.
Cf. A002033, A045778 (strict version), A050322, A050336, A064553, A064554, A064555, A077565, A051731, A005171, A097296, A190938, A216599, A216600, A216601, A216602.
Row sums of A316439 (for n>1).
Cf. A096276 (partial sums).
The additive version is A000041 (integer partitions), strict A000009.
Row sums of A318950.
A002865 counts partitions into parts > 1.
A069016 counts distinct sums of factorizations.
A319000 counts partitions by product and sum, row sums A319916.
A379666 (transpose A380959) counts partitions by sum and product, without 1's A379668, strict A379671.
Cf. A003963, A057567, A057568 (strict A379733), A114324, A301987, A319005, A379734.

a001055 = (map last a066032_tabl !!) . (subtract 1)
-- Reinhard Zumkeller, Oct 01 2012

public class MultiPart {
    public static void main(String[] argV) {
        for (int i=1;i<=100;++i) System.out.println(1+getDivisors(2,i));
    }
    public static int getDivisors(int min,int n) {
        int total = 0;
        for (int i=min;i=i) { ++total; if (n/i>i) total+=getDivisors(i,n/i); }
        return total;
    }
} \\ Scott R. Shannon, Aug 21 2019

with(numtheory):
T := proc(n::integer, m::integer)
        local A, summe, d:
        if isprime(n) then
                if n <= m then
                        return 1;
                end if:
                return 0 ;
        end if:
        A := divisors(n) minus {n, 1}:
        for d in A do
                if d > m then
                        A := A minus {d}:
                end if:
        end do:
        summe := add(T(n/d,d),d=A) ;
        if n <=m then
                summe := summe + 1:
        end if:
        summe ;
end proc:
A001055 := n -> T(n, n):
[seq(A001055(n), n=1..100)]; # Reinhard Zumkeller and Ulrich Schimke (ulrschimke(AT)aol.com)

c[1, r_] := c[1, r]=1; c[n_, r_] := c[n, r] = Module[{ds, i}, ds = Select[Divisors[n], 1 < # <= r &]; Sum[c[n/ds[[i]], ds[[i]]], {i, 1, Length[ds]}]]; a[n_] := c[n, n]; a/@Range[100] (* c[n, r] is the number of factorizations of n with factors <= r. - Dean Hickerson, Oct 28 2002 *)
T[, 1] = T[1, ] = 1;
T[n_, m_] := T[n, m] = DivisorSum[n, Boole[1 < # <= m] * T[n/#, #]&];
a[n_] := T[n, n];
a /@ Range[100] (* Jean-François Alcover, Jan 03 2020 *)

/* factorizations of n with factors <= m (n,m positive integers) */
fcnt(n,m) = {local(s);s=0;if(n == 1,s=1,fordiv(n,d,if(d > 1 & d <= m,s=s+fcnt(n/d,d))));s}
A001055(n) = fcnt(n,n) \\ Michael B. Porter, Oct 29 2009

\\ code using Dirichlet g.f., based on Somos's code for A007896
{a(n) = my(A, v, w, m);
if(
n<1, 0,
\\ define unit vector v = [1, 0, 0, ...] of length n
v = vector(n, k, k==1);
for(k=2, n,
m = #digits(n, k) - 1;
\\ expand 1/(1-x)^k out far enough
A = (1 - x)^ -1 + x * O(x^m);
\\ w = zero vector of length n
w = vector(n);
\\ convert A to a vector
for(i=0, m, w[k^i] = polcoeff(A, i));
\\ build the answer
v = dirmul(v, w)
);
v[n]
)
};
\\ produce the sequence
vector(100,n,a(n)) \\ N. J. A. Sloane, May 26 2014

v=vector(100, k, k==1); for(n=2, #v, v+=dirmul(v, vector(#v, k, (k>1) && n^valuation(k,n)==k)) ); v \\ Max Alekseyev, Jul 16 2014

from sympy import divisors, isprime
def T(n, m):
    if isprime(n): return 1 if n<=m else 0
    A=filter(lambda d: d<=m, divisors(n)[1:-1])
    s=sum(T(n//d, d) for d in A)
    return s + 1 if n<=m else s
def a(n): return T(n, n)
print([a(n) for n in range(1, 106)]) # Indranil Ghosh, Aug 19 2017

The asymptotic behavior of this sequence was studied by Canfield, Erdős & Pomerance and Luca, Mukhopadhyay, & Srinivas. - Jonathan Vos Post, Jul 07 2008
Dirichlet g.f.: Product_{k>=2} 1/(1 - 1/k^s).
If n = p^k for a prime p, a(n) = partitions(k) = A000041(k).
Since the sequence a(n) is the right diagonal of A066032, the given recursive formula for A066032 applies (see Maple program). - Reinhard Zumkeller and Ulrich Schimke (ulrschimke(AT)aol.com)
a(A002110(n)) = A000110(n).
a(p^k*q^k) = A002774(k) if p and q are distinct primes. - R. J. Mathar, Jun 06 2024
a(n) = A028422(n) + 1. - Gus Wiseman, Feb 07 2025

Incorrect assertion about asymptotic behavior deleted by N. J. A. Sloane, Jun 08 2009

A001248 Squares of primes.

Original entry on oeis.org

4, 9, 25, 49, 121, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 16129, 17161, 18769, 19321, 22201, 22801, 24649, 26569, 27889, 29929, 32041, 32761, 36481

Offset: 1

N. J. A. Sloane

Also 4, together with numbers n such that Sum_{d|n}(-1)^d = -A048272(n) = -3. - Benoit Cloitre, Apr 14 2002
Also, all solutions to the equation sigma(x) + phi(x) = 2x + 1. - Farideh Firoozbakht, Feb 02 2005
Unique numbers having 3 divisors (1, their square root, themselves). - Alexandre Wajnberg, Jan 15 2006
Smallest (or first) new number deleted at the n-th step in an Eratosthenes sieve. - Lekraj Beedassy, Aug 17 2006
Subsequence of semiprimes A001358. - Lekraj Beedassy, Sep 06 2006
Integers having only 1 factor other than 1 and the number itself. Every number in the sequence is a multiple of 1 factor other than 1 and the number itself. 4 : 2 is the only factor other than 1 and 4; 9 : 3 is the only factor other than 1 and 9; and so on. - Rachit Agrawal (rachit_agrawal(AT)daiict.ac.in), Oct 23 2007
The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime. - Omar E. Pol, May 06 2008
There are 2 Abelian groups of order p^2 (C_p^2 and C_p x C_p) and no non-Abelian group. - Franz Vrabec, Sep 11 2008
Also numbers n such that phi(n) = n - sqrt(n). - Michel Lagneau, May 25 2012
For n > 1, n is the sum of numbers from A006254(n-1) to A168565(n-1). - Vicente Izquierdo Gomez, Dec 01 2012
A078898(a(n)) = 2. - Reinhard Zumkeller, Apr 06 2015
Let r(n) = (a(n) - 1)/(a(n) + 1); then Product_{n>=1} r(n) = (3/5) * (4/5) * (12/13) * (24/25) * (60/61) * ... = 2/5. - Dimitris Valianatos, Feb 26 2019
Numbers k such that A051709(k) = 1. - Jianing Song, Jun 27 2021

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 5000 terms from N. J. A. Sloane)
Nicholas John Bizzell-Browning, LIE scales: Composing with scales of linear intervallic expansion, Ph. D. Thesis, Brunel Univ. (UK, 2024). See p. 143.
R. P. Boas and N. J. A. Sloane, Correspondence, 1974
Marius Coman, On the special relation between the numbers of the form 505+ 1008k and the squares of primes, 2015.
Brady Haran and Matt Parker, Squaring Primes, Numberphile video (2018).
Eric Weisstein's World of Mathematics, Prime Power.
OEIS Wiki, Index entries for number of divisors
Index to sequences related to prime signature

Cf. A000005, A000040, A001358, A001694, A002033, A005408, A005722, A006254.
Cf. A008864, A033273, A049001, A048272, A051709, A059956, A060800, A062799.
Cf. A078898, A082020, A085548, A087112, A095874, A168565, A192134, A211110.
Cf. A258599.
Subsequence of A000430, A001358, and of A190641.
Cf. A024450 (partial sums), A069482 (first differences).

a001248 n = a001248_list !! (n-1)
a001248_list = map (^ 2) a000040_list -- Reinhard Zumkeller, Sep 23 2011

[p^2: p in PrimesUpTo(300)]; // Vincenzo Librandi, Mar 27 2014

A001248:=n->ithprime(n)^2; seq(A001248(k), k=1..50); # Wesley Ivan Hurt, Oct 11 2013

Prime[Range[30]]^2 (* Zak Seidov, Dec 07 2011 *)
Select[Range[40000], DivisorSigma[0, #] == 3 &] (* Carlos Eduardo Olivieri, Jun 01 2015 *)

forprime(p=2,1e3,print1(p^2", ")) \\ Charles R Greathouse IV, Jun 10 2011

A001248(n)=prime(n)^2  \\ M. F. Hasler, Sep 16 2012

from sympy import prime
def A001248(n): return prime(n)**2 # Chai Wah Wu, Aug 09 2024

[n^2 for n in prime_range(1,301)] # G. C. Greubel, May 02 2024

n such that A062799(n) = 2. - Benoit Cloitre, Apr 06 2002
A000005(a(n)^(k-1)) = A005408(k) for all k>0. - Reinhard Zumkeller, Mar 04 2007
a(n) = A000040(n)^(3-1)=A000040(n)^2, where 3 is the number of divisors of a(n). - Omar E. Pol, May 06 2008
A000005(a(n)) = 3 or A002033(a(n)) = 2. - Juri-Stepan Gerasimov, Oct 10 2009
A033273(a(n)) = 3. - Juri-Stepan Gerasimov, Dec 07 2009
For n > 2: (a(n) + 17) mod 12 = 6. - Reinhard Zumkeller, May 12 2010
A192134(A095874(a(n))) = A005722(n) + 1. - Reinhard Zumkeller, Jun 26 2011
For n > 2: a(n) = 1 (mod 24). - Zak Seidov, Dec 07 2011
A211110(a(n)) = 2. - Reinhard Zumkeller, Apr 02 2012
a(n) = A087112(n,n). - Reinhard Zumkeller, Nov 25 2012
a(n) = prime(n)^2. - Jon E. Schoenfield, Mar 29 2015
Product_{n>=1} a(n)/(a(n)-1) = Pi^2/6. - Daniel Suteu, Feb 06 2017
Sum_{n>=1} 1/a(n) = P(2) = 0.4522474200... (A085548). - Amiram Eldar, Jul 27 2020
From Amiram Eldar, Jan 23 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = zeta(2)/zeta(4) = 15/Pi^2 (A082020).
Product_{n>=1} (1 - 1/a(n)) = 1/zeta(2) = 6/Pi^2 (A059956). (End)

A008578 Prime numbers at the beginning of the 20th century (today 1 is no longer regarded as a prime).

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271

Offset: 1

N. J. A. Sloane

1 together with the primes; also called the noncomposite numbers.
Also largest sequence of nonnegative integers with the property that the product of 2 or more elements with different indices is never a square. - Ulrich Schimke (ulrschimke(AT)aol.com), Dec 12 2001 [Comment corrected by Farideh Firoozbakht, Aug 03 2014]
Numbers k whose largest divisor <= sqrt(k) equals 1. (See also A161344, A161345, A161424.) - Omar E. Pol, Jul 05 2009
Numbers k such that d(k) <= 2. - Juri-Stepan Gerasimov, Oct 17 2009
Also first column of array in A163280. Also first row of array in A163990. - Omar E. Pol, Oct 24 2009
Possible values of A136548(m) in increasing order, where A136548(m) = the largest numbers h such that A000203(h) <= k (k = 1,2,3,...), where A000203(h) = sum of divisors of h. - Jaroslav Krizek, Mar 01 2010
Where record values of A022404 occur: A086332(n)=A022404(a(n)). - Reinhard Zumkeller, Jun 21 2010
Positive integers that have no divisors other than 1 and itself (the old definition of prime numbers). - Omar E. Pol, Aug 10 2012
Conjecture: the sequence contains exactly those k such that sigma(k) > k*BigOmega(k). - Irina Gerasimova, Jun 08 2013
Note on the Gerasimova conjecture: all terms in the sequence obviously satisfy the inequality, because sigma(p) = 1+p and BigOmega(p) = 1 for primes p, so 1+p > p*1. For composites, the (opposite) inequality is heuristically correct at least up to k <= 4400000. The general proof requires to show that BigOmega(k) is an upper limit of the abundancy sigma(k)/k for composite k. This proof is easy for semiprimes k=p1*p2 in general, where sigma(k)=1+p1+p2+p1*p2 and BigOmega(k)=2 and p1, p2 <= 2. - R. J. Mathar, Jun 12 2013
Numbers k such that phi(k) + sigma(k) = 2k. - Farideh Firoozbakht, Aug 03 2014
isA008578(n) <=> k is prime to n for all k in {1,2,...,n-1}. - Peter Luschny, Jun 05 2017
In 1751 Leonhard Euler wrote: "Having so established this sign S to indicate the sum of the divisors of the number in front of which it is placed, it is clear that, if p indicates a prime number, the value of Sp will be 1 + p, except for the case where p = 1, because then we have S1 = 1, and not S1 = 1 + 1. From this we see that we must exclude unity from the sequence of prime numbers, so that unity, being the start of whole numbers, it is neither prime nor composite." - Omar E. Pol, Oct 12 2021
a(1) = 1; for n >= 2, a(n) is the least unused number that is coprime to all previous terms. - Jianing Song, May 28 2022
A number p is preprime if p = a*b ==> a = 1 or b = 1. This sequence lists the preprimes in the commutative monoid IN \ {0}. - Peter Luschny, Aug 26 2022

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 84 at pp. 214-217.
G. Chrystal, Algebra: An Elementary Textbook. Chelsea Publishing Company, 7th edition, (1964), chap. III.7, p. 38.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 11.
H. D. Huskey, Derrick Henry Lehmer [1905-1991]. IEEE Ann. Hist. Comput. 17 (1995), no. 2, 64-68. Math. Rev. 96b:01035
D. H. Lehmer, The sieve problem for all-purpose computers. Math. Tables and Other Aids to Computation, Math. Tables and Other Aids to Computation, 7, (1953). 6-14. Math. Rev. 14:691e
D. N. Lehmer, "List of Prime Numbers from 1 to 10,006,721", Carnegie Institute, Washington, D.C. 1909.
R. F. Lukes, C. D. Patterson and H. C. Williams, Numerical sieving devices: their history and some applications. Nieuw Arch. Wisk. (4) 13 (1995), no. 1, 113-139. Math. Rev. 96m:11082
H. C. Williams and J. O. Shallit, Factoring integers before computers. Mathematics of Computation 1943-1993: a half-century of computational mathematics (Vancouver, BC, 1993), 481-531, Proc. Sympos. Appl. Math., 48, AMS, Providence, RI, 1994. Math. Rev. 95m:11143

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]
Chris K. Caldwell, Angela Reddick, Yeng Xiong, and Wilfrid Keller, The History of the Primality of One: A Selection of Sources, (a dynamic survey), Journal of Integer Sequences, Vol. 15 (2012), #12.9.8.
C. K. Caldwell and Y. Xiong, What is the smallest prime?, arXiv preprint arXiv:1209.2007 [math.HO], 2012, and J. Int. Seq. 15 (2012) #12.9.6
Leonhard Euler, Découverte d’une loi tout extraordinaire des nombres, par rapport à la somme de leurs diviseurs, in Bibliothèque impartiale, 3, 1751, pp. 10-31. Reprinted in Opera Postuma, 1, 1862, p.76-84. Number 175 in the Eneström index.
G. P. Michon, Is 1 a prime number?
Omar E. Pol, Illustration of initial terms
Omar E. Pol, Illustration of initial terms of A008578, A161344, A161345, A161424
PrimeFan, Arguments for and against the primality of 1
A. Reddick and Y. Xiong, The search for one as a prime number: from ancient Greece to modern times, Electronic Journal of Undergraduate Mathematics, Volume 16, 1 - 13, 2012. - From _N. J. A. Sloane_, Feb 03 2013
J. Todd, Review of Lehmer's tables, Mathematical Tables and Other Aids to Computation, Vol. 11, No. 60, (1957) (on JSTOR.org).
Wikipedia, "Complete" sequence. [Wikipedia calls a sequence "complete" (sic) if every positive integer is a sum of distinct terms. This name is extremely misleading and should be avoided. - _N. J. A. Sloane_, May 20 2023]
Wikipedia, Dirichlet convolution

The main entry for this sequence is A000040.
The complement of A002808.
Cf. A000732 (boustrophedon transform).
Cf. A000010, A000203.
Cf. A023626 (self-convolution).

A008578:=Concatenation([1],Filtered([1..10^5],IsPrime)); # Muniru A Asiru, Sep 07 2017

a008578 n = a008578_list !! (n-1)
a008578_list = 1 : a000040_list
-- Reinhard Zumkeller, Nov 09 2011

[1] cat [n: n in PrimesUpTo(271)];  // Bruno Berselli, Mar 05 2011

A008578 := n->if n=1 then 1 else ithprime(n-1); fi :

Join[ {1}, Table[ Prime[n], {n, 1, 60} ] ]
NestList[ NextPrime, 1, 57] (* Robert G. Wilson v, Jul 21 2015 *)
oldPrimeQ[n_] := AllTrue[Range[n-1], CoprimeQ[#, n]&];
Select[Range[271], oldPrimeQ] (* Jean-François Alcover, Jun 07 2017, after Peter Luschny *)

PARI
```
is(n)=isprime(n)||n==1
```

isA008578 = lambda n: all(gcd(k, n) == 1 for k in (1..n-1))
print([n for n in (1..271) if isA008578(n)]) # Peter Luschny, Jun 07 2017

a(n) = A000040(n-1).
m is in the sequence iff sigma(m) + phi(m) = A065387(m) = 2m. - Farideh Firoozbakht, Jan 27 2005
a(n) = A158611(n+1) for n >= 1. - Jaroslav Krizek, Jun 19 2009
In the following formulas (based on emails from Jaroslav Krizek and R. J. Mathar), the star denotes a Dirichlet convolution between two sequences, and "This" is A008578.
This = A030014 * A008683. (Dirichlet convolution using offset 1 with A030014)
This = A030013 * A000012. (Dirichlet convolution using offset 1 with A030013)
This = A034773 * A007427. (Dirichlet convolution)
This = A034760 * A023900. (Dirichlet convolution)
This = A034762 * A046692. (Dirichlet convolution)
This * A000012 = A030014. (Dirichlet convolution using offset 1 with A030014)
This * A008683 = A030013. (Dirichlet convolution using offset 1 with A030013)
This * A000005 = A034773. (Dirichlet convolution)
This * A000010 = A034760. (Dirichlet convolution)
This * A000203 = A034762. (Dirichlet convolution)
A002033(a(n))=1. - Juri-Stepan Gerasimov, Sep 27 2009
a(n) = A181363((2*n-1)*2^k), k >= 0. - Reinhard Zumkeller, Oct 16 2010
a(n) = A001747(n)/2. - Omar E. Pol, Jan 30 2012
A060448(a(n)) = 1. - Reinhard Zumkeller, Apr 05 2012
A086971(a(n)) = 0. - Reinhard Zumkeller, Dec 14 2012
Sum_{n>=1} x^a(n) = (Sum_{n>=1} (A002815(n)*x^n))*(1-x)^2. - L. Edson Jeffery, Nov 25 2013

A002865 Number of partitions of n that do not contain 1 as a part.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 21, 24, 34, 41, 55, 66, 88, 105, 137, 165, 210, 253, 320, 383, 478, 574, 708, 847, 1039, 1238, 1507, 1794, 2167, 2573, 3094, 3660, 4378, 5170, 6153, 7245, 8591, 10087, 11914, 13959, 16424, 19196, 22519, 26252, 30701

Offset: 0

N. J. A. Sloane

Also the number of partitions of n-1, n >= 2, such that the least part occurs exactly once. See A096373, A097091, A097092, A097093. - Robert G. Wilson v, Jul 24 2004 [Corrected by Wolfdieter Lang, Feb 18 2009]
Number of partitions of n+1 where the number of parts is itself a part. Take a partition of n (with k parts) which does not contain 1, remove 1 from each part and add a new part of size k+1. - Franklin T. Adams-Watters, May 01 2006
Number of partitions where the largest part occurs at least twice. - Joerg Arndt, Apr 17 2011
Row sums of triangle A147768. - Gary W. Adamson, Nov 11 2008
From Lewis Mammel (l_mammel(AT)att.net), Oct 06 2009: (Start)
a(n) is the number of sets of n disjoint pairs of 2n things, called a pairing, disjoint with a given pairing (A053871), that are unique under permutations preserving the given pairing.
Can be seen immediately from a graphical representation which must decompose into even numbered cycles of 4 or more things, as connected by pairs alternating between the pairings. Each thing is in a single cycle, so this is a partition of 2n into even parts greater than 2, equivalent to a partition of n into parts greater than 1. (End)
Convolution product (1, 1, 2, 2, 4, 4, ...) * (1, 2, 3, ...) = A058682 starting (1, 3, 7, 13, 23, 37, ...); with row sums of triangle A171239 = A058682. - Gary W. Adamson, Dec 05 2009
Also the number of 2-regular multigraphs with loops forbidden. - Jason Kimberley, Jan 05 2011
Number of appearances of the multiplicity n, n-1, ..., n-k in all partitions of n, for k < n/2. (Only populated by multiplicities of large numbers of 1's.) - William Keith, Nov 20 2011
Also the number of equivalence classes of n X n binary matrices with exactly 2 1's in each row and column, up to permutations of rows and columns (cf. A133687). - N. J. A. Sloane, Sep 16 2013
The q-Catalan numbers ((1-q)/(1-q^(n+1)))[2n,n]q, where [2n,n]_q are the central q-binomial coefficients, match this sequence in their initial segment of length n. - _William J. Keith, Nov 14 2013
Starting at a(2) this sequence gives the number of vertices on a nim tree created in the game of edge removal for a path P_{n} where n is the number of vertices on the path. This is the number of nonisomorphic graphs that can result from the path when the game of edge removal is played. - Lyndsey Wong, Jul 09 2016
The number of different ways to climb a staircase taking at least two stairs at a time. - Mohammad K. Azarian, Nov 20 2016
Let 1,0,1,1,1,... (offset 0) count unlabeled, connected, loopless 1-regular digraphs. This here is the Euler transform of that sequence, counting unlabeled loopless 1-regular digraphs. A145574 is the associated multiset transformation. A000166 are the labeled loopless 1-regular digraphs. - R. J. Mathar, Mar 25 2019
For n > 1, also the number of partitions with no part greater than the number of ones. - George Beck, May 09 2019 [See A187219 which is the correct sequence for this interpretation for n >= 1. - Spencer Miller, Jan 30 2023]
From Gus Wiseman, May 19 2019: (Start)
Conjecture: Also the number of integer partitions of n - 1 that have a consecutive subsequence summing to each positive integer from 1 to n - 1. For example, (32211) is such a partition because we have consecutive subsequences:
  1: (1)
  2: (2)
  3: (3) or (21)
  4: (22) or (211)
  5: (32) or (221)
  6: (2211)
  7: (322)
  8: (3221)
  9: (32211)
(End)
There is a sufficient and necessary condition to characterize the partitions defined by Gus Wiseman. It is that the largest part must be less than or equal to the number of ones plus one. Hence, the number of partitions of n with no part greater than the number of ones is the same as the number of partitions of n-1 that have a consecutive subsequence summing to each integer from 1 to n-1. Gus Wiseman's conjecture can be proved bijectively. - Andrew Yezhou Wang, Dec 14 2019
From Peter Bala, Dec 01 2024: (Start)
Let P(2, n) denote the set of partitions of n into parts k > 1. Then A000041(n) = - Sum_{parts k in all partitions in P(2, n+2)} mu(k). For example, with n = 5, there are 4 partitions of n + 2 = 7 into parts greater than 1, namely, 7, 5 + 2, 4 + 3, 3 + 2 + 2, and mu(7) + (mu(5) + mu(2)) + (mu(4 ) + mu(3)) + (mu(3) + mu(2) + mu(2)) = -7 = - A000041(5). (End)

			a(6) = 4 from 6 = 4+2 = 3+3 = 2+2+2.
G.f. = 1 + x^2 + x^3 + 2*x^4 + 2*x^5 + 4*x^6 + 4*x^7 + 7*x^8 + 8*x^9 + ...
From _Gus Wiseman_, May 19 2019: (Start)
The a(2) = 1 through a(9) = 8 partitions not containing 1 are the following. The Heinz numbers of these partitions are given by A005408.
  (2)  (3)  (4)   (5)   (6)    (7)    (8)     (9)
            (22)  (32)  (33)   (43)   (44)    (54)
                        (42)   (52)   (53)    (63)
                        (222)  (322)  (62)    (72)
                                      (332)   (333)
                                      (422)   (432)
                                      (2222)  (522)
                                              (3222)
The a(2) = 1 through a(9) = 8 partitions of n - 1 whose least part appears exactly once are the following. The Heinz numbers of these partitions are given by A247180.
  (1)  (2)  (3)   (4)   (5)    (6)    (7)     (8)
            (21)  (31)  (32)   (42)   (43)    (53)
                        (41)   (51)   (52)    (62)
                        (221)  (321)  (61)    (71)
                                      (331)   (332)
                                      (421)   (431)
                                      (2221)  (521)
                                              (3221)
The a(2) = 1 through a(9) = 8 partitions of n + 1 where the number of parts is itself a part are the following. The Heinz numbers of these partitions are given by A325761.
  (21)  (22)  (32)   (42)   (52)    (62)    (72)     (82)
              (311)  (321)  (322)   (332)   (333)    (433)
                            (331)   (431)   (432)    (532)
                            (4111)  (4211)  (531)    (631)
                                            (4221)   (4222)
                                            (4311)   (4321)
                                            (51111)  (4411)
                                                     (52111)
The a(2) = 1 through a(8) = 7 partitions of n whose greatest part appears at least twice are the following. The Heinz numbers of these partitions are given by A070003.
  (11)  (111)  (22)    (221)    (33)      (331)      (44)
               (1111)  (11111)  (222)     (2221)     (332)
                                (2211)    (22111)    (2222)
                                (111111)  (1111111)  (3311)
                                                     (22211)
                                                     (221111)
                                                     (11111111)
Nonisomorphic representatives of the a(2) = 1 through a(6) = 4 2-regular multigraphs with n edges and n vertices are the following.
  {12,12}  {12,13,23}  {12,12,34,34}  {12,12,34,35,45}  {12,12,34,34,56,56}
                       {12,13,24,34}  {12,13,24,35,45}  {12,12,34,35,46,56}
                                                        {12,13,23,45,46,56}
                                                        {12,13,24,35,46,56}
The a(2) = 1 through a(9) = 8 partitions of n with no part greater than the number of ones are the following. The Heinz numbers of these partitions are given by A325762.
  (11)  (111)  (211)   (2111)   (2211)    (22111)    (22211)     (33111)
               (1111)  (11111)  (3111)    (31111)    (32111)     (222111)
                                (21111)   (211111)   (41111)     (321111)
                                (111111)  (1111111)  (221111)    (411111)
                                                     (311111)    (2211111)
                                                     (2111111)   (3111111)
                                                     (11111111)  (21111111)
                                                                 (111111111)
(End)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 836.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 115, p*(n).
H. P. Robinson, Letter to N. J. A. Sloane, Jan 04 1974.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
P. G. Tait, Scientific Papers, Cambridge Univ. Press, Vol. 1, 1898, Vol. 2, 1900, see Vol. 1, p. 334.

Andrew van den Hoeven, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
A. P. Akande et al., Computational study of non-unitary partitions, arXiv:2112.03264 [math.CO], 2021.
Colin Albert, Olivia Beckwith, Irfan Demetoglu, Robert Dicks, John H. Smith, and Jasmine Wang, Integer partitions with large Dyson rank, arXiv:2203.08987 [math.NT], 2022.
Max A. Alekseyev and Allan Bickle, Forbidden Subgraphs of Single Graphs, (2024). See p. 6.
Kevin Beanland and Hung Viet Chu, On Schreier-type Sets, Partitions, and Compositions, arXiv:2311.01926 [math.CO], 2023.
G. Dahl and T. A. Haufmann, Zero-one completely positive matrices and the A(R,S) classes, Preprint, 2016.
Atul Dixit, Gaurav Kumar, and Aviral Srivastava, Non-Rascoe partitions and a rank parity function associated to the Rogers-Ramanujan partitions, arXiv:2508.04359 [math.CO], 2025. See references.
R. P. Gallant, G. Gunther, B. L. Hartnell, and D. F. Rall, A game of edge removal on graphs, JCMCC, 57 (2006), 75 - 82.
Edray Herber Goins and Talitha M. Washington, On the generalized climbing stairs problem, Ars Combin. 117 (2014), 183-190. MR3243840 (Reviewed), arXiv:0909.5459 [math.CO], 2009.
H. Gropp, On tactical configurations, regular bipartite graphs and (v,k,even)-designs, Discr. Math., 155 (1996), 81-98.
R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
Cristiano Husu, The butterfly sequence: the second difference sequence of the numbers of integer partitions with distinct parts, its pentagonal number structure, its combinatorial identities and the cyclotomic polynomials 1-x and 1+x+x^2, arXiv:1804.09883 [math.NT], 2018.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 100
Wenwei Li, Approximation of the Partition Number After Hardy and Ramanujan: An Application of Data Fitting Method in Combinatorics, arXiv preprint arXiv:1612.05526 [math.NT], 2016-2018.
Wenwei Li, On the Number of Conjugate Classes of Derangements, arXiv:1612.08186 [math.CO], 2016.
J. L. Nicolas and A. Sárközy, On partitions without small parts, Journal de théorie des nombres de Bordeaux, 12 no. 1 (2000), p. 227-254.
R. A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
H. P. Robinson, Letter to N. J. A. Sloane, Jul 12 1971
H. P. Robinson, Letter to N. J. A. Sloane, Dec 10 1973
H. P. Robinson, Letter to N. J. A. Sloane, Jan 4 1974.
Noah Rubin, Curtis Bright, Kevin K. H. Cheung, and Brett Stevens, Integer and Constraint Programming Revisited for Mutually Orthogonal Latin Squares, arXiv:2103.11018 [cs.DM], 2021.
Miloslav Znojil, Non-Hermitian N-state degeneracies: unitary realizations via antisymmetric anharmonicities, arXiv:2010.15014 [quant-ph], 2020.
Miloslav Znojil, Quantum phase transitions mediated by clustered non-Hermitian degeneracies, arXiv:2102.12272 [quant-ph], 2021.
Miloslav Znojil, Bose-Einstein condensation processes with nontrivial geometric multiplicites realized via PT-symmetric and exactly solvable linear-Bose-Hubbard building blocks, arXiv:2108.07110 [quant-ph], 2021.
Index entries for related partition-counting sequences

First differences of partition numbers A000041. Cf. A053445, A072380, A081094, A081095, A232697.
Pairwise sums seem to be in A027336.
Essentially the same as A085811.
Cf. A025147, A147768, A058682, A171239.
A column of A090824 and of A133687 and of A292508 and of A292622. Cf. A229161.
2-regular not necessarily connected graphs: A008483 (simple graphs), A000041 (multigraphs with loops allowed), this sequence (multigraphs with loops forbidden), A027336 (graphs with loops allowed but no multiple edges). - Jason Kimberley, Jan 05 2011
See also A098743 (parts that do not divide n).
Numbers n such that in the edge-delete game on the path P_{n} the first player does not have a winning strategy: A274161. - Lyndsey Wong, Jul 09 2016
Cf. A002033, A070003, A103295, A247180, A325676, A325684, A325761, A325762, A325763.
Row sums of characteristic array A145573.
Number of partitions of n into parts >= m: A008483 (m = 3), A008484 (m = 4), A185325 - A185329 (m = 5 through 9).

Concatenation([1],List([1..41],n->NrPartitions(n)-NrPartitions(n-1))); # Muniru A Asiru, Aug 20 2018

A41 := func; [A41(n)-A41(n-1):n in [0..50]]; // Jason Kimberley, Jan 05 2011

with(combstruct): ZL1:=[S, {S=Set(Cycle(Z,card>1))}, unlabeled]: seq(count(ZL1,size=n), n=0..50);  # Zerinvary Lajos, Sep 24 2007
G:= {P=Set (Set (Atom, card>1))}: combstruct[gfsolve](G, unlabeled, x): seq  (combstruct[count] ([P, G, unlabeled], size=i), i=0..50);  # Zerinvary Lajos, Dec 16 2007
with(combstruct):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, card>=m))}, unlabeled]; end: A:=a(2):seq(count(A, size=n), n=0..50);  # Zerinvary Lajos, Jun 11 2008
# alternative Maple program:
A002865:= proc(n) option remember; `if`(n=0, 1, add(
      (numtheory[sigma](j)-1)*A002865(n-j), j=1..n)/n)
    end:
seq(A002865(n), n=0..60);  # Alois P. Heinz, Sep 17 2017

Table[ PartitionsP[n + 1] - PartitionsP[n], {n, -1, 50}] (* Robert G. Wilson v, Jul 24 2004 *)
f[1, 1] = 1; f[n_, k_] := f[n, k] = If[n < 0, 0, If[k > n, 0, If[k == n, 1, f[n, k + 1] + f[n - k, k]]]]; Table[ f[n, 2], {n, 50}] (* Robert G. Wilson v *)
Table[SeriesCoefficient[Exp[Sum[x^(2*k)/(k*(1 - x^k)), {k, 1, n}]], {x, 0, n}], {n, 0, 50}] (* Vaclav Kotesovec, Aug 18 2018 *)
CoefficientList[Series[1/QPochhammer[x^2, x], {x,0,50}], x] (* G. C. Greubel, Nov 03 2019 *)
Table[Count[IntegerPartitions[n],?(FreeQ[#,1]&)],{n,0,50}] (* _Harvey P. Dale, Feb 12 2023 *)

{a(n) = if( n<0, 0, polcoeff( (1 - x) / eta(x + x * O(x^n)), n))};

a(n)=if(n,numbpart(n)-numbpart(n-1),1) \\ Charles R Greathouse IV, Nov 26 2012

from sympy import npartitions
def A002865(n): return npartitions(n)-npartitions(n-1) if n else 1 # Chai Wah Wu, Mar 30 2023

def A002865_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/product((1-x^(m+2)) for m in (0..60)) ).list()
A002865_list(50) # G. C. Greubel, Nov 03 2019

G.f.: Product_{m>1} 1/(1-x^m).
a(0)=1, a(n) = p(n) - p(n-1), n >= 1, with the partition numbers p(n) := A000041(n).
a(n) = A085811(n+3). - James Sellers, Dec 06 2005 [Corrected by Gionata Neri, Jun 14 2015]
a(n) = A116449(n) + A116450(n). - Reinhard Zumkeller, Feb 16 2006
a(n) = Sum_{k=2..floor((n+2)/2)} A008284(n-k+1,k-1) for n > 0. - Reinhard Zumkeller, Nov 04 2007
G.f.: 1 + Sum_{n>=2} x^n / Product_{k>=n} (1 - x^k). - Joerg Arndt, Apr 13 2011
G.f.: Sum_{n>=0} x^(2*n) / Product_{k=1..n} (1 - x^k). - Joerg Arndt, Apr 17 2011
a(n) = A090824(n,1) for n > 0. - Reinhard Zumkeller, Oct 10 2012
a(n) ~ Pi * exp(sqrt(2*n/3)*Pi) / (12*sqrt(2)*n^(3/2)) * (1 - (3*sqrt(3/2)/Pi + 13*Pi/(24*sqrt(6)))/sqrt(n) + (217*Pi^2/6912 + 9/(2*Pi^2) + 13/8)/n). - Vaclav Kotesovec, Feb 26 2015, extended Nov 04 2016
G.f.: exp(Sum_{k>=1} (sigma_1(k) - 1)*x^k/k). - Ilya Gutkovskiy, Aug 21 2018
a(0) = 1, a(n) = A232697(n) - 1. - George Beck, May 09 2019
From Peter Bala, Feb 19 2021: (Start)
G.f.: A(q) = Sum_{n >= 0} q^(n^2)/( (1 - q)*Product_{k = 2..n} (1 - q^k)^2 ).
More generally, A(q) = Sum_{n >= 0} q^(n*(n+r))/( (1 - q) * Product_{k = 2..n} (1 - q^k)^2 * Product_{i = 1..r} (1 - q^(n+i)) ) for r = 0,1,2,.... (End)
G.f.: 1 + Sum_{n >= 1} x^(n+1)/Product_{k = 1..n-1} 1 - x^(k+2). - Peter Bala, Dec 01 2024

A008966 a(n) = 1 if n is squarefree, otherwise 0.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0

Offset: 1

N. J. A. Sloane

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3, 1).

The infinite lower triangular matrix with A008966 on the main diagonal and the rest zeros is the square of triangle A143255. - Gary W. Adamson, Aug 02 2008

Daniel Forgues, Table of n, a(n) for n = 1..100000
Eric Weisstein's World of Mathematics, Moebius Function
Index entries for characteristic functions
Index entries for sequences computed from exponents in factorization of n

Cf. A005117, A008836 (Dirichlet inverse), A013928 (partial sums).
Cf. A179211, A179212, A179213, A179214, A179215.
Cf. A020639, A073576, A087188.
Cf. A124010, A212793, A085357, A156552.
Parity of A002033.
Cf. A047999, A036987
Cf. A082020 (Dgf at s=2), A157289 (Dgf at s=3), A157290 (Dgf at s=4).

a008966 = abs . a008683
-- Reinhard Zumkeller, Dec 13 2015, Dec 15 2014, May 27 2012, Jan 25 2012

Magma
```
[ Abs(MoebiusMu(n)) : n in [1..100]];
```

A008966 := proc(n) if numtheory[issqrfree](n) then 1 ; else 0 ; end if; end proc: # R. J. Mathar, Mar 14 2011

A008966[n_] := Abs[MoebiusMu[n]]; Table[A008966[n], {n, 100}] (* Enrique Pérez Herrero, Apr 15 2010 *)
Table[If[SquareFreeQ[n],1,0],{n,100}] (* or *) Boole[SquareFreeQ/@ Range[ 100]] (* Harvey P. Dale, Feb 28 2015 *)

MuPAD
```
func(abs(numlib::moebius(n)), n):
```
PARI
```
a(n)=if(n<1,0,direuler(p=2,n,1+X))[n]
```

a(n)=issquarefree(n) \\ Michel Marcus, Feb 22 2015

from sympy import factorint
def A008966(n): return int(max(factorint(n).values(),default=1)==1) # Chai Wah Wu, Apr 05 2023

Dirichlet g.f.: zeta(s)/zeta(2s).
a(n) = abs(mu(n)), where mu is the Moebius function (A008683).
a(n) = 0^(bigomega(n) - omega(n)), where bigomega(n) and omega(n) are the numbers of prime factors of n with and without repetition (A001222, A001221, A046660). - Reinhard Zumkeller, Apr 05 2003
Multiplicative with p^e -> 0^(e - 1), p prime and e > 0. - Reinhard Zumkeller, Jul 15 2003
a(n) = 0^(A046951(n) - 1). - Reinhard Zumkeller, May 20 2007
a(n) = 1 - A107078(n). - Reinhard Zumkeller, Oct 03 2008
a(n) = floor(rad(n)/n), where rad() is A007947. - Enrique Pérez Herrero, Nov 13 2009
A175046(n) = a(n)*A073311(n). - Reinhard Zumkeller, Apr 05 2010
a(n) = floor(A000005(n^2)/A007425(n)). - Enrique Pérez Herrero, Apr 15 2010
a(A005117(n)) = 1; a(A013929(n)) = 0; a(n) = A013928(n + 1) - A013928(n). - Reinhard Zumkeller, Jul 05 2010
a(n) * A112526(n) = A063524(n). - Reinhard Zumkeller, Sep 16 2011
a(n) = mu(n) * lambda(n) = A008836(n) * A008683(n). - Enrique Pérez Herrero, Nov 29 2013
a(n) = Sum_{d|n} 2^omega(d)*mu(n/d). - Geoffrey Critzer, Feb 22 2015
a(n) = A085357(A156552(n)). - Antti Karttunen, Mar 06 2017
Limit_{n->oo} (1/n)*Sum_{j=1..n} a(j) = 6/Pi^2. - Andres Cicuttin, Aug 13 2017
a(1) = 1; a(n) = -Sum_{d|n, d < n} (-1)^bigomega(n/d) * a(d). - Ilya Gutkovskiy, Mar 10 2021

Deleted an unclear comment. - N. J. A. Sloane, May 30 2021

A074206 Kalmár's [Kalmar's] problem: number of ordered factorizations of n.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 8, 1, 3, 3, 8, 1, 8, 1, 8, 3, 3, 1, 20, 2, 3, 4, 8, 1, 13, 1, 16, 3, 3, 3, 26, 1, 3, 3, 20, 1, 13, 1, 8, 8, 3, 1, 48, 2, 8, 3, 8, 1, 20, 3, 20, 3, 3, 1, 44, 1, 3, 8, 32, 3, 13, 1, 8, 3, 13, 1, 76, 1, 3, 8, 8, 3, 13, 1, 48, 8, 3, 1, 44, 3, 3, 3, 20, 1, 44, 3, 8, 3, 3, 3, 112

Offset: 0

N. J. A. Sloane, Apr 29 2003

a(0)=0, a(1)=1; thereafter a(n) is the number of ordered factorizations of n as a product of integers greater than 1.
Kalmár (1931) seems to be the earliest reference that mentions this sequence (as opposed to A002033). - N. J. A. Sloane, May 05 2016
a(n) is the permanent of the n-1 X n-1 matrix A with (i,j) entry = 1 if j|i+1 and = 0 otherwise. This is because ordered factorizations correspond to nonzero elementary products in the permanent. For example, with n=6, 3*2 -> 1,3,6 [partial products] -> 6,3,1 [reverse list] -> (6,3)(3,1) [partition into pairs with offset 1] -> (5,3)(2,1) [decrement first entry] -> (5,3)(2,1)(1,2)(3,4)(4,5) [append pairs (i,i+1) to get a permutation] -> elementary product A(1,2)A(2,1)A(3,4)A(4,5)A(5,3). - David Callan, Oct 19 2005
This sequence is important in describing the amount of energy in all wave structures in the Universe according to harmonics theory. - Ray Tomes (ray(AT)tomes.biz), Jul 22 2007
a(n) appears to be the number of permutation matrices contributing to the Moebius function. See A008683 for more information. Also a(n) appears to be the Moebius transform of A067824. Furthermore it appears that except for the first term a(n)=A067824(n)*(1/2). Are there other sequences such that when the Moebius transform is applied, the new sequence is also a constant factor times the starting sequence? - Mats Granvik, Jan 01 2009
Numbers divisible by n distinct primes appear to have ordered factorization values that can be found in an n-dimensional summatory Pascal triangle. For example, the ordered factorization values for numbers divisible by two distinct primes can be found in table A059576. - Mats Granvik, Sep 06 2009
Inverse Mobius transform of A174725 and also except for the first term, inverse Mobius transform of A174726. - Mats Granvik, Mar 28 2010
a(n) is a lower bound on the worst-case number of solutions to the probed partial digest problem for n fragments of DNA; see the Newberg & Naor reference, below. - Lee A. Newberg, Aug 02 2011
All integers greater than 1 are perfect numbers over this sequence (for definition of A-perfect numbers, see comment to A175522). - Vladimir Shevelev, Aug 03 2011
If n is squarefree, then a(n) = A000670(A001221(n)) = A000670(A001222(n)). - Vladimir Shevelev and Franklin T. Adams-Watters, Aug 05 2011
A034776 lists the values taken by this sequence. - Robert G. Wilson v, Jun 02 2012
From Gus Wiseman, Aug 25 2020: (Start)
Also the number of strict chains of divisors from n to 1. For example, the a(n) chains for n = 1, 2, 4, 6, 8, 12, 30 are:
  1  2/1  4/1    6/1    8/1      12/1      30/1
          4/2/1  6/2/1  8/2/1    12/2/1    30/2/1
                 6/3/1  8/4/1    12/3/1    30/3/1
                        8/4/2/1  12/4/1    30/5/1
                                 12/6/1    30/6/1
                                 12/4/2/1  30/10/1
                                 12/6/2/1  30/15/1
                                 12/6/3/1  30/6/2/1
                                           30/6/3/1
                                           30/10/2/1
                                           30/10/5/1
                                           30/15/3/1
                                           30/15/5/1
(End)
a(n) is also the number of ways to tile a strip of length log(n) with tiles having lengths {log(k) : k>=2}. - David Bevan, Jan 07 2025

			G.f. = x + x^2 + x^3 + 2*x^4 + x^5 + 3*x^6 + x^7 + 4*x^8 + 2*x^9 + 3*x^10 + ...
Number of ordered factorizations of 8 is 4: 8 = 2*4 = 4*2 = 2*2*2.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 126, see #27.
R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 141.
Kalmár, Laszlo, A "factorisatio numerorum" problemajarol [Hungarian], Matemat. Fizik. Lapok, 38 (1931), 1-15.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 124.

N. J. A. Sloane, Table of n, a(n) for n = 0..20000 (first 10000 terms from T. D. Noe)
David Bevan and Julien Condé, Introducing irrational enumeration: analytic combinatorics for objects of irrational size, arXiv:2412.14682 [math.CO], 2024. See p. 11.
Peter Brown, Number of Ordered Factorizations.
Peter Brown, Number of Ordered Factorizations.
Benny Chor, Paul Lemke, and Ziv Mador, On the number of ordered factorizations of natural numbers, Discrete Math. 214 (2000), no. 1-3, 123-133. MR1743631 (2000m:11093).
Kristin DeVleming and Nikita Singh, Rational unicuspidal plane curves of low degree, arXiv:2311.15922 [math.AG], 2023. See p. 14.
T. M. A. Fink, Properties of the recursive divisor function and the number of ordered factorizations, arXiv:2307.09140 [math.NT], 2023.
E. Hille, A problem in factorisatio numerorum, Acta Arith., 2 (1936), 134-144.
E. Hille, The inversion problem of Möbius, Duke Math. J., 3 (1937), 549-568.
Shikao Ikehara, On Kalmar's Problem in "Factorisatio Numerorum", Proceedings of the Physico-Mathematical Society of Japan. 3rd Series, Vol. 21 (1939) pp. 208-219.
Shikao Ikehara, On Kalmar's Problem in "Factorisatio Numerorum" II, Proceedings of the Physico-Mathematical Society of Japan. 3rd Series, Vol. 23 (1941) pp. 767-774.
Laszlo Kalmár, Über die mittlere Anzahl der Produktdarstellungen der Zahlen. (Erste Mitteilung), Acta Litt. ac Scient. Szeged 5 (1931): 95-107.
M. Klazar and F. Luca, On the maximal order of numbers in the "factorisatio numerorum" problem, arXiv:math/0505352 [math.NT], 2005-2006.
Arnold Knopfmacher and Michael Mays, Ordered and Unordered Factorization of Integers, The Mathematica Journal, Volume 10, Issue 1 p. 72.
Arnau Mir, Francesc Rossello, and Lucia Rotger, Sound Colless-like balance indices for multifurcating trees, arXiv:1805.01329 [q-bio.PE], 2018.
Augustine O. Munagi, Labeled factorization of integers, INTEGERS: The Electronic Journal of Combinatorics 16:1 (2009), #R50.
L. A. Newberg and D. Naor, A lower bound on the number of solutions to the probed partial digest problem, Advances in Applied Mathematics, 14(2), 1993, 172-183. doi: 10.1006/aama.1993.1009.
Ray Tomes, The Maths and Physics of the Harmonics Theory.
Eric Weisstein's World of Mathematics, Perfect Partition.
Eric Weisstein's World of Mathematics, Ordered Factorization.
David W. Wilson, Comments on A074206 and related sequences.
David W. Wilson, Perl program for A074206.
Index entries for "core" sequences

Apart from initial term, same as A002033.
a(A002110) = A000670, row sums of A251683.
A173382 (and A025523) gives partial sums.
Cf. A008683, A050324, A027751, A001221, A034776, A000670, A050369.
A124433 has these as unsigned row sums.
A334996 has these as row sums.
A001055 counts factorizations.
A001222 counts prime factors with multiplicity.
A008480 counts ordered prime factorizations.
A067824 counts strict chains of divisors starting with n.
A122651 counts strict chains of divisors summing to n.
A253249 counts strict chains of divisors.
Cf. A000005, A025487, A046523, A059576, A122408, A124010, A153881 (Dirichlet inverse), A167865, A174725, A174726, A175522, A181819, A320390, A334997, A337105, A361665.

a074206 n | n <= 1 = n
| otherwise = 1 + (sum $ map (a074206 . (div n)) $
tail $ a027751_row n)
-- Reinhard Zumkeller, Oct 01 2012

a := array(1..150): for k from 1 to 150 do a[k] := 0 od: a[1] := 1: for j from 2 to 150 do for m from 1 to j-1 do if j mod m = 0 then a[j] := a[j]+a[m] fi: od: od: for k from 1 to 150 do printf(`%d,`,a[k]) od: # James Sellers, Dec 07 2000
A074206:= proc(n) option remember; if n > 1 then `+`(op(apply(A074206, numtheory[divisors](n)[1..-2]))) else n fi end: # M. F. Hasler, Oct 12 2018

a[0] = 0; a[1] = 1; a[n_] := a[n] = a /@ Most[Divisors[n]] // Total; a /@ Range[20000] (* N. J. A. Sloane, May 04 2016, based on program in A002033 *)
ccc[n_]:=Switch[n,0,{},1,{{1}},,Join@@Table[Prepend[#,n]&/@ccc[d],{d,Most[Divisors[n]]}]]; Table[Length[ccc[n]],{n,0,100}] (* _Gus Wiseman, Aug 25 2020 *)

A=vector(100);A[1]=1; for(n=2,#A,A[n]=1+sumdiv(n,d,A[d])); A/=2; A[1]=1; concat(0,A) \\ Charles R Greathouse IV, Nov 20 2012

{a(n) = if( n<2, n>0, my(A = divisors(n)); sum(k=1, #A-1, a(A[k])))}; /* Michael Somos, Dec 26 2016 */

A074206(n)=if(n>1, sumdiv(n, i, if(iA074206(i))),n) \\ M. F. Hasler, Oct 12 2018

A74206=[1]; A074206(n)={if(#A74206A074206(i)))} \\ Use memoization for computing many values. - M. F. Hasler, Oct 12 2018

first(n) = {my(res = vector(n, i, 1)); for(i = 2, n, for(j = 2, n \ i, res[i*j] += res[i])); concat(0, res)} \\ David A. Corneth, Oct 13 2018

first(n) = {my(res = vector(n, i, 1)); for(i = 2, n, d = divisors(i); res[i] += sum(j = 1, #d-1, res[d[j]])); concat(0, res)} \\ somewhat faster than progs above for finding first terms of n. \\ David A. Corneth, Oct 12 2018

a(n)={if(!n, 0, my(sig=factor(n)[,2], m=vecsum(sig)); sum(k=0, m, prod(i=1, #sig, binomial(sig[i]+k-1, k-1))*sum(r=k, m, binomial(r,k)*(-1)^(r-k))))} \\ Andrew Howroyd, Aug 30 2020

from math import prod
from functools import lru_cache
from sympy import divisors, factorint, prime
@lru_cache(maxsize=None)
def A074206(n): return sum(A074206(d) for d in divisors(prod(prime(i+1)**e for i,e in enumerate(sorted(factorint(n).values(),reverse=True))),generator=True,proper=True)) if n > 1 else n # Chai Wah Wu, Sep 16 2022

@cached_function
def minus_mu(n):
    if n < 2: return n
    return sum(minus_mu(d) for d in divisors(n)[:-1])
# Note that changing the sign of the sum gives the Möbius function A008683.
print([minus_mu(n) for n in (0..96)]) # Peter Luschny, Dec 26 2016

With different offset: a(n) = sum of all a(i) such that i divides n and i < n. - Clark Kimberling
a(p^k) = 2^(k-1) if k>0 and p is a prime.
Dirichlet g.f.: 1/(2-zeta(s)). - Herbert S. Wilf, Apr 29 2003
a(n) = A067824(n)/2 for n>1; a(A122408(n)) = A122408(n)/2. - Reinhard Zumkeller, Sep 03 2006
If p,q,r,... are distinct primes, then a(p*q)=3, a(p^2*q)=8, a(p*q*r)=13, a(p^3*q)=20, etc. - Vladimir Shevelev, Aug 03 2011 [corrected by Charles R Greathouse IV, Jun 02 2012]
a(0) = 0, a(1) = 1; a(n) = [x^n] Sum_{k=1..n-1} a(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Dec 11 2017
a(n) = a(A046523(n)); a(A025487(n)) = A050324(n): a(n) depends only on the nonzero exponents in the prime factorization of n, more precisely prime signature of n, cf. A124010 and A320390. - M. F. Hasler, Oct 12 2018
a(n) = A000670(A001221(n)) for squarefree n. In particular a(A002110(n)) = A000670(n). - Amiram Eldar, May 13 2019
a(n) = A050369(n)/n, for n>=1. - Ridouane Oudra, Aug 31 2019
a(n) = A361665(A181819(n)). - Pontus von Brömssen, Mar 25 2023
From Ridouane Oudra, Nov 02 2023: (Start)
If p,q are distinct primes, and n,m>0 then we have:
a(p^n*q^m) = Sum_{k=0..min(n,m)} 2^(n+m-k-1)*binomial(n,k)*binomial(m,k);
More generally: let tau[k](n) denote the number of ordered factorizations of n as a product of k terms, also named the k-th Piltz function (see A007425), then we have for n>1:
a(n) = Sum_{j=1..bigomega(n)} Sum_{k=1..j} (-1)^(j-k)*binomial(j,k)*tau[k](n), or
a(n) = Sum_{j=1..bigomega(n)} Sum_{k=0..j-1} (-1)^k*binomial(j,k)*tau[j-k](n). (End)

Originally this sequence was merged with A002033, the number of perfect partitions. Herbert S. Wilf suggested that it warrants an entry of its own.

cp's OEIS Frontend

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