A000688 -id:A000688 - cp's OEIS frontend

A005117 Squarefree numbers: numbers that are not divisible by a square greater than 1.

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113

Offset: 1

N. J. A. Sloane

1 together with the numbers that are products of distinct primes.
Also smallest sequence with the property that a(m)*a(k) is never a square for k != m. - Ulrich Schimke (ulrschimke(AT)aol.com), Dec 12 2001
Numbers k such that there is only one Abelian group with k elements, the cyclic group of order k (the numbers such that A000688(k) = 1). - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 25 2001
Numbers k such that A007913(k) > phi(k). - Benoit Cloitre, Apr 10 2002
a(n) is the smallest m with exactly n squarefree numbers <= m. - Amarnath Murthy, May 21 2002
k is squarefree <=> k divides prime(k)# where prime(k)# = product of first k prime numbers. - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 30 2004
Numbers k such that omega(k) = Omega(k) = A072047(k). - Lekraj Beedassy, Jul 11 2006
The LCM of any finite subset is in this sequence. - Lekraj Beedassy, Jul 11 2006
This sequence and the Beatty Pi^2/6 sequence (A059535) are "incestuous": the first 20000 terms are bounded within (-9, 14). - Ed Pegg Jr, Jul 22 2008
Let us introduce a function D(n) = sigma_0(n)/2^(alpha(1) + ... + alpha(r)), sigma_0(n) number of divisors of n (A000005), prime factorization of n = p(1)^alpha(1) * ... * p(r)^alpha(r), alpha(1) + ... + alpha(r) is sequence (A001222). Function D(n) splits the set of positive integers into subsets, according to the value of D(n). Squarefree numbers (A005117) has D(n)=1, other numbers are "deviated" from the squarefree ideal and have 0 < D(n) < 1. For D(n)=1/2 we have A048109, for D(n)=3/4 we have A060687. - Ctibor O. Zizka, Sep 21 2008
Numbers k such that gcd(k,k')=1 where k' is the arithmetic derivative (A003415) of k. - Giorgio Balzarotti, Apr 23 2011
Numbers k such that A007913(k) = core(k) = k. - Franz Vrabec, Aug 27 2011
Numbers k such that sqrt(k) cannot be simplified. - Sean Loughran, Sep 04 2011
Indices m where A057918(m)=0, i.e., positive integers m for which there are no integers k in {1,2,...,m-1} such that k*m is a square. - John W. Layman, Sep 08 2011
It appears that these are numbers j such that Product_{k=1..j} (prime(k) mod j) = 0 (see Maple code). - Gary Detlefs, Dec 07 2011. - This is the same claim as Mohammed Bouayoun's Mar 30 2004 comment above. To see why it holds: Primorial numbers, A002110, a subsequence of this sequence, are never divisible by any nonsquarefree number, A013929, and on the other hand, the index of the greatest prime dividing any n is less than n. Cf. A243291. - Antti Karttunen, Jun 03 2014
Conjecture: For each n=2,3,... there are infinitely many integers b > a(n) such that Sum_{k=1..n} a(k)*b^(k-1) is prime, and the smallest such an integer b does not exceed (n+3)*(n+4). - Zhi-Wei Sun, Mar 26 2013
The probability that a random natural number belongs to the sequence is 6/Pi^2, A059956 (see Cesàro reference). - Giorgio Balzarotti, Nov 21 2013
Booker, Hiary, & Keating give a subexponential algorithm for testing membership in this sequence without factoring. - Charles R Greathouse IV, Jan 29 2014
Because in the factorizations into prime numbers these a(n) (n >= 2) have exponents which are either 0 or 1 one could call the a(n) 'numbers with a fermionic prime number decomposition'. The levels are the prime numbers prime(j), j >= 1, and the occupation numbers (exponents) e(j) are 0 or 1 (like in Pauli's exclusion principle). A 'fermionic state' is then denoted by a sequence with entries 0 or 1, where, except for the zero sequence, trailing zeros are omitted. The zero sequence stands for a(1) = 1. For example a(5) = 6 = 2^1*3^1 is denoted by the 'fermionic state' [1, 1], a(7) = 10 by [1, 0, 1]. Compare with 'fermionic partitions' counted in A000009. - Wolfdieter Lang, May 14 2014
From Vladimir Shevelev, Nov 20 2014: (Start)
The following is an Eratosthenes-type sieve for squarefree numbers. For integers > 1:
1) Remove even numbers, except for 2; the minimal non-removed number is 3.
2) Replace multiples of 3 removed in step 1, and remove multiples of 3 except for 3 itself; the minimal non-removed number is 5.
3) Replace multiples of 5 removed as a result of steps 1 and 2, and remove multiples of 5 except for 5 itself; the minimal non-removed number is 6.
4) Replace multiples of 6 removed as a result of steps 1, 2 and 3 and remove multiples of 6 except for 6 itself; the minimal non-removed number is 7.
5) Repeat using the last minimal non-removed number to sieve from the recovered multiples of previous steps.
Proof. We use induction. Suppose that as a result of the algorithm, we have found all squarefree numbers less than n and no other numbers. If n is squarefree, then the number of its proper divisors d > 1 is even (it is 2^k - 2, where k is the number of its prime divisors), and, by the algorithm, it remains in the sequence. Otherwise, n is removed, since the number of its squarefree divisors > 1 is odd (it is 2^k-1).
(End)
The lexicographically least sequence of integers > 1 such that each entry has an even number of proper divisors occurring in the sequence (that's the sieve restated). - Glen Whitney, Aug 30 2015
0 is nonsquarefree because it is divisible by any square. - Jon Perry, Nov 22 2014, edited by M. F. Hasler, Aug 13 2015
The Heinz numbers of partitions with distinct parts. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product_{j=1..r} prime(j) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] the Heinz number is 2*2*3*7*29 = 2436. The number 30 (= 2*3*5) is in the sequence because it is the Heinz number of the partition [1,2,3]. - Emeric Deutsch, May 21 2015
It is possible for 2 consecutive terms to be even; for example a(258)=422 and a(259)=426. - Thomas Ordowski, Jul 21 2015. [These form a subsequence of A077395 since their product is divisible by 4. - M. F. Hasler, Aug 13 2015]
There are never more than 3 consecutive terms. Runs of 3 terms start at 1, 5, 13, 21, 29, 33, ... (A007675). - Ivan Neretin, Nov 07 2015
a(n) = product of row n in A265668. - Reinhard Zumkeller, Dec 13 2015
Numbers without excess, i.e., numbers k such that A001221(k) = A001222(k). - Juri-Stepan Gerasimov, Sep 05 2016
Numbers k such that b^(phi(k)+1) == b (mod k) for every integer b. - Thomas Ordowski, Oct 09 2016
Boreico shows that the set of square roots of the terms of this sequence is linearly independent over the rationals. - Jason Kimberley, Nov 25 2016 (reference found by Michael Coons).
Numbers k such that A008836(k) = A008683(k). - Enrique Pérez Herrero, Apr 04 2018
The prime zeta function P(s) "has singular points along the real axis for s=1/k where k runs through all positive integers without a square factor". See Wolfram link. - Maleval Francis, Jun 23 2018
Numbers k such that A007947(k) = k. - Kyle Wyonch, Jan 15 2021
The Schnirelmann density of the squarefree numbers is 53/88 (Rogers, 1964). - Amiram Eldar, Mar 12 2021
Comment from Isaac Saffold, Dec 21 2021: (Start)
Numbers k such that all groups of order k have a trivial Frattini subgroup [Dummit and Foote].
Let the group G have order n. If n is squarefree and n > 1, then G is solvable, and thus by Hall's Theorem contains a subgroup H_p of index p for all p | n. Each H_p is maximal in G by order considerations, and the intersection of all the H_p's is trivial. Thus G's Frattini subgroup Phi(G), being the intersection of G's maximal subgroups, must be trivial. If n is not squarefree, the cyclic group of order n has a nontrivial Frattini subgroup. (End)
Numbers for which the squarefree divisors (A206778) and the unitary divisors (A077610) are the same; moreover they are also the set of divisors (A027750). - Bernard Schott, Nov 04 2022
0 = A008683(a(n)) - A008836(a(n)) = A001615(a(n)) - A000203(a(n)). - Torlach Rush, Feb 08 2023
From Robert D. Rosales, May 20 2024: (Start)
Numbers n such that mu(n) != 0, where mu(n) is the Möbius function (A008683).
Solutions to the equation Sum_{d|n} mu(d)*sigma(d) = mu(n)*n, where sigma(n) is the sum of divisors function (A000203). (End)
a(n) is the smallest root of x = 1 + Sum_{k=1..n-1} floor(sqrt(x/a(k))) greater than a(n-1). - Yifan Xie, Jul 10 2024
Number k such that A001414(k) = A008472(k). - Torlach Rush, Apr 14 2025
To elaborate on the formula from Greathouse (2018), the maximum of a(n) - floor(n*Pi^2/6 + sqrt(n)/17) is 10 at indices n = 48715, 48716, 48721, and 48760. The maximum is 11, at the same indices, if floor is taken individually for the two addends and the square root. If the value is rounded instead, the maximum is 9 at 10 indices between 48714 and 48765. - M. F. Hasler, Aug 08 2025

Jean-Marie De Koninck, Ces nombres qui nous fascinent, Entry 165, p. 53, Ellipses, Paris, 2008.
David S. Dummit and Richard M. Foote, Abstract algebra. Vol. 1999. Englewood Cliffs, NJ: David S.Prentice Hall, 1991.
Ivan M. Niven and Herbert S. Zuckerman, An Introduction to the Theory of Numbers. 2nd ed., Wiley, NY, 1966, p. 251.
Michael Pohst and Hans J. Zassenhaus, Algorithmic Algebraic Number Theory, Cambridge Univ. Press, page 432.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Daniel Forgues, Table of n, a(n) for n = 1..60794 (first 10000 terms from T. D. Noe)
Zenon B. Batang, Squarefree integers and the abc conjecture, arXiv:2109.10226 [math.GM], 2021.
Thomas Bloom, Problem 145, Erdős Problems.
Andrew R. Booker, Ghaith A. Hiary and Jon P. Keating, Detecting squarefree numbers, Duke Mathematical Journal, Vol. 164, No. 2 (2015), pp. 235-275; arXiv preprint, arXiv:1304.6937 [math.NT], 2013-2015.
Iurie Boreico, Linear independence of radicals, The Harvard College Mathematics Review 2(1), 87-92, Spring 2008.
Ernesto Cesàro, La serie di Lambert in aritmetica assintotica, Rendiconto della Reale Accademia delle Scienze di Napoli, Serie 2, Vol. 7 (1893), pp. 197-204.
Henri Cohen, Francois Dress, and Mohamed El Marraki, Explicit estimates for summatory functions linked to the Möbius μ-function, Functiones et Approximatio Commentarii Mathematici 37 (2007), part 1, pp. 51-63.
Hubert Gent, Letter to N. J. A. Sloane, Nov 27 1975.
Andrew Granville, ABC means we can count squarefrees, International Mathematical Research Notices 19 (1998), 991-1009.
Pentti Haukkanen, Mika Mattila, Jorma K. Merikoski and Timo Tossavainen, Can the Arithmetic Derivative be Defined on a Non-Unique Factorization Domain?, Journal of Integer Sequences, Vol. 16 (2013), Article 13.1.2.
Aaron Krowne, squarefree number, PlanetMath.org.
Louis Marmet, First occurrences of squarefree gaps and an algorithm for their computation.
Louis Marmet, First occurrences of square-free gaps and an algorithm for their computation, arXiv preprint arXiv:1210.3829 [math.NT], 2012.
Srinivasa Ramanujan, Irregular numbers, J. Indian Math. Soc., Vol. 5 (1913), pp. 105-106.
Kenneth Rogers, The Schnirelmann density of the squarefree integers, Proceedings of the American Mathematical Society, Vol. 15, No. 4 (1964), pp. 515-516.
J. A. Scott, Square-freedom revisited, The Mathematical Gazette, Vol. 90, No. 517 (2006), pp. 112-113.
Vladimir Shevelev, Set of all densities of exponentially S-numbers, arXiv preprint arXiv:1511.03860 [math.NT], 2015.
Terence Tao, Erdős problem database, see no. 145.
Ognian Trifonov, On the Squarefree Problem II, Math. Balkanica, Vol. 3 (1989), Fasc. 3-4.
Eric Weisstein's World of Mathematics, Squarefree.
Eric Weisstein's World of Mathematics, Prime Zeta Function.
Wikipedia, Squarefree integer.
Index entries for "core" sequences.

Complement of A013929. Subsequence of A072774 and A209061.
Characteristic function: A008966 (mu(n)^2, where mu = A008683).
Subsequences: A000040, A002110, A235488.
Cf. A001414, A008472, A076259 (first differences), A173143 (partial sums), A000688, A003277, A013928, A020753, A020754, A020755, A030059, A030229, A033197, A034444, A039956, A048672, A053797, A057918, A059956, A071403, A072284, A120992, A133466, A136742, A136743, A160764, A243289, A243347, A243348, A243351, A215366, A046660, A265668, A265675.
Cf. A027750, A077610, A206778.
Subsequences: numbers j such that j*a(k) is squarefree where k > 1: A056911 (k = 2), A261034 (k = 3), A274546 (k = 5), A276378 (k = 6).

a005117 n = a005117_list !! (n-1)
a005117_list = filter ((== 1) . a008966) [1..]
-- Reinhard Zumkeller, Aug 15 2011, May 10 2011

[ n : n in [1..1000] | IsSquarefree(n) ];

with(numtheory); a := [ ]; for n from 1 to 200 do if issqrfree(n) then a := [ op(a), n ]; fi; od:
t:= n-> product(ithprime(k),k=1..n): for n from 1 to 113 do if(t(n) mod n = 0) then print(n) fi od; # Gary Detlefs, Dec 07 2011
A005117 := proc(n) option remember; if n = 1 then 1; else for a from procname(n-1)+1 do if numtheory[issqrfree](a) then return a; end if; end do: end if; end proc:  # R. J. Mathar, Jan 09 2013

Select[ Range[ 113], SquareFreeQ] (* Robert G. Wilson v, Jan 31 2005 *)
Select[Range[150], Max[Last /@ FactorInteger[ # ]] < 2 &] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 26 2006 *)
NextSquareFree[n_, k_: 1] := Block[{c = 0, sgn = Sign[k]}, sf = n + sgn; While[c < Abs[k], While[ ! SquareFreeQ@ sf, If[sgn < 0, sf--, sf++]]; If[ sgn < 0, sf--, sf++]; c++]; sf + If[ sgn < 0, 1, -1]]; NestList[ NextSquareFree, 1, 70] (* Robert G. Wilson v, Apr 18 2014 *)
Select[Range[250], MoebiusMu[#] != 0 &] (* Robert D. Rosales, May 20 2024 *)

bnd = 1000; L = vector(bnd); j = 1; for (i=1,bnd, if(issquarefree(i),L[j]=i; j=j+1)); L

{a(n)= local(m,c); if(n<=1,n==1, c=1; m=1; while( cMichael Somos, Apr 29 2005 */

list(n)=my(v=vectorsmall(n,i,1),u,j); forprime(p=2,sqrtint(n), forstep(i=p^2, n, p^2, v[i]=0)); u=vector(sum(i=1,n,v[i])); for(i=1,n,if(v[i],u[j++]=i)); u \\ Charles R Greathouse IV, Jun 08 2012

for(n=1, 113, if(core(n)==n, print1(n, ", "))); \\ Arkadiusz Wesolowski, Aug 02 2016

S(n) = my(s); forsquarefree(k=1,sqrtint(n),s+=n\k[1]^2*moebius(k)); s;
a(n) = my(min=1, max=231, k=0, sc=0); if(n >= 144, min=floor(zeta(2)*n - 5*sqrt(n)); max=ceil(zeta(2)*n + 5*sqrt(n))); while(min <= max, k=(min+max)\2; sc=S(k); if(abs(sc-n) <= sqrtint(n), break); if(sc > n, max=k-1, if(sc < n, min=k+1, break))); while(!issquarefree(k), k-=1); while(sc != n, my(j=1); if(sc > n, j = -1); k += j; sc += j; while(!issquarefree(k), k += j)); k; \\ Daniel Suteu, Jul 07 2022

first(n)=my(v=vector(n),i); forsquarefree(k=1,if(n<268293,(33*n+30)\20,(n*Pi^2/6+0.058377*sqrt(n))\1), if(i++>n, return(v)); v[i]=k[1]); v \\ Charles R Greathouse IV, Jan 10 2023

A5117=[1..3]; A005117(n)={if(n>#A5117, my(N=#A5117); A5117=Vec(A5117, max(n+999, N*5\4)); iferr(forsquarefree(k=A5117[N]+1, #A5117*Pi^2\6+sqrtint(#A5117)\17+11, A5117[N++]=k[1]),E,)); A5117[n]} \\ M. F. Hasler, Aug 08 2025

from sympy.ntheory.factor_ import core
def ok(n): return core(n, 2) == n
print(list(filter(ok, range(1, 114)))) # Michael S. Branicky, Jul 31 2021

from itertools import count, islice
from sympy import factorint
def A005117_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:all(x == 1 for x in factorint(n).values()),count(max(startvalue,1)))
A005117_list = list(islice(A005117_gen(),20)) # Chai Wah Wu, May 09 2022

from math import isqrt
from sympy import mobius
def A005117(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Jul 22 2024

Limit_{n->oo} a(n)/n = Pi^2/6 (see A013661). - Benoit Cloitre, May 23 2002
Equals A039956 UNION A056911. - R. J. Mathar, May 16 2008
A122840(a(n)) <= 1; A010888(a(n)) < 9. - Reinhard Zumkeller, Mar 30 2010
a(n) = A055229(A062838(n)) and a(n) > A055229(m) for m < A062838(n). - Reinhard Zumkeller, Apr 09 2010
A008477(a(n)) = 1. - Reinhard Zumkeller, Feb 17 2012
A055653(a(n)) = a(n); A055654(a(n)) = 0. - Reinhard Zumkeller, Mar 11 2012
A008966(a(n)) = 1. - Reinhard Zumkeller, May 26 2012
Sum_{n>=1} 1/a(n)^s = zeta(s)/zeta(2*s). - Enrique Pérez Herrero, Jul 07 2012
A056170(a(n)) = 0. - Reinhard Zumkeller, Dec 29 2012
A013928(a(n)+1) = n. - Antti Karttunen, Jun 03 2014
A046660(a(n)) = 0. - Reinhard Zumkeller, Nov 29 2015
Equals {1} UNION A000040 UNION A006881 UNION A007304 UNION A046386 UNION A046387 UNION A067885 UNION A123321 UNION A123322 UNION A115343 ... - R. J. Mathar, Nov 05 2016
|a(n) - n*Pi^2/6| < 0.058377*sqrt(n) for n >= 268293; this result can be derived from Cohen, Dress, & El Marraki, see links. - Charles R Greathouse IV, Jan 18 2018
From Amiram Eldar, Jul 07 2021: (Start)
Sum_{n>=1} (-1)^(a(n)+1)/a(n)^2 = 9/Pi^2.
Sum_{k=1..n} 1/a(k) ~ (6/Pi^2) * log(n).
Sum_{k=1..n} (-1)^(a(k)+1)/a(k) ~ (2/Pi^2) * log(n).
(all from Scott, 2006) (End)

A013929 Numbers that are not squarefree. Numbers that are divisible by a square greater than 1. The complement of A005117.

Original entry on oeis.org

4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 64, 68, 72, 75, 76, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 104, 108, 112, 116, 117, 120, 121, 124, 125, 126, 128, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160

Offset: 1

Henri Lifchitz

Sometimes misnamed squareful numbers, but officially those are given by A001694.
This is different from the sequence of numbers k such that A007913(k) < phi(k). The two sequences differ at the values: 420, 660, 780, 840, 1320, 1560, 4620, 5460, 7140, ..., which is essentially A070237. - Ant King, Dec 16 2005
Numbers k such that Sum_{d|k} (d/phi(d))*mu(k/d) = 0. - Benoit Cloitre, Apr 28 2002
Also, k with at least one x < k such that A007913(x) = A007913(k). - Benoit Cloitre, Apr 28 2002
Numbers k for which there exists a partition into two parts p and q such that p + q = k and p*q is a multiple of k. - Amarnath Murthy, May 30 2003
Numbers k such that there is a solution 0 < x < k to x^2 == 0 (mod k). - Franz Vrabec, Aug 13 2005
Numbers k such that moebius(k) = 0.
a(n) = k such that phi(k)/k = phi(m)/m for some m < k. - Artur Jasinski, Nov 05 2008
Appears to be numbers such that when a column with index equal to a(n) in A051731 is deleted, there is no impact on the result in the first column of A054525. - Mats Granvik, Feb 06 2009
Numbers k such that the number of prime divisors of (k+1) is less than the number of nonprime divisors of (k+1). - Juri-Stepan Gerasimov, Nov 10 2009
Orders for which at least one non-cyclic finite abelian group exists: A000688(a(n)) > 1. This follows from the fact that not all exponents in the prime factorization of a(n) are 1 (moebius(a(n)) = 0). The number of such groups of order a(n) is A192005(n) = A000688(a(n)) - 1. - Wolfdieter Lang, Jul 29 2011
Subsequence of A193166; A192280(a(n)) = 0. - Reinhard Zumkeller, Aug 26 2011
It appears that terms are the numbers m such that Product_{k=1..m} (prime(k) mod m) <> 0. See Maple code. - Gary Detlefs, Dec 07 2011
A008477(a(n)) > 1. - Reinhard Zumkeller, Feb 17 2012
A057918(a(n)) > 0. - Reinhard Zumkeller, Mar 27 2012
A056170(a(n)) > 0. - Reinhard Zumkeller, Dec 29 2012
Numbers k such that A001221(k) != A001222(k). - Felix Fröhlich, Aug 13 2014
Numbers k such that A001222(k) > A001221(k), since in this case at least one prime factor of k occurs more than once, which implies that k is divisible by at least one perfect square > 1. - Carlos Eduardo Olivieri, Aug 02 2015
Lexicographically least sequence such that each term has a positive even number of proper divisors not occurring in the sequence, cf. the sieve characterization of A005117. - Glen Whitney, Aug 30 2015
There are arbitrarily long runs of consecutive terms. Record runs start at 4, 8, 48, 242, ... (A045882). - Ivan Neretin, Nov 07 2015
A number k is a term if 0 < min(A000010(k) + A023900(k), A000010(k) - A023900(k)). - Torlach Rush, Feb 22 2018
Every squareful number > 1 is nonsquarefree, but the converse is false and the nonsquarefree numbers that are not squareful (see first comment) are in A332785. - Bernard Schott, Apr 11 2021
Integers m where at least one k < m exists such that m divides k^m. - Richard R. Forberg, Jul 31 2021
Consider the Diophantine equation S(x,y) = (x+y) + (x-y) + (x*y) + (x/y) = z, when x and y are both positive integers with y | x. Then, there is a solution (x,y) iff z is a term of this sequence; in this case, if x = K*y, then z = S(K*y,y) = K*(y+1)^2 (see A351381, link and references Perelman); example: S(12,4) = 75 = a(28). The number of solutions for S(x,y) = a(n) is A353282(n). - Bernard Schott, Mar 29 2022
For each positive integer m, the number of unitary divisors of m = the number of squarefree divisors of m (see A034444); but only for the terms of this sequence does the set of unitary divisors differ from the set of squarefree divisors. Example: the set of unitary divisors of 20 is {1, 4, 5, 20}, while the set of squarefree divisors of 20 is {1, 2, 5, 10}. - Bernard Schott, Oct 15 2022

			For the terms up to 20, we compute the squares of primes up to floor(sqrt(20)) = 4. Those squares are 4 and 9. For every such square s, put the terms s*k^2 for k = 1 to floor(20 / s). This gives after sorting and removing duplicates the list 4, 8, 9, 12, 16, 18, 20. - _David A. Corneth_, Oct 25 2017

I. Perelman, L'Algèbre récréative, Deux nombres et quatre opérations, Editions en langues étrangères, Moscou, 1959, pp. 101-102.
Ya. I. Perelman, Algebra can be fun, Two numbers and four operations, Mir Publishers Moscow, 1979, pp. 131-132.

David A. Corneth, Table of n, a(n) for n = 1..100000 (first 1000 terms from T. D. Noe)
H. Gent, Letter to N. J. A. Sloane, Nov 27 1975.
Louis Marmet, First occurrences of square-free gaps and an algorithm for their computation, arXiv:1210.3829 [math.NT], 2012.
Ya. I. Perelman, Algebra Can Be Fun, Chapter IV, Diophantine Equations, Two numbers and four operations, Mir Publishers Moscow, 1979, pp. 131-132.
Srinivasa Ramanujan, Irregular numbers, J. Indian Math. Soc. 5 (1913) 105-106.
Eric Weisstein's World of Mathematics, Smarandache Near-to-Primorial Function, Squarefree, Squareful, Moebius Function.

Complement of A005117. Subsequences: A130897, A190641, A332785.
Cf. A001694, A008683, A034444, A038109, A351381, A353282.
Partitions into: A114374, A256012.

a013929 n = a013929_list !! (n-1)
a013929_list = filter ((== 0) . a008966) [1..]
-- Reinhard Zumkeller, Apr 22 2012

[ n : n in [1..1000] | not IsSquarefree(n) ];

a := n -> `if`(numtheory[mobius](n)=0,n,NULL); seq(a(i),i=1..160); # Peter Luschny, May 04 2009
t:= n-> product(ithprime(k),k=1..n): for n from 1 to 160 do (if t(n) mod n <>0) then print(n) fi od; # Gary Detlefs, Dec 07 2011
with(NumberTheory): isQuadrateful := n -> irem(Radical(n), n) <> 0:
select(isQuadrateful, [`$`(1..160)]);  # Peter Luschny, Jul 12 2022

Union[ Flatten[ Table[ n i^2, {i, 2, 20}, {n, 1, 400/i^2} ] ] ]
Select[ Range[2, 160], (Union[Last /@ FactorInteger[ # ]][[ -1]] > 1) == True &] (* Robert G. Wilson v, Oct 11 2005 *)
Cases[Range[160], n_ /; !SquareFreeQ[n]] (* Jean-François Alcover, Mar 21 2011 *)
Select[Range@160, ! SquareFreeQ[#] &] (* Robert G. Wilson v, Jul 21 2012 *)
Select[Range@160, PrimeOmega[#] > PrimeNu[#] &] (* Carlos Eduardo Olivieri, Aug 02 2015 *)
Select[Range[200], MoebiusMu[#] == 0 &] (* Alonso del Arte, Nov 07 2015 *)

{a(n)= local(m,c); if(n<=1,4*(n==1), c=1; m=4; while( cMichael Somos, Apr 29 2005 */

for(n=1, 1e3, if(omega(n)!=bigomega(n), print1(n, ", "))) \\ Felix Fröhlich, Aug 13 2014

upto(n)=my(res = List()); forprime(p = 2, sqrtint(n), for(k = 1, n \ p^2, listput(res, k * p^2))); listsort(res, 1); res \\ David A. Corneth, Oct 25 2017

from sympy.ntheory.factor_ import core
def ok(n): return core(n, 2) != n
print(list(filter(ok, range(1, 161)))) # Michael S. Branicky, Apr 08 2021

from math import isqrt
from sympy import mobius
def A013929(n):
    def f(x): return n+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Jul 20 2024

A008966(a(n)) = 0. - Reinhard Zumkeller, Apr 22 2012
Sum_{n>=1} 1/a(n)^s = (zeta(s)*(zeta(2*s)-1))/zeta(2*s). - Enrique Pérez Herrero, Jul 07 2012
a(n) ~ n/k, where k = 1 - 1/zeta(2) = 1 - 6/Pi^2 = A229099. - Charles R Greathouse IV, Sep 13 2013
A001222(a(n)) > A001221(a(n)). - Carlos Eduardo Olivieri, Aug 02 2015
phi(a(n)) > A003958(a(n)). - Juri-Stepan Gerasimov, Apr 09 2019

More terms from Erich Friedman

More terms from Franz Vrabec, Aug 13 2005

A001970 Functional determinants; partitions of partitions; Euler transform applied twice to all 1's sequence.

Original entry on oeis.org

1, 1, 3, 6, 14, 27, 58, 111, 223, 424, 817, 1527, 2870, 5279, 9710, 17622, 31877, 57100, 101887, 180406, 318106, 557453, 972796, 1688797, 2920123, 5026410, 8619551, 14722230, 25057499, 42494975, 71832114, 121024876, 203286806, 340435588, 568496753, 946695386

Offset: 0

N. J. A. Sloane

a(n) = number of partitions of n, when for each k there are p(k) different copies of part k. E.g., let the parts be 1, 2a, 2b, 3a, 3b, 3c, 4a, 4b, 4c, 4d, 4e, ... Then the a(4) = 14 partitions of 4 are: 4 = 4a = 4b = ... = 4e = 3a+1 = 3b+1 = 3c+1 = 2a+2a = 2a+2b = 2b+2b = 2a+1 = 2b+1 = 1+1+1+1.
Equivalently (Cayley), a(n) = number of 2-dimensional partitions of n. E.g., for n = 4 we have:
  4   31   3   22   2   211   21   2    2   1111   111   11    11   1
           1        2         1    11   1          1     11    1    1
                                        1                      1    1
                                                                    1
Also total number of different species of singularity for conjugate functions with n letters (Sylvester).
According to [Belmans], this sequence gives "[t]he number of Segre symbols for the intersection of two quadrics in a fixed dimension". - Eric M. Schmidt, Sep 02 2017
From Gus Wiseman, Jul 30 2022: (Start)
Also the number of non-isomorphic multiset partitions of weight n with all constant blocks. The strict case is A089259. For example, non-isomorphic representatives of the a(1) = 1 through a(3) = 6 multiset partitions are:
  {{1}}  {{1,1}}    {{1,1,1}}
         {{1},{1}}  {{1},{1,1}}
         {{1},{2}}  {{1},{2,2}}
                    {{1},{1},{1}}
                    {{1},{2},{2}}
                    {{1},{2},{3}}
A000688 counts factorizations into prime powers.
A007716 counts non-isomorphic multiset partitions by weight.
A279784 counts twice-partitions of type PPR, factorizations A295935.
Constant partitions are ranked by prime-powers: A000961, A023894, A054685, A246655, A355743.
(End)

			G.f. = 1 + x + 3*x^2 + 6*x^3 + 15*x^4 + 28*x^5 + 66*x^6 + 122*x^7 + ...
a(3) = 6 because we have (111) = (111) = (11)(1) = (1)(1)(1), (12) = (12) = (1)(2), (3) = (3).
The a(4)=14 multiset partitions whose total sum of parts is 4 are:
((4)),
((13)), ((1)(3)),
((22)), ((2)(2)),
((112)), ((1)(12)), ((2)(11)), ((1)(1)(2)),
((1111)), ((1)(111)), ((11)(11)), ((1)(1)(11)), ((1)(1)(1)(1)). - _Gus Wiseman_, Dec 19 2016

A. Cayley, Recherches sur les matrices dont les termes sont des fonctions linéaires d'une seule indéterminée, J. Reine angew. Math., 50 (1855), 313-317; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, p. 219.
V. A. Liskovets, Counting rooted initially connected directed graphs. Vesci Akad. Nauk. BSSR, ser. fiz.-mat., No 5, 23-32 (1969), MR44 #3927.
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).
J. J. Sylvester, An Enumeration of the Contacts of Lines and Surfaces of the Second Order, Phil. Mag. 1 (1851), 119-140. Reprinted in Collected Papers, Vol. 1. See p. 239, where one finds a(n)-2, but with errors.
J. J. Sylvester, Note on the 'Enumeration of the Contacts of Lines and Surfaces of the Second Order', Phil. Mag., Vol. VII (1854), pp. 331-334. Reprinted in Collected Papers, Vol. 2, pp. 30-33.

Reinhard Zumkeller, Table of n, a(n) for n = 0..5000 (first 500 terms from T. D. Noe)
Pieter Belmans, Segre symbols, 2016.
Philip Boalch, Counting the fission trees and nonabelian Hodge graphs, arXiv:2410.23358 [math.AG], 2024. See pp. 10, 16.
P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 148
R. Kaneiwa, An asymptotic formula for Cayley's double partition function p(2; n), Tokyo J. Math. 2, 137-158 (1979).
L. Kaylor and D. Offner, Counting matrices over a finite field with all eigenvalues in the field, Involve, a Journal of Mathematics, Vol. 7 (2014), No. 5, 627-645. [DOI]
M. Kozek, F. Luca, P. Pollack, and C. Pomerance, Harmonious pairs, 2014.
M. Kozek, F. Luca, P. Pollack, and C. Pomerance, Harmonious numbers, IJNT, to appear.
XiKun Li, JunLi Li, Bin Liu and CongFeng Qiao, The parametric symmetry and numbers of the entangled class of 2 × M × N system, Science China Physics, Mechanics & Astronomy, Volume 54, Number 8, 1471-1475, DOI: 10.1007/s11433-011-4395-9.
Jessie Pitsillides, Segre Characteristic Equivalence, arXiv:2506.12065 [math.GM], 2025.
Paul Pollack and Carl Pomerance, Some problems of Erdős on the sum-of-divisors function, For Richard Guy on his 99th birthday: May his sequence be unbounded, Trans. Amer. Math. Soc. Ser. B, Vol. 3 (2016), pp. 1-26; Errata.
N. J. A. Sloane, Transforms.
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.
J. J. Sylvester, The collected mathematical papers of James Joseph Sylvester, vol. 2, vol. 3, vol. 4.
Index entries for sequences related to rooted trees

Related to A001383 via generating function.
The multiplicative version (factorizations) is A050336.
The ordered version (sequences of partitions) is A055887.
Row-sums of A061260.
Main diagonal of A055885.
We have A271619(n) <= a(n) <= A063834(n).
Column k=3 of A290353.
The strict case is A316980.
Cf. A000041, A061259, A006171, A061255, A061256, A061257, A089292, A000219.
Cf. A089300.
Cf. A001055, A072233, A112798, A275024.

Following Vladeta Jovovic:
a001970 n = a001970_list !! (n-1)
a001970_list = 1 : f 1 [1] where
   f x ys = y : f (x + 1) (y : ys) where
            y = sum (zipWith (*) ys a061259_list) `div` x
-- Reinhard Zumkeller, Oct 31 2015

with(combstruct); SetSetSetU := [T, {T=Set(S), S=Set(U,card >= 1), U=Set(Z,card >=1)},unlabeled];
# second Maple program:
with(numtheory): with(combinat):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      numbpart(d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Dec 19 2016

m = 32; f[x_] = Product[1/(1-x^k)^PartitionsP[k], {k, 1, m}]; CoefficientList[ Series[f[x], {x, 0, m-1}], x] (* Jean-François Alcover, Jul 19 2011, after g.f. *)

{a(n) = if( n<0, 0, polcoeff( 1 / prod(k=1, n, 1 - numbpart(k) * x^k + x * O(x^n)), n))}; /* Michael Somos, Dec 20 2016 */

from sympy.core.cache import cacheit
from sympy import npartitions, divisors
@cacheit
def a(n): return 1 if n == 0 else sum([sum([d*npartitions(d) for d in divisors(j)])*a(n - j) for j in range(1, n + 1)]) / n
[a(n) for n in range(51)]  # Indranil Ghosh, Aug 19 2017, after Maple code
# (Sage) # uses[EulerTransform from A166861]
b = BinaryRecurrenceSequence(0, 1, 1)
a = EulerTransform(EulerTransform(b))
print([a(n) for n in range(36)]) # Peter Luschny, Nov 17 2022

G.f.: Product_{k >= 1} 1/(1-x^k)^p(k), where p(k) = number of partitions of k = A000041. [Cayley]
a(n) = (1/n)*Sum_{k = 1..n} a(n-k)*b(k), n > 1, a(0) = 1, b(k) = Sum_{d|k} d*numbpart(d), where numbpart(d) = number of partitions of d, cf. A061259. - Vladeta Jovovic, Apr 21 2001
Logarithmic derivative yields A061259 (equivalent to above formula from Vladeta Jovovic). - Paul D. Hanna, Sep 05 2012
a(n) = Sum_{k=1..A000041(n)} A001055(A215366(n,k)) = number of factorizations of Heinz numbers of integer partitions of n. - Gus Wiseman, Dec 19 2016
a(n) = |{m>=1 : n = Sum_{k=1..A001222(m)} A056239(A112798(m,k)+1)}| = number of normalized twice-prime-factored multiset partitions (see A275024) whose total sum of parts is n. - Gus Wiseman, Dec 19 2016

Additional comments from Valery A. Liskovets

Sylvester references from Barry Cipra, Oct 07 2003

A000001 Number of groups of order n.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, 2, 2, 1, 15, 2, 2, 5, 4, 1, 4, 1, 51, 1, 2, 1, 14, 1, 2, 2, 14, 1, 6, 1, 4, 2, 2, 1, 52, 2, 5, 1, 5, 1, 15, 2, 13, 2, 2, 1, 13, 1, 2, 4, 267, 1, 4, 1, 5, 1, 4, 1, 50, 1, 2, 3, 4, 1, 6, 1, 52, 15, 2, 1, 15, 1, 2, 1, 12, 1, 10, 1, 4, 2

Offset: 0

N. J. A. Sloane

Also, number of nonisomorphic subgroups of order n in symmetric group S_n. - Lekraj Beedassy, Dec 16 2004
Also, number of nonisomorphic primitives (antiderivatives) of the combinatorial species Lin[n-1], or X^{n-1}; see Rajan, Summary item (i). - Nicolae Boicu, Apr 29 2011
In (J. H. Conway, Heiko Dietrich and E. A. O'Brien, 2008), a(n) is called the "group number of n", denoted by gnu(n), and the first occurrence of k is called the "minimal order attaining k", denoted by moa(k) (see A046057). - Daniel Forgues, Feb 15 2017
It is conjectured in (J. H. Conway, Heiko Dietrich and E. A. O'Brien, 2008) that the sequence n -> a(n) -> a(a(n)) = a^2(n) -> a(a(a(n))) = a^3(n) -> ... -> consists ultimately of 1s, where a(n), denoted by gnu(n), is called the "group number of n". - Muniru A Asiru, Nov 19 2017
MacHale (2020) shows that there are infinitely many values of n for which there are more groups than rings of that order (cf. A027623). He gives n = 36355 as an example. It would be nice to have enough values of n to create an OEIS entry for them. - N. J. A. Sloane, Jan 02 2021
I conjecture that a(i) * a(j) <= a(i*j) for all nonnegative integers i and j. - Jorge R. F. F. Lopes, Apr 21 2024

			Groups of orders 1 through 10 (C_n = cyclic, D_n = dihedral of order n, Q_8 = quaternion, S_n = symmetric):
1: C_1
2: C_2
3: C_3
4: C_4, C_2 X C_2
5: C_5
6: C_6, S_3=D_6
7: C_7
8: C_8, C_4 X C_2, C_2 X C_2 X C_2, D_8, Q_8
9: C_9, C_3 X C_3
10: C_10, D_10

S. R. Blackburn, P. M. Neumann, and G. Venkataraman, Enumeration of Finite Groups, Cambridge, 2007.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 302, #35.
J. H. Conway et al., The Symmetries of Things, Peters, 2008, p. 209.
H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 134.
CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 150.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, A Foundation for Computer Science, Addison-Wesley Publ. Co., Reading, MA, 1989, Section 6.6 'Fibonacci Numbers' pp. 281-283.
M. Hall, Jr. and J. K. Senior, The Groups of Order 2^n (n <= 6). Macmillan, NY, 1964.
D. Joyner, 'Adventures in Group Theory', Johns Hopkins Press. Pp. 169-172 has table of groups of orders < 26.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIII.24, p. 481.
M. F. Newman and E. A. O'Brien, A CAYLEY library for the groups of order dividing 128. Group theory (Singapore, 1987), 437-442, de Gruyter, Berlin-New York, 1989.
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).

H. U. Besche and Ivan Panchenko, Table of n, a(n) for n = 0..2047 [Terms 1 through 2015 copied from Small Groups Library mentioned below. Terms 2016 - 2047 added by _Ivan Panchenko_, Aug 29 2009. 0 prepended by _Ray Chandler_, Sep 16 2015. a(1024) corrected by _Benjamin Przybocki_, Jan 06 2022]
H. A. Bender, A determination of the groups of order p^5, Ann. of Math. (2) 29, pp. 61-72 (1927).
Hans Ulrich Besche and Bettina Eick, Construction of finite groups, Journal of Symbolic Computation, Vol. 27, No. 4, Apr 15 1999, pp. 387-404.
Hans Ulrich Besche and Bettina Eick, The groups of order at most 1000 except 512 and 768, Journal of Symbolic Computation, Vol. 27, No. 4, Apr 15 1999, pp. 405-413.
H. U. Besche, B. Eick and E. A. O'Brien, The groups of order at most 2000, Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 1-4.
H. U. Besche, B. Eick, E. A. O'Brien and Max Horn, The Small Groups Library
H. U. Besche, B. Eick and E. A. O'Brien, Number of isomorphism types of finite groups of given order [gives incorrect a(1024)]
H. U. Besche, B. Eick and E. A. O'Brien, A Millennium Project: Constructing Small Groups, Internat. J. Algebra and Computation, 12 (2002), 623-644.
Henry Bottomley, Illustration of initial terms
David Burrell, On the number of groups of order 1024, Communications in Algebra, 2021, 1-3.
J. H. Conway, Heiko Dietrich and E. A. O'Brien, Counting groups: gnus, moas and other exotica, Math. Intell., Vol. 30, No. 2, Spring 2008.
Marek Dančo, Mikoláš Janota, Michael Codish, and João Jorge Araújo, Complete Symmetry Breaking for Finite Models, arXiv:2502.10155 [cs.LO], 2025. See p. 7. Mentions this sequence.
Yang-Hui He and Minhyong Kim, Learning Algebraic Structures: Preliminary Investigations, arXiv:1905.02263 [cs.LG], 2019.
Otto Hölder, Die Gruppen der Ordnungen p^3, pq^2, pqr, p^4, Math. Ann. 43 pp. 301-412 (1893).
Max Horn, Numbers of isomorphism types of finite groups of given order
Rodney James, The groups of order p^6 (p an odd prime), Math. Comp. 34 (1980), 613-637.
Rodney James and John Cannon, Computation of isomorphism classes of p-groups, Mathematics of Computation 23.105 (1969): 135-140.
Olexandr Konovalov, Crowdsourcing project for the database of numbers of isomorphism types of finite groups, Github (a list of gnu(n) for many n < 50000).
Desmond MacHale, Are There More Finite Rings than Finite Groups?, Amer. Math. Monthly (2020) Vol. 127, Issue 10, 936-938.
Mehdi Makhul, Josef Schicho, and Audie Warren, On Galois groups of type-1 minimally rigid graphs, arXiv:2306.04392 [math.CO], 2023.
G. A. Miller, Determination of all the groups of order 64, Amer. J. Math., 52 (1930), 617-634.
László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003.
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003 [Cached copy, with permission (pdf only)]
Dayanand S. Rajan, The equations D^kY=X^n in combinatorial species, Discrete Mathematics 118 (1993) 197-206 North-Holland.
E. Rodemich, The groups of order 128, J. Algebra 67 (1980), no. 1, 129-142.
Gordon Royle, Combinatorial Catalogues. See subpage "Generators of small groups" for explicit generators for most groups of even order < 1000. [broken link]
D. Rusin, Asymptotics [Cached copy of lost web page]
Eric Weisstein's World of Mathematics, Finite Group
Wikipedia, Finite group
M. Wild, The groups of order sixteen made easy, Amer. Math. Monthly, 112 (No. 1, 2005), 20-31.
Gang Xiao, SmallGroup
Index entries for sequences related to groups
Index entries for "core" sequences

The main sequences concerned with group theory are A000001 (this one), A000679, A001034, A001228, A005180, A000019, A000637, A000638, A002106, A005432, A000688, A060689, A051532.
Cf. A046058, A046059, A023675, A023676.
A003277 gives n for which A000001(n) = 1, A063756 (partial sums).
A046057 gives first occurrence of each k.
A027623 gives the number of rings of order n.

A000001 := Concatenation([0], List([1..500], n -> NumberSmallGroups(n))); # Muniru A Asiru, Oct 15 2017

D:=SmallGroupDatabase(); [ NumberOfSmallGroups(D, n) : n in [1..1000] ]; // John Cannon, Dec 23 2006

GroupTheory:-NumGroups(n); # with(GroupTheory); loads this command - N. J. A. Sloane, Dec 28 2017

FiniteGroupCount[Range[100]] (* Harvey P. Dale, Jan 29 2013 *)
a[ n_] := If[ n < 1, 0, FiniteGroupCount @ n]; (* Michael Somos, May 28 2014 *)

From Mitch Harris, Oct 25 2006: (Start)
For p, q, r primes:
a(p) = 1, a(p^2) = 2, a(p^3) = 5, a(p^4) = 14, if p = 2, otherwise 15.
a(p^5) = 61 + 2*p + 2*gcd(p-1,3) + gcd(p-1,4), p >= 5, a(2^5)=51, a(3^5)=67.
a(p^e) ~ p^((2/27)e^3 + O(e^(8/3))).
a(p*q) = 1 if gcd(p,q-1) = 1, 2 if gcd(p,q-1) = p. (p < q)
a(p*q^2) is one of the following:
   ---------------------------------------------------------------------------
  | a(p*q^2) |            p*q^2 of the form             |  Sequences (p*q^2)  |
   ---------- ------------------------------------------ ---------------------
  | (p+9)/2  |          q == 1 (mod p), p odd           |      A350638        |
  |    5     |         p=3, q=2 =>  p*q^2 = 12          |Special case with A_4|
  |    5     |               p=2, q odd                 |      A143928        |
  |    5     |             p == 1 (mod q^2)             |      A350115        |
  |    4     | p == 1 (mod q), p > 3, p !== 1 (mod q^2) |      A349495        |
  |    3     |       q == -1 (mod p), p and q odd       |      A350245        |
  |    2     |  q !== +-1 (mod p) and p !== 1 (mod q)   |      A350422        |
   ---------------------------------------------------------------------------
[Table from Bernard Schott, Jan 18 2022]
a(p*q*r) (p < q < r) is one of the following:
  q == 1 (mod p)  r == 1 (mod p)  r == 1 (mod q)  a(p*q*r)
  --------------  --------------  --------------  --------
        No              No              No            1
        No              No              Yes           2
        No              Yes             No            2
        No              Yes             Yes           4
        Yes             No              No            2
        Yes             No              Yes           3
        Yes             Yes             No           p+2
        Yes             Yes             Yes          p+4
[table from Derek Holt].
(End)
a(n) = A000688(n) + A060689(n). - R. J. Mathar, Mar 14 2015

More terms from Michael Somos

Typo in b-file description fixed by David Applegate, Sep 05 2009

A055887 Number of ordered partitions of partitions.

Original entry on oeis.org

1, 1, 3, 8, 22, 59, 160, 431, 1164, 3140, 8474, 22864, 61697, 166476, 449210, 1212113, 3270684, 8825376, 23813776, 64257396, 173387612, 467856828, 1262431711, 3406456212, 9191739970, 24802339472, 66924874539, 180585336876, 487278670744, 1314838220172

Offset: 0

Christian G. Bower, Jun 09 2000

nonn

Jordan matrices are upper bidiagonal matrices such that (A) the diagonal entries are in sorted order, (B) there are only 1's and 0's on the superdiagonal, (C) for each superdiagonal 1, the two diagonal entries to the left and below it must be equal. Let J(N) be the number of N X N Jordan matrices where the diagonal values are, without loss of generality, taken to be a prefix of some fixed strictly increasing sequence x_1, x_2, x_3, ... If Jordan blocks sorted by eigenvalue with ties broken by block size during the sorting, then J(1, 2, 3, ...) is this sequence. - Warren D. Smith, Jan 28 2002
Number of compositions of n into parts k >= 1 where there are A000041(k) sorts of part k. - Joerg Arndt, Sep 30 2012
Also number of chains of multisets that partition a normal multiset of weight n, where a multiset is normal if it spans an initial interval of positive integers. - Gus Wiseman, Oct 28 2015
From Gus Wiseman, Jul 31 2022: (Start)
Also the number of ways to choose a multiset partition into constant multisets of a multiset of length n covering an initial interval of positive integers. This interpretation involves only multisets, not sequences. For example, the a(1) = 1 through a(3) = 8 multiset partitions are:
  {{1}}  {{1,1}}    {{1,1,1}}
         {{1},{1}}  {{1},{1,1}}
         {{1},{2}}  {{1},{2,2}}
                    {{2},{1,1}}
                    {{1},{1},{1}}
                    {{1},{1},{2}}
                    {{1},{2},{2}}
                    {{1},{2},{3}}
Factorizations into prime powers, are counted by A000688.
The strongly normal case is A063834.
The strongly normal strict case is A270995.
Twice-partitions of type PPR are counted by A279784, factorizations A295935.
The strict case is A304969.
(End)

			The a(4) = 22 chains of multisets, where notation x-y means "y is a submultiset of x", are: (o-o-o-o) (oo-o-o) (oo-oo) (ooo-o) (oooo) (oe-o-o) (ooe-o) (oooe) (oe-oe) (ooe-e) (oee-o) (ooee) (oei-o) (ooei) (oe-e-e) (oee-e) (oeee) (oei-e) (oeei) (oei-i) (oeii) (oeis).
From _Gus Wiseman_, Jul 31 2022: (Start)
a(n) is the number of ways to choose an integer partition of each part of an integer composition of n. The a(0) = 1 through a(3) = 8 choices are:
  ()  ((1))  ((2))     ((3))
             ((11))    ((21))
             ((1)(1))  ((111))
                       ((1)(2))
                       ((2)(1))
                       ((1)(11))
                       ((11)(1))
                       ((1)(1)(1))
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..2320
N. J. A. Sloane, Transforms
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89, DOI:10.1007/s11083-004-9460-9.
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.

Cf. A055888, A083355, A095975, A246828.
Row sums of A060642.
Cf. A326346.
The unordered version is A001970, row-sums of A061260.
A000041 counts integer partitions, strict A000009.
A011782 counts integer compositions.
A072233 counts partitions by sum and length.
Cf. A001055, A050361, A063834, A279784, A355743.

with(combstruct); SeqSetSetU := [T, {T=Sequence(S), S=Set(U,card >= 1), U=Set(Z,card >=1)},unlabeled];
P := (x) -> product( 1/(1-x^k), k=1..20 ) - 1; F := (x) -> series( 1/(1-P(x)) - 1, x, 21 ); # F(x) is g.f. for this sequence # Warren D. Smith, Jan 28 2002
A055887rec:= proc(n::integer) local k; option remember; with(combinat): if n = 0 then 1 else add(numbpart(k) *procname(n - k), k=1..n); end if; end proc: seq (A055887rec(n), n=0..10); # Thomas Wieder, Nov 26 2007

a = 1/Product[(1 - x^k), {k, 1, \[Infinity]}] - 1; CoefficientList[Series[1/(1 - a), {x, 0, 20}], x] (* Geoffrey Critzer, Dec 23 2010 *)
(1/(2 - 1/QPochhammer[x]) + O[x]^30)[[3]] (* Vladimir Reshetnikov, Sep 22 2016 *)
Table[Sum[Times@@PartitionsP/@c,{c,Join@@Permutations/@IntegerPartitions[n]}],{n,0,10}] (* Gus Wiseman, Jul 31 2022 *)

Vec(1/(2-1/eta(x+O(x^66)))) \\ Joerg Arndt, Sep 30 2012

Invert transform of partitions numbers A000041.
Let p(k) be the number of integer partitions of k. Furthermore, set a(0)=1. Then a(n) = Sum_{k=1..n} p(k)*a(n-k). - Thomas Wieder, Nov 26 2007
G.f.: 1/( 1 - Sum_{k>=1} p(k)*x^k ) where p(k) = A000041(k) is the number of integer partitions of k. - Joerg Arndt, Sep 30 2012
a(n) ~ c * d^n, where d = 2.698329106474211231263998666188376330713465125913986356769... (see A246828) and c = 0.414113793172792357745578049739573823627306487211379286647... - Vaclav Kotesovec, Mar 29 2014
G.f.: 1/(2 - 1/(x)inf), where (x)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Sep 22 2016

cp's OEIS Frontend

A346213 Number of iterations of A000688 needed to reach 1 starting at n (n is counted).

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A369171 Numbers k such that A000688(k) = A046660(k+1).

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A138286 a(n) = A058162(n) / A000688(n).

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A208663 Non-Abelian numbers: n such that A000001(n)/A000688(n) is a new record.

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A369763 Decimal expansion of the asymptotic mean of the ratio A000688(k)/A038538(k).

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A005117 Squarefree numbers: numbers that are not divisible by a square greater than 1.

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A013929 Numbers that are not squarefree. Numbers that are divisible by a square greater than 1. The complement of A005117.

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