A033273 -id:A033273 - cp's OEIS frontend

A324594 a(n) is the smallest number k such that n consecutive integers starting at k have the same number of nonprime divisors (A033273).

1, 1, 1, 19940, 204323, 294590, 310042685, 2587701932494, 2587701932494

Offset: 1

Ilya Gutkovskiy, Sep 03 2019

			19940 has 9 nonprime divisors {1, 4, 10, 20, 1994, 3988, 4985, 9970, 19940}, 19941 has 9 nonprime divisors {1, 51, 69, 289, 391, 867, 1173, 6647, 19941}, 19942 has 9 nonprime divisors {1, 26, 118, 169, 338, 767, 1534, 9971, 19942} and 19943 has 9 nonprime divisors {1, 49, 77, 259, 407, 539, 1813, 2849, 19943}. These the first 4 consecutive numbers with the same number of nonprime divisors, so a(4) = 19940.

Rémy Sigrist, C program for A324594

Cf. A006558, A033273, A045983, A045984, A323253, A324593.

C
```
See Links section.
```

a(7) from Rémy Sigrist, Sep 04 2019

a(8)-a(9) from Giovanni Resta, Sep 04 2019

A180040 Record values in A033273.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 10, 13, 15, 17, 21, 27, 28, 29, 32, 33, 36, 37, 44, 45, 56, 60, 68, 76, 80, 86, 91, 92, 96, 104, 115, 116, 123, 139, 140, 155, 163, 175, 187, 195, 211, 219, 234, 235, 250, 251, 282, 283, 314, 330, 331, 354, 378, 394, 426, 442, 474, 498, 506, 570, 594

Offset: 1

Ray Chandler, Aug 07 2010

nonn

Amiram Eldar, Table of n, a(n) for n = 1..571 (terms 1..146 from Ray Chandler)

Cf. A033273, A059992.

lista(nn) = {my(m=0, nb); for (n=1, nn, nb = sumdiv(n, d, !isprime(d)); if (nb > m, m = nb; print1(nb, ", ")););} \\ Michel Marcus, Jul 16 2019

a(n) = A033273(A059992(n)).

A352256 a(n) is the least k such that A033273(k) is equal to (A033273(n*k + 1) - 1)/n where A033273(n) is the number of nonprime divisors of n.

Original entry on oeis.org

3, 13, 5, 41, 11, 2479, 23, 73, 103, 2249, 19, 7177, 211, 691, 3089, 1289, 53263, 726493, 41, 1597, 2243, 64406, 13129, 31351, 983, 1579, 197, 43037, 1411, 38246575, 389, 3607, 15403, 61286, 709, 1638349, 3587, 16249, 3585641, 1017119, 1292839, 132347, 593, 32203, 51963

Offset: 1

Juri-Stepan Gerasimov, Mar 09 2022

nonn

			a(2) = 13 because A033273(13) = (A033273(2*13 + 1) - 1)/2 = (A033273(27) - 1)/2 = 1.

Cf. A000005, A001221, A033273, A350516.

f[n_] := DivisorSigma[0, n] - PrimeNu[n]; a[n_] := Module[{k = 2}, While[f[k] != (f[n*k + 1] - 1)/n, k++]; k]; Array[a, 29] (* Amiram Eldar, Mar 10 2022 *)

f(n) = numdiv(n) - omega(n); \\ A033273
a(n) = my(k=2); while (f(k) != (f(n*k + 1) - 1)/n, k++); k; \\ Michel Marcus, Mar 10 2022

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)

A005171 Characteristic function of nonprimes: 0 if n is prime, else 1.

Original entry on oeis.org

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

Offset: 1

Russ Cox

Number of orbits of length n in map whose periodic points are A023890. - Thomas Ward
Characteristic function of nonprimes A018252. - Jonathan Vos Post, Dec 30 2007
Triangle A157423 = A005171 in every column. A052284 = INVERT transform of A005171, and the eigensequence of triangle A157423. - Gary W. Adamson, Feb 28 2009

Douglas Hofstadter, Fluid Concepts and Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought.

Antti Karttunen, Table of n, a(n) for n = 1..65537
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
Yash Puri and Thomas Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Yash Puri and Thomas Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.
Index entries for characteristic functions

Cf. A010051, A018252, A023890.

Cf. A157423, A157424, A052284, A050374, A033273, A008683.

a005171 = (1 -) . a010051  -- Reinhard Zumkeller, Mar 30 2014

A005171 := proc(n)
    if isprime(n) then
        0 ;
    else
        1 ;
    end if;
end proc: # R. J. Mathar, May 26 2017

a[n_] := If[PrimeQ@ n, 0, 1]; Array[a, 105] (* Robert G. Wilson v, Jun 20 2011 *)
nn = 105; t[n_, k_] :=  t[n, k] = If[n == k, 1, If[k == 1, 1 - Product[t[n, k + i], {i, 1, n - 1}], If[Mod[n, k] == 0, t[n/k, 1], 1], 1]]; Table[t[n, 1], {n, 1, nn}] (* Mats Granvik, Sep 21 2013 *)

a(n)=if(n<1, 0, !isprime(n)) /* Michael Somos, Jun 08 2005 */

from sympy import isprime
def a(n): return int(not isprime(n))
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Oct 28 2021

a(n) = (1/n)* Sum_{ d divides n } mu(d)*A023890(n/d). E.g., a(6) = 1 since the 6th term of A023890 is 7 and the first term is 1. [edited by Michel Marcus, Dec 14 2023]
a(n) = 1 - A010051(n). - Jonathan Vos Post, Dec 30 2007
a(n) equals the first column in a table T defined by the recurrence: If n = k then T(n,k) = 1 else if k = 1 then T(n,k) = 1 - Product_{k divides n} of T(n,k), else if k divides n then T(n,k) = T(n/k,1). This is true since T(n,k) = 0 when k divides n and n/k is prime which results in Product_{k divides n} = 0 for the composite numbers and where k ranges from 2 to n. Therefore there is a remaining 1 in the expression 1-Product_{k divides n}, in the first column. Provided below is a Mathematica program as an illustration. - Mats Granvik, Sep 21 2013
a(n) = A057427(A239968(n)). - Reinhard Zumkeller, Mar 30 2014
a(n) = Sum_{d|n} A033273(d)*A008683(n/d). - Ridouane Oudra, Jul 03 2025

A034836 Number of ways to write n as n = xyz with 1 <= x <= y <= z.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 3, 4, 1, 5, 1, 5, 2, 2, 2, 8, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 9, 2, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 10, 1, 2, 4, 7, 2, 5, 1, 4, 2, 5, 1, 12, 1, 2, 4, 4, 2, 5, 1, 9, 4, 2, 1, 10, 2, 2, 2, 6, 1, 10, 2, 4, 2, 2, 2, 12, 1, 4, 4, 8

Offset: 1

Erich Friedman

nonn

Number of boxes with integer edge lengths and volume n.
Starts the same as, but is different from, A033273. First values of n such that a(n) differs from A033273(n) are 36,48,60,64,72,80,84,90,96,100. - Benoit Cloitre, Nov 25 2002
a(n) depends only on the signature of n; the sorted exponents of n. For instance, a(12) and a(18) are the same because both 12 and 18 have signature (1,2). - T. D. Noe, Nov 02 2011
Number of 3D grids of n congruent cubes, in a box, modulo rotation (cf. A007425 and A140773 for boxes instead of cubes; cf. A038548 for the 2D case). - Manfred Boergens, Apr 06 2021

			a(12) = 4 because we can write 12 = 1*1*12 = 1*2*6 = 1*3*4 = 2*2*3.
a(36) = 8 because we can write 36 = 1*1*36 = 1*2*18 = 1*3*12 = 1*4*9 = 1*6*6 = 2*2*9 = 2*3*6 = 3*3*4.
For n = p*q, p < q primes: a(n) = 2 because we can write n = 1*1*pq = 1*p*q.
For n = p^2, p prime: a(n) = 2 because we can write n = 1*1*p^2 = 1*p*p.

T. D. Noe, Table of n, a(n) for n = 1..10000
Dorin Andrica and Eugen J. Ionascu, On the number of polynomials with coefficients in [n], An. Şt. Univ. Ovidius Constanţa, Vol. 22, No. 1 (2013), pp. 13-23; alternative link.
Index entries for sequences computed from exponents in factorization of n.

See also: writing n = x*y (A038548, unordered, A000005, ordered), n = x*y*z (this sequence, unordered, A007425, ordered), n = w*x*y*z (A007426, ordered)
Cf. A001358, A008578, A010057, A046951, A088432, A088433, A088434.
Differs from A033273 and A226378 for the first time at n=36.

f:=proc(n) local t1,i,j,k; t1:=0; for i from 1 to n do for j from i to n do for k from j to n do if i*j*k = n then t1:=t1+1; fi; od: od: od: t1; end;
# second Maple program:
A034836:=proc(n)
   local a,b,i;
   a:=0;
   b:=(l,x,h)->l<=x and x<=h;
   for i in select(`<=`,NumberTheory:-Divisors(n),iroot(n,3)) do
      a:=a+nops(select[2](b,i,NumberTheory:-Divisors(n/i),isqrt(n/i)))
   od;
   return a
end proc;
seq(A034836(n),n=1..100); # Felix Huber, Oct 02 2024

Table[c=0; Do[If[i<=j<=k && i*j*k==n,c++],{i,t=Divisors[n]},{j,t},{k,t}]; c,{n,100}] (* Jayanta Basu, May 23 2013 *)
(* Similar to the first Mathematica code but with fewer steps in Do[..] *)
b=0; d=Divisors[n]; r=Length[d];
Do[If[d[[h]] d[[i]] d[[j]]==n, b++], {h, r}, {i, h, r}, {j, i, r}]; b (* Manfred Boergens, Apr 06 2021 *)
a[1] = 1; a[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[IntegerQ[Surd[n, 3]], 1/3, 0] + (Times @@ ((e + 1)*(e + 2)/2))/6 + (Times @@ (Floor[e/2] + 1))/2]; Array[a, 100] (* Amiram Eldar, Apr 19 2024 *)

A038548(n)=sumdiv(n, d, d*d<=n) /* <== rhs from A038548 (Michael Somos) */
a(n)=sumdiv(n, d, if(d^3<=n, A038548(n/d) - sumdiv(n/d, d0, d0Rick L. Shepherd, Aug 27 2006

a(n) = {my(e = factor(n)[,2]); (2 * ispower(n, 3) + vecprod(apply(x -> (x+1)*(x+2)/2, e)) + 3 * vecprod(apply(x -> x\2 + 1, e))) / 6;} \\ Amiram Eldar, Apr 19 2024

From Ton Biegstraaten, Jan 04 2016: (Start)
Given a number n, let s(1),...,s(m) be the signature list of n, and a(n) the resulting number in the sequence.
Then np = Product_{k=1..m} binomial(2+s(k),2) is the total number of products solely based on the combination of exponents. The multiplicity of powers is not taken into account (e.g., all combinations of 1,2,4 (6 times) but (2,2,2) only once). See next formulas to compute corrections for 3rd and 2nd powers.
Let ntp = Product_{k=1..m} (floor((s(k) - s(k) mod(3))/s(k))) if the number is a 3rd power or not resulting in 1 or 0.
Let nsq = Product_{k=1..m} (floor(s(k)/2) + 1) is the number of squares.
Conjecture: a(n) = (np + 3*(nsq - ntp) + 5*ntp)/6 = (np + 3*nsq + 2*ntp)/6.
Example: n = 1728; s = [3,6]; np = 10*28 = 280; nsq = 2*4 = 8; ntp = 1 so a(1728)=51 (as in the b-file).
(End)
a(n) >= A226378(n) for all n >= 1. - Antti Karttunen, Aug 30 2017
From Bernard Schott, Dec 12 2021: (Start)
a(n) = 1 iff n = 1 or n is prime (A008578).
a(n) = 2 iff n is semiprime (A001358) (see examples). (End)
a(n) = (2 * A010057(n) + A007425(n) + 3 * A046951(n))/6 (Andrica and Ionascu, 2013, p. 19, eq. 11). - Amiram Eldar, Apr 19 2024

Definition simplified by Jonathan Sondow, Oct 03 2013

A033987 Numbers that are divisible by at least 4 primes (counted with multiplicity).

Original entry on oeis.org

16, 24, 32, 36, 40, 48, 54, 56, 60, 64, 72, 80, 81, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 128, 132, 135, 136, 140, 144, 150, 152, 156, 160, 162, 168, 176, 180, 184, 189, 192, 196, 198, 200, 204, 208, 210, 216, 220, 224, 225, 228, 232, 234, 240, 243

Offset: 1

N. J. A. Sloane

Complement of A037144: A001222(a(n)) > 3; A117358(a(n)) > 1. - Reinhard Zumkeller, Mar 10 2006
Also numbers such that no permutation of all proper divisors exists with coprime adjacent elements: A178254(a(n)) = 0. - Reinhard Zumkeller, May 24 2010
Also, numbers that can be written as a product of at least two composites, i.e., admit a nontrivial factorization into composites. - Felix Fröhlich, Dec 22 2018

T. D. Noe, Table of n, a(n) for n=1..1000

Subsequence of A033942; A178212 is a subsequence.

Cf. A014613, A001055, A033273.

with(numtheory): A033987:=n->`if`(bigomega(n)>3, n, NULL): seq(A033987(n), n=1..300); # Wesley Ivan Hurt, May 26 2015

Select[Range[300],PrimeOmega[#]>3&] (* Harvey P. Dale, Mar 20 2016 *)

is(n)=bigomega(n)>3 \\ Charles R Greathouse IV, May 26 2015

from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A033987(n):
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n+primepi(x)+sum(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,i)) for i in range(2,4)))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 23 2024

Product p_i^e_i with Sum e_i >= 4.
A001055(a(n)) > A033273(a(n)). - Juri-Stepan Gerasimov, Nov 09 2009
a(n) ~ n. - Charles R Greathouse IV, Jul 11 2024

More terms from Patrick De Geest, Jun 15 1998

A083399 Number of divisors of n that are not divisors of other divisors of n.

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 3, 2, 2, 3, 2, 3, 3, 3, 2, 3, 2, 3, 2, 3, 2, 4, 2, 2, 3, 3, 3, 3, 2, 3, 3, 3, 2, 4, 2, 3, 3, 3, 2, 3, 2, 3, 3, 3, 2, 3, 3, 3, 3, 3, 2, 4, 2, 3, 3, 2, 3, 4, 2, 3, 3, 4, 2, 3, 2, 3, 3, 3, 3, 4, 2, 3, 2, 3, 2, 4, 3, 3, 3, 3, 2, 4, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 2, 4

Offset: 1

Reinhard Zumkeller, Jun 12 2003

a(n) <= tau(n); a(n) = tau(n) iff n is prime or n=1 (A008578, A000040); a(n)=tau(n)-1 iff n is semiprime (A001358).
Number of noncomposite divisors of n, (cf. A008578). - Jaroslav Krizek, Nov 25 2009
From Wilf A. Wilson, Jul 21 2017: (Start)
a(n) is the number of maximal subsemigroups of the annular Jones monoid of degree n.
a(n) is the number of maximal subsemigroups of the monoid of orientation-preserving mappings on a set with n elements.
a(n) + 1 is the number of maximal subsemigroups of the monoid of orientation-preserving partial mappings on a set with n elements.
(End)
This is the restricted growth sequence transform of A001221 (and thus also of A007875, A034444, A082476, A292586 and many other sequences). This follows from the formula a(n) = 1+A001221(n), and from the fact that for any n, A001221(n) <= 1+A001221(k) for all k = 1..(n-1). A067003 gives the ordinal transform of A001221. See also A292582, A292583, A292585. - Antti Karttunen, Sep 25 2017

			{1,2,3,4,6,8,12,24} are the divisors of n=24: 1, 2, 3, 4 and 6 divide not only 24, but also 8 or 12, therefore a(24) = 3.
{1,2,3,4,6,8,12,24} are the divisors of n=24: 1, 2 and 3 are noncomposites, therefore a(24) = 3. - _Jaroslav Krizek_, Nov 25 2009

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
James East, Jitender Kumar, James D. Mitchell, and Wilf A. Wilson, Maximal subsemigroups of finite transformation and partition monoids, arXiv:1706.04967 [math.GR], 2017. [_Wilf A. Wilson_, Jul 21 2017]

Cf. A000005, A001221, A067003, A055212, A077761.

a083399 = (+ 1) . a001221  -- Reinhard Zumkeller, Sep 14 2015

[(#(PrimeDivisors(n)))+1: n in [1..100]]; // Vincenzo Librandi, Feb 15 2015

A083399 := proc(n)
    1+nops(numtheory[factorset](n)) ;
end proc:
seq(A083399(n),n=1..100) ; # R. J. Mathar, Sep 26 2017

A083399[n_Integer]:=1+PrimeNu[n]; (* Enrique Pérez Herrero, Jun 25 2010 *)
Rest@ CoefficientList[ Series[(1/(1 - x)) + Sum[1/(1 - x^Prime[j]), {j, 200}], {x, 0, 111}], x] (* Robert G. Wilson v, Aug 16 2011 *)

a(n)=#factor(n)~+1 \\ Charles R Greathouse IV, Sep 14 2015

a(n) = omega(n) + 1, where omega = A001221.
a(n) = tau(n) - A055212(n) = A000005(n)-A055212(n).
a(n) = A000005(n) - A033273(n) + 1. - Jaroslav Krizek, Nov 25 2009
a(n) = A010553(A007947(n)) = A000005(A000005(A007947(n))) = tau_2(tau_2(rad(n))). - Enrique Pérez Herrero, Jun 25 2010
G.f.: x/(1 - x) + Sum_{k>=1} x^prime(k)/(1 - x^prime(k)). - Ilya Gutkovskiy, Mar 21 2017
Sum_{k=1..n} a(k) = n * (log(log(n)) + B + 1) + O(n/log(n)), where B is Mertens's constant (A077761). - Amiram Eldar, Sep 29 2024

A364531 Positive integers with no prime index equal to the sum of prime indices of any nonprime divisor.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 76, 77

Offset: 1

Gus Wiseman, Aug 01 2023

nonn

First differs from A299702 (knapsack) in having 525: {2,3,3,4}.
First differs from A325778 in lacking 462: {1,2,4,5}.
These are the Heinz numbers of partitions whose parts are disjoint from their own non-singleton subset-sums.

Partitions of this type are counted by A237667, strict A364349.
The binary version is A364462, complement A364461.
The complement is A364532, counted by A237668.
A000005 counts divisors, nonprime A033273, composite A055212.
A299701 counts distinct subset-sums of prime indices.
A299702 ranks knapsack partitions, counted by A108917, complement A299729.
A363260 counts partitions disjoint from differences, complement A364467.
Cf. A002865, A236912, A237113, A320340, A326083, A363225, A364347.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Intersection[prix[#],Total/@Subsets[prix[#],{2,Length[prix[#]]}]]=={}&]

A364532 Positive integers with a prime index equal to the sum of prime indices of some nonprime divisor. Heinz numbers of a variation of sum-full partitions.

Original entry on oeis.org

12, 24, 30, 36, 40, 48, 60, 63, 70, 72, 80, 84, 90, 96, 108, 112, 120, 126, 132, 140, 144, 150, 154, 156, 160, 165, 168, 180, 189, 192, 198, 200, 204, 210, 216, 220, 224, 228, 240, 252, 264, 270, 273, 276, 280, 286, 288, 300, 308, 312, 315, 320, 324, 325, 330

Offset: 1

Gus Wiseman, Aug 01 2023

nonn

First differs from A299729 (non-knapsack) in lacking 525: {2,3,3,4}.
First differs from A325777 in having 462: {1,2,4,5} and lacking 675:{2,2,2,3,3}.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
These are the Heinz numbers of partitions containing the sum of some non-singleton submultiset.

			The terms together with their prime indices begin:
  12: {1,1,2}
  24: {1,1,1,2}
  30: {1,2,3}
  36: {1,1,2,2}
  40: {1,1,1,3}
  48: {1,1,1,1,2}
  60: {1,1,2,3}
  63: {2,2,4}
  70: {1,3,4}
  72: {1,1,1,2,2}
  80: {1,1,1,1,3}
  84: {1,1,2,4}
  90: {1,2,2,3}
  96: {1,1,1,1,1,2}

Partitions not of this type are counted by A237667, strict A364349.
Partitions of this type are counted by A237668, strict A364272.
The binary complement is A364461, re-usable A364347 (counted by A364345).
The binary version is A364462, re-usable A364348 (counted by A363225).
The complement is A364531.
Subsets of this type are counted by A364534, complement A151897.
A000005 counts divisors, nonprime A033273, composite A055212.
A001222 counts prime indices.
A108917 counts knapsack partitions, strict A275972, for subsets A325864.
A112798 lists prime indices, sum A056239.
A299701 counts distinct subset-sums of prime indices.
A299702 ranks knapsack partitions, complement A299729.
Cf. A085489, A088809, A093971, A236912, A237113, A301900, A326083, A364350.

Select[Range[100],Intersection[prix[#],Total/@Subsets[prix[#],{2,Length[prix[#]]}]]!={}&]

cp's OEIS Frontend

A324594 a(n) is the smallest number k such that n consecutive integers starting at k have the same number of nonprime divisors (A033273).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

C

Extensions

A180040 Record values in A033273.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A352256 a(n) is the least k such that A033273(k) is equal to (A033273(n*k + 1) - 1)/n where A033273(n) is the number of nonprime divisors of n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

A001248 Squares of primes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

Python

SageMath

Formula

A005171 Characteristic function of nonprimes: 0 if n is prime, else 1.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Formula

A034836 Number of ways to write n as n = x*y*z with 1 <= x <= y <= z.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

A033987 Numbers that are divisible by at least 4 primes (counted with multiplicity).

Views

Author

Keywords

A034836 Number of ways to write n as n = xyz with 1 <= x <= y <= z.