A004101 -id:A004101 - cp's OEIS frontend

A046951 a(n) is the number of squares dividing n.

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

Offset: 1

Simon Colton (simonco(AT)cs.york.ac.uk)

Rediscovered by the HR automatic theory formation program.
a(n) depends only on prime signature of n (cf. A025487, A046523). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3, 1).
First differences of A013936. Average value tends towards Pi^2/6 = 1.644934... (A013661, A013679). - Henry Bottomley, Aug 16 2001
We have a(n) = A159631(n) for all n < 125, but a(125) = 2 < 3 = A159631(125). - Steven Finch, Apr 22 2009
Number of 2-generated Abelian groups of order n, if n > 1. - Álvar Ibeas, Dec 22 2014 [In other words, number of order-n abelian groups with rank <= 2. Proof: let b(n) be such number. A finite abelian group is the inner direct product of all Sylow-p subgroups, so {b(n)} is multiplicative. Obviously b(p^e) = floor(e/2)+1 (corresponding to the groups C_(p^r) X C_(p^(e-r)) for 0 <= r <= floor(e/2)), hence b(n) = a(n) for all n. - Jianing Song, Nov 05 2022]
Number of ways of writing n = r*s such that r|s. - Eric M. Schmidt, Jan 08 2015
The number of divisors of the square root of the largest square dividing n. - Amiram Eldar, Jul 07 2020
The number of unordered factorizations of n into cubefree powers of primes (1, primes and squares of primes, A166684). - Amiram Eldar, Jun 12 2025

			a(16) = 3 because the squares 1, 4, and 16 divide 16.
G.f. = x + x^2 + x^3 + 2*x^4 + x^5 + x^6 + x^7 + 2*x^8 + 2*x^9 + x^10 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Antonio Amariti, Claudius Klare, Domenico Orlando and Susanne Reffert, The M-theory origin of global properties of gauge theories, Nuclear Physics B, Vol. 901 (2015), pp. 318-337, arXiv preprint, arXiv:1507.04743 [hep-th], 2015 (see (A.13)).
Simon Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2 (1999), Article 99.1.2.
Simon Colton, HR - Automatic Theory Formation in Pure Mathematics
Ian G. Connell, A number theory problem concerning finite groups and rings, Canad. Math. Bull, 7 (1964), 23-34. See delta(n).
Andrew V. Lelechenko, Average number of squares dividing mn, arXiv preprint arXiv:1407.1222 [math.NT], 2014.
Werner Georg Nowak and László Tóth, On the average number of subgroups of the group Z_m X Z_n, International Journal of Number Theory, Vol. 10, No. 2 (2014), pp. 363-374, arXiv preprint, arXiv:1307.1414 [math.NT], 2013.
N. J. A. Sloane, Transforms.
Index entries for sequences computed from exponents in factorization of n.

Cf. A000005, A000188, A004101, A005117 (positions of 1's), A008619, A008833, A013936 (partial sums), A038538, A046952, A052304, A056595, A159631, A007814, A010052, A027750, A239930, A007862, A046523, A064989, A065704, A130279, A156552, A166684, A278161, A307445 (Dirichlet inverse).
One more than A071325.
Differs from A096309 for the first time at n=32, where a(32) = 3, while A096309(32) = 2 (and also A185102(32) = 2).
Sum of the k-th powers of the square divisors of n for k=0..10: this sequence (k=0), A035316 (k=1), A351307 (k=2), A351308 (k=3), A351309 (k=4), A351310 (k=5), A351311 (k=6), A351313 (k=7), A351314 (k=8), A351315 (k=9), A351315 (k=10).
Sequences of the form n^k * Sum_{d^2|n} 1/d^k for k = 0..10: this sequence (k=0), A340774 (k=1), A351600 (k=2), A351601 (k=3), A351602 (k=4), A351603 (k=5), A351604 (k=6), A351605 (k=7), A351606 (k=8), A351607 (k=9), A351608 (k=10).
Cf. A082293 (a(n)==2), A082294 (a(n)==3).

a046951 = sum . map a010052 . a027750_row
-- Reinhard Zumkeller, Dec 16 2013

[#[d: d in Divisors(n)|IsSquare(d)]:n in [1..120]]; // Marius A. Burtea, Jan 21 2020

A046951 := proc(n)
    local a,s;
    a := 1 ;
    for p in ifactors(n)[2] do
        a := a*(1+floor(op(2,p)/2)) ;
    end do:
    a ;
end proc: # R. J. Mathar, Sep 17 2012
# Alternatively:
isbidivisible := (n, d) -> igcd(n, d) = d and igcd(n/d, d) = d:
a := n -> nops(select(k -> isbidivisible(n, k), [seq(1..n)])): # Peter Luschny, Jun 13 2025

a[n_] := Length[ Select[ Divisors[n], IntegerQ[Sqrt[#]]& ] ]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Jun 26 2012 *)
Table[Length[Intersection[Divisors[n], Range[10]^2]], {n, 100}] (* Alonso del Arte, Dec 10 2012 *)
a[ n_] := If[ n < 1, 0, Sum[ Mod[ DivisorSigma[ 0, d], 2], {d, Divisors @ n}]]; (* Michael Somos, Jun 13 2014 *)
a[ n_] := If[ n < 2, Boole[ n == 1], Times @@ (Quotient[ #[[2]], 2] + 1 & /@ FactorInteger @ n)]; (* Michael Somos, Jun 13 2014 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^k^2 / (1 - x^k^2), {k, Sqrt @ n}], {x, 0, n}]]; (* Michael Somos, Jun 13 2014 *)
f[p_, e_] := 1 + Floor[e/2]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 15 2020 *)

a(n)=my(f=factor(n));for(i=1,#f[,1],f[i,2]\=2);numdiv(factorback(f)) \\ Charles R Greathouse IV, Dec 11 2012

a(n) = direuler(p=2, n, 1/((1-X^2)*(1-X)))[n]; \\ Michel Marcus, Mar 08 2015

a(n)=factorback(apply(e->e\2+1, factor(n)[,2])) \\ Charles R Greathouse IV, Sep 17 2015

from math import prod
from sympy import factorint
def A046951(n): return prod((e>>1)+1 for e in factorint(n).values()) # Chai Wah Wu, Aug 04 2024

def is_bidivisible(n, d) -> bool: return gcd(n, d) == d and gcd(n//d, d) == d
def aList(n) -> list[int]: return [k for k in range(1, n+1) if is_bidivisible(n, k)]
print([len(aList(n)) for n in range(1, 126)])  # Peter Luschny, Jun 13 2025

(definec (A046951 n) (if (= 1 n) 1 (* (A008619 (A007814 n)) (A046951 (A064989 n)))))
(define (A008619 n) (+ 1 (/ (- n (modulo n 2)) 2)))
;; Antti Karttunen, Nov 14 2016

a(p^k) = A008619(k) = [k/2] + 1. a(A002110(n)) = 1 for all n. (This is true for any squarefree number, A005117). - Original notes clarified by Antti Karttunen, Nov 14 2016
a(n) = |{(i, j) : i*j = n AND i|j}| = |{(i, j) : i*j^2 = n}|. Also tau(A000188(n)), where tau = A000005.
Multiplicative with p^e --> floor(e/2) + 1, p prime. - Reinhard Zumkeller, May 20 2007
a(A130279(n)) = n and a(m) <> n for m < A130279(n); A008966(n)=0^(a(n) - 1). - Reinhard Zumkeller, May 20 2007
Inverse Moebius transform of characteristic function of squares (A010052). Dirichlet g.f.: zeta(s)*zeta(2s).
G.f.: Sum_{k > 0} x^(k^2)/(1 - x^(k^2)). - Vladeta Jovovic, Dec 13 2002
a(n) = Sum_{k=1..A000005(n)} A010052(A027750(n,k)). - Reinhard Zumkeller, Dec 16 2013
a(n) = Sum_{k = 1..n} ( floor(n/k^2) - floor((n-1)/k^2) ). - Peter Bala, Feb 17 2014
From Antti Karttunen, Nov 14 2016: (Start)
a(1) = 1; for n > 1, a(n) = A008619(A007814(n)) * a(A064989(n)).
a(n) = A278161(A156552(n)). (End)
G.f.: Sum_{k>0}(theta(q^k)-1)/2, where theta(q)=1+2q+2q^4+2q^9+2q^16+... - Mamuka Jibladze, Dec 04 2016
From  Antti Karttunen, Nov 12 2017: (Start)
a(n) = A000005(n) - A056595(n).
a(n) = 1 + A071325(n).
a(n) = 1 + A001222(A293515(n)). (End)
L.g.f.: -log(Product_{k>=1} (1 - x^(k^2))^(1/k^2)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Jul 30 2018
a(n) = Sum_{d|n} A000005(d) * A008836(n/d). - Torlach Rush, Jan 21 2020
a(n) = A000005(sqrt(A008833(n))). - Amiram Eldar, Jul 07 2020
a(n) = Sum_{d divides n} mu(core(d)^2), where core(n) = A007913(n). - Peter Bala, Jan 24 2024

Data section filled up to 125 terms and wrong claim deleted from Crossrefs section by Antti Karttunen, Nov 14 2016

A038538 Number of semisimple rings with n elements.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 6, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 8, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 6, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 13, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 6, 6, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 8, 1, 2, 2

Offset: 1

Paolo Dominici (pl.dm(AT)libero.it)

Enumeration uses Wedderburn-Artin theorem and fact that a finite division ring is a field.

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).

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.1 Abelian group enumeration constants, pp. 274-276.
John Knopfmacher, Abstract analytic number theory, North-Holland, 1975, pp. 63-64.
T. Y. Lam, A First Course in Noncommutative Rings, Springer-Verlag, 2001.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Catalina Calderón and María José Zárate, The Number of Semisimple Rings of Order at most x, Extracta mathematicae, Vol. 7, No. 2-3 (1992), pp. 144-147.
J. Duttlinger, Eine Bemerkung zu einer asymptotischen Formel von Herrn Knopfmacher, Journal für die reine und angewandte Mathematik, Vol. 1974, No. 266 (1974), pp. 104-106.
John Knopfmacher, Arithmetical properties of finite rings and algebras, and analytic number theory, Journal für die reine und angewandte Mathematik, Volume 1972, No. 252 (1972), pp. 16-43.
Werner Georg Nowak, On the value distribution of a class of arithmetic functions, Commentationes Mathematicae Universitatis Carolinae, Vol. 37, No. 1 (1996), pp. 117-134.
Index entries for sequences computed from exponents in factorization of n.

Cf. A002110, A004101, A005117, A025487, A027623, A052305, A123030 (partial sums), A244285.

With[{emax = 7}, f[e_] := f[e] = Coefficient[Series[Product[1/(1 - x^(j*k^2)), {k, 1, Floor[Sqrt[emax]] + 1}, {j, 1, Floor[emax/k^2] + 1}], {x, 0, emax}], x, e]; a[1] = 1; a[n_] := Times @@ f /@ FactorInteger[n][[;; , 2]]; Array[a, 2^emax]] (* Amiram Eldar, Jan 31 2024, using code by Vaclav Kotesovec at A004101 *)

v004101from1 = [1, 2, 3, 6, 8, 13, 18, 29, 40, 58, 79, 115, 154, 213, 284, 391, 514, 690, 900, 1197]; \\ From the data-section of A004101.
A004101(n) = v004101from1[n];
vecproduct(v) = { my(m=1); for(i=1,#v,m *= v[i]); m; };
A038538(n) = vecproduct(apply(e -> A004101(e), factorint(n)[, 2])); \\ Antti Karttunen, Nov 18 2017

Multiplicative with a(p^k) = A004101(k).
For all n, a(A002110(n)) = a(A005117(n)) = 1.
From Amiram Eldar, Jan 31 2024: (Start)
Dirichlet g.f.: Product_{k,m>=1} zeta(k*m^2*s).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2.499616... = A244285 (see A123030 for a more precise asymptotic formula). (End)

A280451 G.f.: Product_{k>=1, j>=1} (1+x^(j*k^2)).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 9, 13, 16, 20, 27, 34, 42, 53, 67, 82, 102, 125, 153, 188, 227, 274, 332, 401, 478, 574, 686, 815, 969, 1147, 1356, 1600, 1884, 2210, 2597, 3040, 3547, 4141, 4824, 5607, 6508, 7546, 8732, 10100, 11656, 13431, 15473, 17793, 20429, 23436

Offset: 0

Vaclav Kotesovec, Jan 03 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A004101, A006171, A107742, A280663, A280664.

nmax = 100; CoefficientList[Series[Product[(1 + x^(j*k^2)), {k, 1, Floor[Sqrt[nmax]+1]}, {j, 1, Floor[nmax/k^2] + 1}], {x, 0, nmax}], x]

my(N=66, x='x+O('x^N)); Vec(prod(k=1, sqrt(N), eta(x^(2*k^2))/eta(x^(k^2)))) \\ Seiichi Manyama, Apr 29 2021

a(n) ~ exp(Pi^2*sqrt(n/2)/3 + sqrt(3) * (sqrt(2)-1) * Zeta(1/2) * Zeta(3/2) * n^(1/4) / (2^(3/4) * sqrt(Pi)) - 9*((sqrt(2)-1) * Zeta(1/2) * Zeta(3/2))^2 / (16*Pi^3)) * sqrt(Pi) / (2^(3/2) * sqrt(3) * n^(3/4)).

A320236 G.f.: Product_{k>=1, j>=1} 1/(1 - x^(k*j))^2.

Original entry on oeis.org

1, 2, 7, 16, 41, 86, 193, 384, 787, 1504, 2899, 5338, 9852, 17586, 31330, 54490, 94350, 160370, 271171, 451776, 748460, 1225106, 1993860, 3212378, 5146851, 8175114, 12915747, 20252564, 31595134, 48964310, 75515995, 115777684, 176696336, 268231986, 405436258

Offset: 0

Vaclav Kotesovec, Oct 08 2018

nonn

Self-convolution of A006171.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A006171, A004101, A320235.

nmax = 50; CoefficientList[Series[Product[1/(1-x^(k*j))^2, {k, 1, nmax}, {j, 1, Floor[nmax/k] + 1}], {x, 0, nmax}], x]

Conjecture: log(a(n)) ~ Pi * sqrt(2*n*log(n)/3).

A136406 Triangle read by rows: T(n,k) is the number of bi-partitions of the pair (n,k) into pairs (n_i,k_i) of positive integers such that sum k_i = k and sum n_i*k_i^2 = n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 3, 1, 1, 1, 3, 4, 3, 1, 1, 1, 3, 5, 4, 3, 1, 1, 1, 5, 6, 8, 4, 3, 1, 1, 1, 4, 10, 8, 8, 4, 3, 1, 1, 1, 5, 10, 14, 11, 8, 4, 3, 1, 1, 1, 5, 12, 16, 17, 11, 8, 4, 3, 1, 1, 1, 7, 14, 23, 21, 21, 11, 8, 4, 3, 1, 1, 1, 6, 17, 25, 32, 24, 21, 11, 8, 4, 3, 1, 1

Offset: 1

Benoit Jubin, Apr 13 2008

T(n,1) = T(n,n) = 1.

T(n,n-k) does not depend on k if k <= floor(n/2).

			Triangle begins:
  1,
  1, 1;
  1, 1,  1;
  1, 3,  1,  1;
  1, 2,  3,  1,  1;
  1, 3,  4,  3,  1,  1;
  1, 3,  5,  4,  3,  1, 1;
  1, 5,  6,  8,  4,  3, 1, 1;
  1, 4, 10,  8,  8,  4, 3, 1, 1;
  1, 5, 10, 14, 11,  8, 4, 3, 1, 1;
  1, 5, 12, 16, 17, 11, 8, 4, 3, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Row sums are A004101.

Cf. A136405, A137504.

P(k, w, n)={prod(i=1, k, 1 - x^(i*w) + O(x*x^(n-k*w)))}
T(n)={Vecrev(polcoef(prod(w=1, sqrtint(n), sum(k=0, n\w^2, (x^w*y)^(k*w) / P(k,w^2,n))), n)/y)}
{ for(n=1, 10, print(T(n))) } \\ Andrew Howroyd, Oct 23 2019

Terms a(68) and beyond from Andrew Howroyd, Oct 22 2019

A280662 G.f.: Product_{k>=1, j>=1} 1/(1 - x^(j*k^4)).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 232, 298, 387, 493, 632, 799, 1013, 1270, 1597, 1988, 2478, 3066, 3795, 4666, 5739, 7018, 8582, 10442, 12699, 15379, 18614, 22443, 27039, 32470, 38957, 46601, 55694, 66383, 79047, 93901, 111432

Offset: 0

Vaclav Kotesovec, Jan 07 2017

nonn

In general, if m>=3 and g.f. = Product_{k>=1, j>=1} 1/(1-x^(j*k^m)), then a(n, m) ~ exp(Pi*sqrt(2*Zeta(m)*n/3) + Pi^(-1/m) * Gamma(1+1/m) * Zeta(1+1/m) * Zeta(1/m) * (6*n/Zeta(m))^(1/(2*m))) * 2^(m/4 - 7/8) * Pi^(m/4) * Zeta(m)^(1/8) / (3^(1/8) * n^(5/8)).

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A006171 (m=1), A004101 (m=2), A280661 (m=3).

Cf. A280664.

nmax = 100; CoefficientList[Series[1/Product[1-x^(j*k^4), {k, 1, Floor[nmax^(1/4)]+1}, {j, 1, Floor[nmax/k^4]+1}], {x, 0, nmax}], x]

a(n) ~ exp(Pi^3 * sqrt(n/15)/3 + 2^(-7/4) * 3^(3/8) * 5^(1/8) * Pi^(-3/4) * Gamma(1/4) * Zeta(5/4) * Zeta(1/4) * n^(1/8)) * Pi^(3/2) / (3^(3/8) * 5^(1/8) * n^(5/8)).

A123030 Partial sums of A038538.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 11, 13, 14, 15, 17, 18, 19, 20, 26, 27, 29, 30, 32, 33, 34, 35, 38, 40, 41, 44, 46, 47, 48, 49, 57, 58, 59, 60, 64, 65, 66, 67, 70, 71, 72, 73, 75, 77, 78, 79, 85, 87, 89, 90, 92, 93, 96, 97, 100, 101, 102, 103, 105, 106, 107, 109, 122, 123, 124, 125, 127, 128

Offset: 1

Jonathan Vos Post, Jul 07 2008

nonn

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.1 Abelian group enumeration constants, pp. 274-276.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A038538, A244285 (A_1*B_1).

With[{emax = 7}, f[e_] := f[e] = Coefficient[Series[Product[1/(1 - x^(j*k^2)), {k, 1, Floor[Sqrt[emax]] + 1}, {j, 1, Floor[emax/k^2] + 1}], {x, 0, emax}], x, e]; a[1] = 1; a[n_] := Times @@ f /@ FactorInteger[n][[;; , 2]]; Accumulate@ Array[a, 2^emax]] (* Amiram Eldar, Jan 31 2024, using code by Vaclav Kotesovec at A004101 *)

a(n) = A_1*B_1*n + A_2*B_2*n^(1/2) + A_3*B_3*n^(1/3) + O(n^(50/199+eps)), where A_k = Product_{m>=1, m!=k} zeta(m/k) and B_k = Product_{r>=1, m>=2} zeta(r*m^2/k) (Finch, 2003). - Amiram Eldar, Jan 31 2024

A280661 G.f.: Product_{k>=1, j>=1} 1/(1 - x^(j*k^3)).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 23, 31, 44, 59, 82, 108, 146, 191, 255, 329, 431, 552, 714, 907, 1159, 1461, 1853, 2318, 2911, 3622, 4515, 5582, 6912, 8499, 10464, 12801, 15667, 19079, 23236, 28168, 34142, 41222, 49755, 59836, 71926, 86190, 103218, 123262, 147091

Offset: 0

Vaclav Kotesovec, Jan 07 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A006171, A004101, A280662.

nmax = 100; CoefficientList[Series[1/Product[1-x^(j*k^3), {k, 1, Floor[nmax^(1/3)]+1}, {j, 1, Floor[nmax/k^3]+1}], {x, 0, nmax}], x]

a(n) ~ exp(Pi*sqrt(2*Zeta(3)*n/3) + Pi^(-1/3) * Gamma(4/3) * Zeta(4/3) * Zeta(1/3) * (6*n/Zeta(3))^(1/6)) * Pi^(3/4) * Zeta(3)^(1/8) / (6^(1/8) * n^(5/8)).

A327744 Expansion of Product_{i>=1, j>=1} 1 / (1 - x^(ij(j + 1)/2)).

Original entry on oeis.org

1, 1, 2, 4, 6, 9, 17, 23, 35, 54, 77, 108, 163, 221, 309, 436, 593, 800, 1109, 1470, 1968, 2642, 3482, 4566, 6052, 7848, 10204, 13276, 17092, 21924, 28245, 35949, 45762, 58231, 73609, 92789, 117140, 146799, 183826, 229995, 286483, 356040, 442566, 547489

Offset: 0

Ilya Gutkovskiy, Sep 23 2019

nonn

Euler transform of A007862.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A004101, A007862, A327745.

nmax = 43; CoefficientList[Series[Product[1/(1 - x^k)^Length[Select[Divisors[k], IntegerQ[Sqrt[8 # + 1]] &]], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d Length[Select[Divisors[d], IntegerQ[Sqrt[8 # + 1]] &]], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 43}]
nmax = 50; CoefficientList[Series[Product[1/QPochhammer[x^(k*(k + 1)/2)], {k, 1, Sqrt[2*nmax]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 24 2019 *)

G.f.: Product_{k>=1} 1 / (1 - x^k)^A007862(k).

A280453 G.f.: Product_{k>=1, j>=1} 1/(1 - x^(j^2*k^2)).

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 3, 3, 6, 8, 8, 8, 12, 16, 16, 16, 24, 30, 33, 33, 45, 53, 59, 59, 75, 93, 102, 106, 126, 154, 166, 174, 204, 242, 270, 282, 331, 379, 423, 439, 507, 583, 643, 681, 768, 886, 962, 1022, 1138, 1300, 1425, 1507, 1687, 1893, 2086, 2190, 2434

Offset: 0

Vaclav Kotesovec, Jan 03 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A006171, A004101, A280452.

nmax = 100; CoefficientList[Series[Product[1/(1-x^(j^2*k^2)), {k, 1, Sqrt[nmax]+1}, {j, 1, Floor[Sqrt[nmax/k^2]] + 1}], {x, 0, nmax}], x]

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A046951 a(n) is the number of squares dividing n.

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A038538 Number of semisimple rings with n elements.

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A280451 G.f.: Product_{k>=1, j>=1} (1+x^(j*k^2)).

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A320236 G.f.: Product_{k>=1, j>=1} 1/(1 - x^(k*j))^2.

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A136406 Triangle read by rows: T(n,k) is the number of bi-partitions of the pair (n,k) into pairs (n_i,k_i) of positive integers such that sum k_i = k and sum n_i*k_i^2 = n.

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A280662 G.f.: Product_{k>=1, j>=1} 1/(1 - x^(j*k^4)).

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A123030 Partial sums of A038538.

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A327744 Expansion of Product_{i>=1, j>=1} 1 / (1 - x^(ij(j + 1)/2)).