A203640 -id:A203640 - cp's OEIS frontend

A384914 The number of unordered factorizations of n into numbers of the form p^(k^2) where p is prime and k >= 0 (A323520).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Jun 12 2025

First differs from A203640, A295658 and A365333 at n = 64, from A043289 and A053164 at n = 81, and from A063775 at n = 512.

			a(16) = 2 since 4 has 2 factorizations: 2^1 * 2^1 * 2^1 * 2^1 and 2^4, with exponents 1 and 4 that are squares.

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

Cf. A000290, A001156, A197680, A323520.
Cf. A043289, A053164, A063775, A203640, A295658, A365333.
Similar sequences: A000688, A046951, A050361, A050377, A188581, A188585, A322885, A370256, A384912, A384913, A384915, A384916.

s[n_] := s[n] = If[n == 0, 1, Sum[Sum[d * Boole[IntegerQ[Sqrt[d]]], {d, Divisors[j]}] * s[n-j], {j, 1, n}] / n];
f[p_, e_] := s[e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(n) = if(n < 1, 1, sum(j = 1, n, sumdiv(j, d, d*issquare(d)) * s(n-j))/n);
a(n) = vecprod(apply(s, factor(n)[, 2]));

Multiplicative with a(p^e) = A001156(e).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 1.08451356983124311685..., where f(x) = (1-x) / Product_{k>=1} (1-x^(k^2)).

A203639 Multiplicative with a(p^e) = e*p^(e-1).

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 12, 6, 1, 1, 4, 1, 1, 1, 32, 1, 6, 1, 4, 1, 1, 1, 12, 10, 1, 27, 4, 1, 1, 1, 80, 1, 1, 1, 24, 1, 1, 1, 12, 1, 1, 1, 4, 6, 1, 1, 32, 14, 10, 1, 4, 1, 27, 1, 12, 1, 1, 1, 4, 1, 1, 6, 192, 1, 1, 1, 4, 1, 1, 1, 72, 1, 1, 10, 4, 1, 1, 1, 32, 108, 1, 1, 4, 1, 1, 1, 12, 1, 6, 1, 4, 1, 1, 1, 80, 1, 14, 6, 40

Offset: 1

R. J. Mathar, Jan 04 2012

Antti Karttunen, Table of n, a(n) for n = 1..10000
Krassimir Atanassov, New integer functions, related to φ and σ functions, Bull. Number Theory Related Topics, Vol. 11, No. 1 (1987), pp. 3-26.
G. L. Cohen, D. E. Iannucci, Derived Sequences, J. Int. Seq. 6 (2003) #03.1.1
Vaclav Kotesovec, Graph - the asymptotic ratio (500000000 terms).
J. Sandor, B. Crstici, Handbook of Number Theory II, Kluwer, 2004, page 337.

Cf. A005361, A007947, A203640 (cycles).

A203639 := proc(n)
    local a,f,e ;
    a :=1;
    for f in ifactors(n)[2] do
        e := op(2,f) ;
        p := op(1,f) ;
        a := a*e*p^(e-1) ;
    end do;
    a;
end proc; # R. J. Mathar, Jan 11 2012

Table[n*Times @@ Transpose[FactorInteger[n]][[2]] / Last[Select[Divisors[n], SquareFreeQ]], {n, 1, 100}] (* Vaclav Kotesovec, Dec 18 2019 *)
f[p_, e_] := e*p^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 21 2020 *)

a(n)=my(f=factor(n)); n*prod(i=1,#f~, f[i,2]/f[i,1]) \\ Charles R Greathouse IV, Dec 09 2016

for(n=1, 100, print1(direuler(p=2, n, (1 + X/(1 - p*X)^2))[n], ", ")) \\ Vaclav Kotesovec, Jun 14 2020
(Scheme, with memoization-macro definec)
(definec (A203639 n) (if (= 1 n) n (* (A067029 n) (expt (A020639 n) (+ -1 (A067029 n))) (A203639 (A028234 n)))))
;; Antti Karttunen, Sep 13 2017

a(n) = n*A005361(n)/A007947(n).
a(n)=1 for all squarefree n.
Dirichlet g.f.: zeta^2(s-1)*product_{primes p} (1-2*p^(1-s)+p^(2-2s)+p^(-s)). - R. J. Mathar, Jan 19 2012
a(n) = A005361(n)*A003557(n). - Vaclav Kotesovec, Jun 20 2020

Terms a(1)-a(24) confirmed and terms a(25)-a(100) added by John W. Layman, Jan 04 2012

A365333 The number of exponentially odd coreful divisors of the largest square dividing n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Sep 01 2023

First differs from A043289, A053164, A063775, A203640 and A295658 at n = 64.
The number of squares dividing the largest exponentially odd divisor of n is A325837(n).
The sum of the exponentially odd divisors of the largest square dividing n is A365334(n). [corrected, Sep 08 2023]
The number of exponentially odd divisors of the largest square dividing n is the same as the number of squares dividing n, A046951(n). - Amiram Eldar, Sep 08 2023

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

Cf. A008833, A046100, A046951, A325837, A365334.

Cf. A043289, A053164, A063775, A203640, A295658.

f[p_, e_] := Max[1, Floor[e/2]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> max(1, x\2), factor(n)[, 2]));

a(n) = A325837(A008833(n)).
a(n) = 1 if and only if n is a biquadratefree number (A046100).
Multiplicative with a(p^e) = max(1, floor(e/2)).
Dirichlet g.f.: zeta(s) * zeta(4*s) * zeta(6*s) / zeta(12*s).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 15015/(1382*Pi^2) = 1.100823... .

Name corrected by Amiram Eldar, Sep 08 2023

cp's OEIS Frontend

A384914 The number of unordered factorizations of n into numbers of the form p^(k^2) where p is prime and k >= 0 (A323520).

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A203639 Multiplicative with a(p^e) = e*p^(e-1).

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A365333 The number of exponentially odd coreful divisors of the largest square dividing n.

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