A064549 -id:A064549 - cp's OEIS frontend

A380988 Sorted positions of first appearances in A290106 (product of prime indices divided by product of distinct prime indices).

1, 9, 25, 27, 81, 121, 125, 169, 243, 289, 625, 675, 729, 841, 961, 1125, 1331, 1681, 1849, 2025, 2187, 2197, 2209, 3125, 3267, 3481, 4489, 4913, 5329, 5625, 6075, 6241, 6561, 6889, 7803, 9801, 10125, 10201, 11881, 11979, 12769, 14641, 15125, 15625, 16129

Offset: 1

Gus Wiseman, Feb 18 2025

nonn

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.

All terms are odd.

			The prime indices of 225 are {2,2,3,3}, with image A290106(225) = 6. The prime indices of 169 are {6,6}, also with image 6. Since the latter is the first with image 6, 169 is in the sequence, and 225 is not.
The terms together with their prime indices begin:
     1: {}
     9: {2,2}
    25: {3,3}
    27: {2,2,2}
    81: {2,2,2,2}
   121: {5,5}
   125: {3,3,3}
   169: {6,6}
   243: {2,2,2,2,2}
   289: {7,7}
   625: {3,3,3,3}
   675: {2,2,2,3,3}
   729: {2,2,2,2,2,2}
   841: {10,10}
   961: {11,11}
  1125: {2,2,3,3,3}
  1331: {5,5,5}
  1681: {13,13}
  1849: {14,14}
  2025: {2,2,2,2,3,3}

For factors instead of indices we have A001694 (unsorted A064549), firsts of A003557.
Sorted firsts of A290106.
The additive version is A380957 (sorted A380956), firsts of A380955.
For difference instead of quotient see A380986.
The unsorted version is A380987.
The additive version for factors is A381075 (unsorted A280286), firsts of A280292.
A000040 lists the primes, differences A001223.
A003963 gives product of prime indices, distinct A156061.
A005117 lists squarefree numbers, complement A013929.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, length A001222.
A304038 lists distinct prime indices, sum A066328, length A001221.
Cf. A000720, A007947, A046660, A066503, A071625, A136565, A178503, A175508, A325034, A379681.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
q=Table[Times@@prix[n]/Times@@Union[prix[n]],{n,1000}];
Select[Range[Length[q]],FreeQ[Take[q,#-1],q[[#]]]&]

A304409 If n = Product (p_j^k_j) then a(n) = Product (p_j*(k_j + 1)).

Original entry on oeis.org

1, 4, 6, 6, 10, 24, 14, 8, 9, 40, 22, 36, 26, 56, 60, 10, 34, 36, 38, 60, 84, 88, 46, 48, 15, 104, 12, 84, 58, 240, 62, 12, 132, 136, 140, 54, 74, 152, 156, 80, 82, 336, 86, 132, 90, 184, 94, 60, 21, 60, 204, 156, 106, 48, 220, 112, 228, 232, 118, 360, 122, 248, 126, 14, 260

Offset: 1

Ilya Gutkovskiy, May 12 2018

			a(12) = a(2^2*3) = 2*(2 + 1) * 3*(1 + 1) = 36.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Ilya Gutkovskiy, Scatter plot of a(n) up to n=50000.
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)
Index entries for sequences computed from exponents in factorization of n.
Index entries for sequences computed from indices in prime factorization.

Cf. A000005, A000026, A000040, A001221, A005117, A007947, A016754 (numbers n such that a(n) is odd), A034444, A038040, A064549, A299822, A304407, A304408, A304410 (fixed points), A304411, A304412.

a[n_] := Times @@ (#[[1]] (#[[2]] + 1) & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 65}]
Table[DivisorSigma[0, n] Last[Select[Divisors[n], SquareFreeQ]], {n, 65}]

a(n)={numdiv(n)*factorback(factorint(n)[, 1])} \\ Andrew Howroyd, Jul 24 2018

a(n) = A000005(n)*A007947(n).
a(p^k) = p*(k + 1) where p is a prime and k > 0.
a(n) = 2^omega(n)*n if n is a squarefree (A005117), where omega() = A001221.
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 + 2/p^(s-1) - 2/p^s - 1/p^(2*s-1) + 1/p^(2*s)). - Amiram Eldar, Sep 17 2023
From Vaclav Kotesovec, Jun 06 2025: (Start)
Let f(s) = Product_{p prime} (1 - 1/p^(2*s-1) + 2/p^(s-1) + 1/p^(2*s) - 2/p^s) * ((p^s - p)/(p^s - 1))^2.
Dirichlet g.f.: zeta(s-1)^2 * f(s).
Sum_{k=1..n} a(k) ~ ((2*log(n) + 4*gamma - 1)*f(2) + 2*f'(2)) * n^2/4, where
f(2) = Product_{p prime} (1 - (3*p^2 + p - 1)/(p^2 * (p+1)^2)) = 0.40693068229776748114138817391056656864938379...,
f'(2) = f(2) * Sum_{p prime} 2*(3*p^4-3*p^2+1) * log(p) / ((p-1)*(p+1)*(p^4+2*p^3-2*p^2-p+1)) = f(2) * 2.2612432627709318567813765271568350301741329636853...
and gamma is the Euler-Mascheroni constant A001620. (End)

A304412 If n = Product (p_j^k_j) then a(n) = Product ((p_j + 1)*(k_j + 1)).

Original entry on oeis.org

1, 6, 8, 9, 12, 48, 16, 12, 12, 72, 24, 72, 28, 96, 96, 15, 36, 72, 40, 108, 128, 144, 48, 96, 18, 168, 16, 144, 60, 576, 64, 18, 192, 216, 192, 108, 76, 240, 224, 144, 84, 768, 88, 216, 144, 288, 96, 120, 24, 108, 288, 252, 108, 96, 288, 192, 320, 360, 120, 864, 124, 384, 192, 21, 336, 1152, 136, 324

Offset: 1

Ilya Gutkovskiy, May 12 2018

			a(36) = a(2^2*3^2) = (2 + 1)*(2 + 1) * (3 + 1)*(2 + 1) = 108.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Ilya Gutkovskiy, Scatter plot of a(n) up to n=50000.
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)
Index entries for sequences computed from exponents in factorization of n.
Index entries for sequences computed from indices in prime factorization.

Cf. A000005, A000026, A000040, A000203, A000302 (numbers n such that a(n) is odd), A001221, A001615, A003959, A005117, A007947, A008864, A045967, A048250, A064549, A064840, A304407, A304408, A304409, A304411.

Cf. A065463.

a[n_] := Times @@ ((#[[1]] + 1) (#[[2]] + 1) & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 68}]
Table[DivisorSigma[0, n] Total[Select[Divisors[n], SquareFreeQ]], {n, 68}]

a(n)={numdiv(n)*sumdiv(n, d, moebius(d)^2*d)} \\ Andrew Howroyd, Jul 24 2018

a(n) = A000005(n)*A048250(n) = A000005(n)*A000203(A007947(n)).
a(p^k) = (p + 1)*(k + 1) where p is a prime and k > 0.
a(n) = 2^omega(n)*Product_{p|n} (p + 1) if n is a squarefree (A005117), where omega() = A001221.
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 + 2/p^(s-1) - 1/p^(2*s-1)). - Amiram Eldar, Sep 17 2023
From Vaclav Kotesovec, May 06 2025: (Start)
Let f(s) = Product_{p prime} (p^s-p)^2 * (p^(2*s)+2*p^(s+1)-p) / (p^(2*s) * (p^s-1)^2).
Dirichlet g.f.: zeta(s-1)^2 * f(s).
Sum_{k=1..n} a(k) ~ ((2*log(n) + 4*gamma - 1)*f(2) + 2*f'(2)) * n^2/4, where
f(2) = A065463 = Product_{p prime} (1 - 1/(p*(p+1))) = 0.704442200999165592736603350326637210188586431417098049414226842591...,
f'(2) = f(2) * Sum_{p prime} 2*(2*p^2-1)*log(p) / ((p^2-1)*(p^2+p-1)) = f(2) * 1.799151495460164053607059266860868724519705035904425832307664926571...
and gamma is the Euler-Mascheroni constant A001620. (End)

A357669 a(n) is the number of divisors of the powerful part of n.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 4, 3, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 4, 3, 1, 4, 3, 1, 1, 1, 6, 1, 1, 1, 9, 1, 1, 1, 4, 1, 1, 1, 3, 3, 1, 1, 5, 3, 3, 1, 3, 1, 4, 1, 4, 1, 1, 1, 3, 1, 1, 3, 7, 1, 1, 1, 3, 1, 1, 1, 12, 1, 1, 3, 3, 1, 1, 1, 5, 5, 1, 1, 3, 1, 1, 1

Offset: 1

Amiram Eldar, Oct 08 2022

The corresponding sum of divisors of the powerful part of n is A295294.

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

Cf. A000005, A001694, A003557, A005117, A056671, A057521, A064549, A295294.

f[p_, e_] := If[e == 1, 1, e + 1]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(e = factor(n)[,2]); prod(i=1, #e, if(e[i] == 1, 1, e[i] + 1))};

a(n) = A000005(A057521(n)).
a(n) = A000005(n)/A056671(n).
a(n) = A000005(A064549(A003557(n))).
a(n) = 1 iff n is squarefree (A005117).
a(n) = A000005(n) iff n is powerful (A001694).
Multiplicative with a(p^e) = 1 if e = 1 and e+1 otherwise.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} ((p^3 - p^2 + 2*p - 1)/(p^2*(p - 1))) = 2.71098009471568319328... .
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^s + 2/p^(2*s) - 1/p^(3*s)). - Amiram Eldar, Sep 09 2023

A087687 Number of solutions to x^2 + y^2 + z^2 == 0 (mod n).

Original entry on oeis.org

1, 4, 9, 8, 25, 36, 49, 32, 99, 100, 121, 72, 169, 196, 225, 64, 289, 396, 361, 200, 441, 484, 529, 288, 725, 676, 891, 392, 841, 900, 961, 256, 1089, 1156, 1225, 792, 1369, 1444, 1521, 800, 1681, 1764, 1849, 968, 2475, 2116, 2209, 576, 2695, 2900, 2601

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 27 2003

To show that a(n) is multiplicative is simple number theory. If gcd(n,m)=1, then any solution of x^2 + y^2 + z^2 == 0 (mod n) and any solution (mod m) can combined to a solution (mod nm) using the Chinese Remainder Theorem and any solution (mod nm) gives solutions (mod n) and (mod m). Hence a(nm) = a(n)*a(m). - Torleiv Kløve, Jan 26 2009

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 80 terms from Robert Gerbicz)
C. Calderón and M. J. De Velasco, On divisors of a quadratic form, Boletim da Sociedade Brasileira de Matemática, Vol. 31, No. 1 (2000), pp. 81-91; alternative link.
László Toth, Counting Solutions of Quadratic Congruences in Several Variables Revisited, J. Int. Seq., Vol. 17 (2014), Article # 14.11.6; arXiv preprint, arXiv:1404.4214 [math.NT], 2014.
Index to sequences related to sums of squares

Cf. A086933, A062775.

Different from A064549.

A087687 := proc(n)
    a := 1;
    for pe in ifactors(n)[2] do
        p := op(1,pe) ;
        e := op(2,pe) ;
        if p = 2 then
            a := a*p^(e+ceil(e/2)) ;
        elif type(e,'odd') then
            a := a*p^((3*e-1)/2)*(p^((e+1)/2)+p^((e-1)/2)-1) ;
        else
            a := a*p^(3*e/2-1)*(p^(e/2+1)+p^(e/2)-1) ;
        end if;
    end do:
    a ;
end proc:
seq(A087687(n),n=1..100) ; # R. J. Mathar, Jun 25 2018

a[n_] := Module[{k=1}, Do[{p, e} = pe; k = k*If[p == 2, p^(e + Ceiling[ e/2]), If[OddQ[e], p^((3*e-1)/2)*(p^((e+1)/2) + p^((e-1)/2) - 1), p^(3*e/2 - 1)*(p^(e/2 + 1) + p^(e/2) - 1)]], {pe, FactorInteger[n]}]; k];
Array[a, 100] (* Jean-François Alcover, Jul 10 2018, after R. J. Mathar *)

a(n)=local(v=vector(n),w);for(i=1,n,v[i^2%n+1]++);w=vector(n,i,sum(j=1,n,v[j]*v[(i-j)%n+1]));sum(j=1,n,w[j]*v[(1-j)%n+1]) \\ Martin Fuller

a(n)={my(f=factor(n)); prod(i=1, #f~, my([p,e]=f[i,]); if(p==2, 2^(e+(e+1)\2), p^(e+(e-1)\2)*(p^(e\2)*(p+1) - 1)))} \\ Andrew Howroyd, Aug 06 2018

a(2^k) = 2^(k + ceiling(k/2)). For odd primes p, a(p^(2k-1)) = p^(3k-2)*(p^k + p^(k-1) - 1) and a(p^(2k)) = p^(3k-1)*(p^(k+1) + p^k - 1). - Martin Fuller, Jan 26 2009

Sum_{k=1..n} a(k) ~ (4*zeta(3))/(15*zeta(4)) * n^3 + O(n^2 * log(n)) (Calderón and de Velasco, 2000). - Amiram Eldar, Mar 04 2021

More terms from Robert Gerbicz, Aug 22 2006

Edited by Steven Finch, Feb 06 2009, Feb 12 2009

A318474 Multiplicative with a(p^e) = 2^A000045(e+1).

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 2, 8, 4, 4, 2, 8, 2, 4, 4, 32, 2, 8, 2, 8, 4, 4, 2, 16, 4, 4, 8, 8, 2, 8, 2, 256, 4, 4, 4, 16, 2, 4, 4, 16, 2, 8, 2, 8, 8, 4, 2, 64, 4, 8, 4, 8, 2, 16, 4, 16, 4, 4, 2, 16, 2, 4, 8, 8192, 4, 8, 2, 8, 4, 8, 2, 32, 2, 4, 8, 8, 4, 8, 2, 64, 32, 4, 2, 16, 4, 4, 4, 16, 2, 16, 4, 8, 4, 4, 4, 512, 2, 8, 8, 16, 2, 8, 2, 16, 8

Offset: 1

Antti Karttunen, Aug 29 2018

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n

Cf. A000045, A318465, A318472, A318473.

Array[Apply[Times, 2^Fibonacci[# + 1] & /@ FactorInteger[#][[All, -1]]] - Boole[# == 1] &, 105] (* Michael De Vlieger, Sep 02 2018 *)

A318474(n) = factorback(apply(e -> 2^fibonacci(1+e),factor(n)[,2]));

a(n) = 2^A318473(n).
a(n) = A318472(A064549(n)).
a(A064549(n)) = a(n)*A318472(n).

A318655 The 2-adic valuation of A318649, the numerators of "Dirichlet Square Root" of squares.

Original entry on oeis.org

0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0

Offset: 1

Antti Karttunen, Sep 01 2018

nonn

Probably also the 2-adic valuation of A318511.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A318511, A318649, A318651, A318652, A318654 (the positions of nonzero terms).

A318655(n) = valuation(A318649(n),2); \\ Needs also code from A318649.

a(n) = A007814(A318649(n)).

It seems that for all n >= 1, a(n) <= A007814(A064549(n)) <= A007814(A000290(n)).

A306458 a(n) = A001694(n)/A007947(A001694(n)), the powerful numbers divided by their squarefree kernel.

Original entry on oeis.org

1, 2, 4, 3, 8, 5, 9, 16, 6, 7, 32, 12, 27, 10, 18, 11, 25, 64, 24, 13, 14, 20, 36, 15, 81, 128, 48, 17, 54, 49, 19, 28, 40, 72, 21, 22, 50, 256, 23, 96, 125, 108, 45, 26, 243, 56, 80, 29, 144, 30, 31, 44, 162, 100, 512, 33, 75, 192, 34, 35, 216, 63, 121, 52

Offset: 1

Amiram Eldar, Feb 17 2019

A permutation of the positive integers.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001694, A007947, A064549.

N:= 10^4: # to get terms corresponding to powerful numbers <= N
rad:= n -> convert(numtheory:-factorset(n), `*`):
S:= {1}:
p:= 1:
do
  p:= nextprime(p);
  if p^2 > N then break fi;
  S:= S union map(t -> seq(t*p^i,i=2..floor(log[p](N/t))),select(`<=`,S,N/p^2));
od:
map(t -> t/rad(t), sort(convert(S,list))); # Robert Israel, Mar 20 2019

p=Join[{1}, Select[ Range@ 12500, Min@ FactorInteger[#][[All, 2]] > 1 &]]; rad[n_] := Times @@ (First@# & /@ FactorInteger@ n);  p/(rad/@p) (* after Harvey P. Dale at A001694 and Robert G. Wilson v at A007947 *)

apply(x->(x/factorback(factorint(x)[, 1])), select(x->ispowerful(x), vector(1600, k, k))) \\ Michel Marcus, Feb 17 2019

from math import isqrt, prod
from sympy import mobius, integer_nthroot, primefactors
def A306458(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, l = n+x-squarefreepi(integer_nthroot(x,3)[0]), 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        return c+l
    return (m:=bisection(f,n,n))//prod(primefactors(m)) # Chai Wah Wu, Sep 14 2024

A064549(a(n)) = A001694(n).

A318680 a(n) = n * A318653(n).

Original entry on oeis.org

1, 2, 9, 4, 25, 18, 49, 8, 27, 50, 121, 36, 169, 98, 225, 48, 289, 54, 361, 100, 441, 242, 529, 72, -125, 338, 405, 196, 841, 450, 961, 96, 1089, 578, 1225, 108, 1369, 722, 1521, 200, 1681, 882, 1849, 484, 675, 1058, 2209, 432, -1029, -250, 2601, 676, 2809, 810, 3025, 392, 3249, 1682, 3481, 900, 3721, 1922, 1323, 320

Offset: 1

Antti Karttunen, Sep 02 2018

Dirichlet convolution of a(n)/A299150(n) with itself gives A064549 [= n * Product_{primes p|n} p], like gives also the self-convolution of A318511(n)/A318512(n), as it is the same ratio reduced to its lowest terms. However, in contrast to A318511, this sequence is multiplicative as both A000027 and A318653 are multiplicative sequences (also, because A064549 and A299150 are both multiplicative).

A007814 gives the 2-adic valuation of this sequence, because there are no even terms in A318653.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A007947, A064549, A299150, A318511, A318512, A318653, A318681.

rad[n_] := Times @@ (First@# & /@ FactorInteger[n]); f[1] = 1; f[n_] := f[n] = (rad[n] - DivisorSum[n, f[#]*f[n/#] &, 1 < # < n &])/2; a[n_] := n * Numerator [f[n]]; Array[a, 100] (* Amiram Eldar, Dec 07 2020 *)

up_to = 65537;
A007947(n) = factorback(factorint(n)[, 1]);
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA007947(n)));
A318653(n) = numerator(v318653_aux[n]);
A318680(n) = (n*A318653(n));

a(n) = n * A318653(n).

a(n)/A299150(n) = A318511(n)/A318512(n).

A355038 a(n) = n^2 times the squarefree kernel of n.

Original entry on oeis.org

1, 8, 27, 32, 125, 216, 343, 128, 243, 1000, 1331, 864, 2197, 2744, 3375, 512, 4913, 1944, 6859, 4000, 9261, 10648, 12167, 3456, 3125, 17576, 2187, 10976, 24389, 27000, 29791, 2048, 35937, 39304, 42875, 7776, 50653, 54872, 59319, 16000, 68921, 74088, 79507, 42592, 30375

Offset: 1

Peter Munn, Jun 16 2022

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

The range of values is A335988.

Cf. A007947, A064549, A065463, A102631, A356191.

a[n_] := n^2 * Times @@ FactorInteger[n][[;; , 1]]; Array[a, 50] (* Amiram Eldar, Jun 18 2022 *)

PARI
```
a(n) = n^2 * factorback(factor(n)[,1]);
```

Multiplicative with a(p^e) = p^(2e+1).
a(n) = n^2 * A007947(n).
a(n) = A064549(n^2). - Amiram Eldar, Jun 20 2022
Sum_{k=1..n} a(k) ~ c * n^4, where c = (1/4) * Product_{p prime} (1 - 1/(p*(p+1))) = A065463 / 4 = 0.1761105502... . - Amiram Eldar, Nov 13 2022
a(n) = A356191(n^2). - Amiram Eldar, Nov 30 2023

cp's OEIS Frontend

A380988 Sorted positions of first appearances in A290106 (product of prime indices divided by product of distinct prime indices).

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A304409 If n = Product (p_j^k_j) then a(n) = Product (p_j*(k_j + 1)).

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A304412 If n = Product (p_j^k_j) then a(n) = Product ((p_j + 1)*(k_j + 1)).

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A357669 a(n) is the number of divisors of the powerful part of n.

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A087687 Number of solutions to x^2 + y^2 + z^2 == 0 (mod n).

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A318474 Multiplicative with a(p^e) = 2^A000045(e+1).

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A318655 The 2-adic valuation of A318649, the numerators of "Dirichlet Square Root" of squares.

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PARI

Formula

A306458 a(n) = A001694(n)/A007947(A001694(n)), the powerful numbers divided by their squarefree kernel.

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