A057521 -id:A057521 - cp's OEIS frontend

A295294 Sum of the divisors of the powerful part of n: a(n) = A000203(A057521(n)).

1, 1, 1, 7, 1, 1, 1, 15, 13, 1, 1, 7, 1, 1, 1, 31, 1, 13, 1, 7, 1, 1, 1, 15, 31, 1, 40, 7, 1, 1, 1, 63, 1, 1, 1, 91, 1, 1, 1, 15, 1, 1, 1, 7, 13, 1, 1, 31, 57, 31, 1, 7, 1, 40, 1, 15, 1, 1, 1, 7, 1, 1, 13, 127, 1, 1, 1, 7, 1, 1, 1, 195, 1, 1, 31, 7, 1, 1, 1, 31, 121, 1, 1, 7, 1, 1, 1, 15, 1, 13, 1, 7, 1, 1, 1, 63

Offset: 1

Antti Karttunen, Nov 25 2017

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sums of divisors.

Cf. A000203, A001694, A003557, A005117, A057521, A064549, A092261, A295295.

Array[DivisorSigma[1, #/Denominator[#/Apply[Times, FactorInteger[#][[All, 1]]]^2] ] &, 96] (* Michael De Vlieger, Nov 26 2017, after Jean-François Alcover at A057521 *)
f[p_, e_] := If[e == 1, 1, (p^(e+1)-1)/(p-1)]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 08 2022 *)

a(n) = {my(f = factor(n), p = f[,1], e = f[,2]); prod(i=1, #p, if(e[i] == 1, 1, (p[i]^(e[i]+1)-1)/(p[i]-1)))}; \\ Amiram Eldar, Oct 08 2022

from math import prod
from sympy import factorint
def A295294(n): return prod((p**(e+1)-1)//(p-1) for p, e in factorint(n).items() if e > 1) # Chai Wah Wu, Nov 14 2022

(define (A295294 n) (A000203 (A057521 n)))
;; With memoization-macro definec:
(definec (A295294 n) (if (= 1 n) n (let ((p (A020639 n)) (e (A067029 n))) (* (if (= e 1) 1 (/ (- (expt p (+ 1 e)) 1) (- p 1))) (A295294 (A028234 n))))))

Multiplicative with a(p) = 1 and a(p^e) = (p^(e+1)-1)/(p-1) for e > 1.
a(n) = A000203(A057521(n)) = A000203(A064549(A003557(n))).
a(n) = A000203(n) / A092261(n).
From Amiram Eldar, Oct 08 2022: (Start)
a(n) = 1 iff n is squarefree (A005117).
a(n) = A000203(n) iff n is powerful (A001694). (End)
Dirichlet g.f.: zeta(s) * zeta(s-1) * Product_{p prime} (1 - 1/p^(s-1) + 1/p^(2*s-2) + 1/p^(2*s-1) - 1/p^(3*s-2)). - Amiram Eldar, Sep 09 2023

A295295 Sum of squarefree divisors of the powerful part of n: a(n) = A048250(A057521(n)).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 6, 1, 4, 3, 1, 1, 1, 3, 1, 1, 1, 12, 1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 3, 8, 6, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 1, 1, 4, 3, 1, 1, 1, 3, 1, 1, 1, 12, 1, 1, 6, 3, 1, 1, 1, 3, 4, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 1, 8, 4, 18, 1, 1, 1, 3, 1

Offset: 1

Antti Karttunen, Nov 25 2017

The sum of the squarefree divisors of n whose square divides n. - Amiram Eldar, Oct 13 2023

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sums of divisors.

Cf. A000203, A003557, A048250, A057521, A064549, A092261, A295294, A344137.

Cf. A001620, A013661, A073002.

Array[DivisorSum[#/Denominator[#/Apply[Times, FactorInteger[#][[All, 1]]]^2], # &, SquareFreeQ] &, 105] (* Michael De Vlieger, Nov 26 2017, after Jean-François Alcover at A057521 *)
f[p_, e_] := If[e == 1, 1, p+1] ; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 18 2023 *)

a(n) = my(f=factor(n)); for (i=1, #f~, if (f[i,2]==1, f[i,1]=1)); sumdiv(factorback(f), d, d*issquarefree(d)); \\ Michel Marcus, Jan 29 2021

Multiplicative with a(p) = 1 and a(p^e) = (p+1) for e > 1.
a(n) = A048250(A057521(n)) = A048250(A064549(A003557(n))).
a(n) = A048250(n) / A092261(n).
a(n) = Sum_{d^2|n} d * mu(d)^2. - Wesley Ivan Hurt, Feb 13 2022
From Amiram Eldar, Sep 18 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-1) / zeta(4*s-2).
Sum_{k=1..n} a(k) ~ (3*n/Pi^2) * (log(n) + 3*gamma - 1 - 4*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620). (End)
a(n) = A048250(n) - A344137(n). - Amiram Eldar, Oct 13 2023

A328013 Decimal expansion of the growth constant for the partial sums of powerful part of n (A057521).

Original entry on oeis.org

3, 5, 1, 9, 5, 5, 5, 0, 5, 8, 4, 1, 7, 1, 0, 6, 6, 4, 7, 1, 9, 7, 5, 2, 9, 4, 0, 3, 6, 9, 8, 5, 7, 8, 1, 7, 1, 8, 6, 0, 3, 9, 8, 0, 8, 2, 2, 5, 4, 0, 7, 8, 1, 4, 7, 1, 1, 4, 6, 4, 0, 3, 1, 4, 5, 4, 1, 7, 8, 3, 9, 8, 4, 7, 9, 7, 3, 5, 4, 0, 8, 9, 7, 7, 1, 3, 5, 8, 0, 3, 7, 5, 3, 6, 4, 6, 1, 6, 2, 0, 1, 1, 4, 5, 5

Offset: 1

Amiram Eldar, Oct 01 2019

			3.51955505841710664719752940369857817186039808225407...

D. Suryanarayana and P. Subrahmanyam, The maximal k-full divisor of an integer, Indian J. Pure Appl. Math., Vol. 12, No. 2 (1981), pp. 175-190.

Maurice-Étienne Cloutier, Les parties k-puissante et k-libre d’un nombre, Thèse de doctorat, Université Laval (2018).
Maurice-Étienne Cloutier, Jean-Marie De Koninck, and Nicolas Doyon, On the powerful and squarefree parts of an integer, Journal of Integer Sequences, Vol. 17 (2014), Article 14.6.6.
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.11, pp. 31-32.

Cf. A055231, A057521, A090699, A191622.

$MaxExtraPrecision = 500; m = 500; c = LinearRecurrence[{0, 0, -2, 0, 1}, {0, 0, 6, 0, -5}, m]; RealDigits[(1 + 2/2^(3/2) - 1/2^(5/2))*(1 + 2/3^(3/2) - 1/3^(5/2))* Exp[NSum[Indexed[c, n]*(PrimeZetaP[n/2] - 1/2^(n/2) - 1/3^(n/2))/n, {n, 3, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

prodeulerrat(1 + 2/p^3 - 1/p^5, 1/2) \\ Amiram Eldar, Jun 29 2023

The constant d1 in the paper by Cloutier et al. such that Sum_{k=1..x} A057521(k) = (d1/3)*x^(3/2) + O(x^(4/3)).
Sum_{k=1..x} 1/A055231(k) = d1*x^(1/2) + O(x^(1/3)).
Equals Product_{primes p} (1 + 2/p^(3/2) - 1/p^(5/2)).
Equals (zeta(3/2)/zeta(3)) * Product_{p prime} (1 + (sqrt(p)-1)/(p*(p-sqrt(p)+1))). - Amiram Eldar, Dec 26 2024

More terms from Vaclav Kotesovec, May 29 2020

A370902 Partial sums of the powerful part function (A057521).

Original entry on oeis.org

1, 2, 3, 7, 8, 9, 10, 18, 27, 28, 29, 33, 34, 35, 36, 52, 53, 62, 63, 67, 68, 69, 70, 78, 103, 104, 131, 135, 136, 137, 138, 170, 171, 172, 173, 209, 210, 211, 212, 220, 221, 222, 223, 227, 236, 237, 238, 254, 303, 328, 329, 333, 334, 361, 362, 370, 371, 372, 373

Offset: 1

Amiram Eldar, Mar 05 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
Maurice-Étienne Cloutier, Les parties k-puissante et k-libre d’un nombre, Thèse de doctorat, Université Laval (2018); alternative link.
Maurice-Étienne Cloutier, Jean-Marie De Koninck, and Nicolas Doyon, On the powerful and squarefree parts of an integer, Journal of Integer Sequences, Vol. 17 (2014), Article 14.6.6.
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

Cf. A057521, A328013, A370903.

f[p_, e_] := If[e == 1, 1, p^e]; pfp[n_] := Times @@ f @@@ FactorInteger[n]; pfp[1] = 1; Accumulate[Array[pfp[#] &, 100]]

pfp(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1, 1, f[i, 1]^f[i, 2]));}
lista(kmax) = {my(s = 0); for(k = 1, kmax, s += pfp(k); print1(s, ", "))};

a(n) = Sum_{k=1..n} A057521(k).
a(n) = c_1 * n^(3/2) / 3 + c_2 * n^(4/3) / 4 + O(n^(6/5)), where c_1 = A328013 and c_2 are positive constants (Tóth, 2017).
c_2 = zeta(2/3) * Product_{p prime} (1 + 1/p^(4/3) - 2/p^2 - 1/p^(7/3) + 1/p^3) = -2.59305556147555965163... (László Tóth, personal communication). - Amiram Eldar, Mar 07 2024

A370903 Partial alternating sums of the powerful part function (A057521).

Original entry on oeis.org

1, 0, 1, -3, -2, -3, -2, -10, -1, -2, -1, -5, -4, -5, -4, -20, -19, -28, -27, -31, -30, -31, -30, -38, -13, -14, 13, 9, 10, 9, 10, -22, -21, -22, -21, -57, -56, -57, -56, -64, -63, -64, -63, -67, -58, -59, -58, -74, -25, -50, -49, -53, -52, -79, -78, -86, -85

Offset: 1

Amiram Eldar, Mar 05 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

Cf. A057521, A370902.
Cf. A002117, A013661, A078434.
Similar sequences: A068762, A068773, A307704, A357817, A362028.

f[p_, e_] := If[e == 1, 1, p^e]; pfp[n_] := Times @@ f @@@ FactorInteger[n]; pfp[1] = 1; Accumulate[Array[(-1)^(# + 1) * pfp[#] &, 100]]

pfp(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1, 1, f[i, 1]^f[i, 2]));}
lista(kmax) = {my(s = 0); for(k = 1, kmax, s += (-1)^(k+1) * pfp(k); print1(s, ", "))};

a(n) = c_1 * n^(3/2) + c_2 * n^(4/3) + O(n^(6/5)), where c_1 = (zeta(3/2)/(3*zeta(3))) * ((9-12*sqrt(2))/23) * Product_{p prime} (1 + (sqrt(p)-1)/(p*(p-sqrt(p)+1))) = -0.40656281796860400941..., and c_2 = (zeta(4/3)/(4*zeta(2))) * ((2^(5/3)-3*2^(1/3)-1)/(2^(5/3)-2^(1/3)+1)) * Product_{p prime} (1 + (p^(1/3)-1)/(p*(p^(2/3)-p^(1/3)+1))) = -0.52513876339565998938... (Tóth, 2017).

A375142 Numbers whose powerful part (A057521) is a power of a squarefree number that is larger than 1 (A072777).

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, 75, 76, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 104, 112, 116, 117, 120, 121, 124, 125, 126, 128, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 160, 162, 164

Offset: 1

Amiram Eldar, Aug 01 2024

Subsequence of A013929 and first differs from it at n = 27: A013929(27) = 72 = 2^3 * 3^2 is not a term of this sequence.
Numbers whose prime factorization has one distinct exponent that does not equal 1.
Numbers that are a product of a squarefree number (A005117) and a power of a different squarefree number that is not squarefree.
The asymptotic density of this sequence is Sum_{k>=2} (d(k)-1)/zeta(2) = 0.36113984820338109927..., where d(k) = zeta(k) * Product_{p prime} (1 + Sum_{i=k+1..2*k-1} (-1)^i/p^i), if k is even, and d(k) = (zeta(2*k)/zeta(k)) * Product_{p prime} (1 + 2/p^k + Sum_{i=k+1..2*k-1} (-1)^(i+1)/p^i) if k is odd > 1.

			12 = 2^2 * 3 is a term because its powerful part, 4 = 2^2, is a power of a squarefree number, 2, that is larger than 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical notes. III. Certain equally distributed sets of integers, Pacific Journal of Mathematics, Vol. 12, No. 1 (1962), pp. 77-84.
Index entries for sequences computed from exponents in factorization of n.

Cf. A005117, A057521.
Subsequence of A013929.
Subsequences: A067259, A072777, A190641, A336591.

q[n_] := Count[Union[FactorInteger[n][[;; , 2]]], _?(# > 1 &)] == 1; Select[Range[200], q]

is(k) = {my(e = select(x -> (x > 1), Set(factor(k)[,2]))); #e == 1;}

A326185 a(n) = sigma(n) - A057521(n) - n.

Original entry on oeis.org

-1, 0, 0, -1, 0, 5, 0, -1, -5, 7, 0, 12, 0, 9, 8, -1, 0, 12, 0, 18, 10, 13, 0, 28, -19, 15, -14, 24, 0, 41, 0, -1, 14, 19, 12, 19, 0, 21, 16, 42, 0, 53, 0, 36, 24, 25, 0, 60, -41, 18, 20, 42, 0, 39, 16, 56, 22, 31, 0, 104, 0, 33, 32, -1, 18, 77, 0, 54, 26, 73, 0, 51, 0, 39, 24, 60, 18, 89, 0, 90, -41, 43, 0, 136, 22, 45, 32, 84, 0

Offset: 1

Antti Karttunen, Jul 11 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000203, A001065, A057521, A326184, A326186.

Cf. also A326188.

A057521(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); }; \\ From A057521.
A326185(n) = ((sigma(n)-A057521(n))-n);

a(n) = A326184(n) - n = sigma(n) - A057521(n) - n.

a(n) = A001065(n) - A057521(n).

A328014 Numbers whose powerful part (A057521) is larger than their powerfree part (A055231).

Original entry on oeis.org

4, 8, 9, 12, 16, 18, 24, 25, 27, 32, 36, 40, 45, 48, 49, 50, 54, 56, 63, 64, 72, 75, 80, 81, 96, 98, 100, 108, 112, 121, 125, 128, 135, 144, 147, 150, 160, 162, 169, 175, 176, 180, 189, 192, 196, 200, 208, 216, 224, 225, 240, 242, 243, 245, 250, 252, 256, 270

Offset: 1

Amiram Eldar, Oct 01 2019

nonn

Differs from A122145(n) at n >= 25.

Cloutier et al. showed that the number of terms of this sequence below x is D0 * x^(3/4) + O(x^(2/3)*log(x)), where D0 is a constant given in A328015.

			12 is in the sequence since A057521(12) = 4 > A055231(12) = 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Maurice-Étienne Cloutier, Jean-Marie De Koninck, and Nicolas Doyon, On the powerful and squarefree parts of an integer, Journal of Integer Sequences, Vol. 17 (2014), Article 14.6.6.

Cf. A057521, A055231, A328015.

funp[p_, e_] := If[e > 1, p^e, 1]; pow[n_] := Times @@ (funp @@@ FactorInteger[n]); aQ[n_] := pow[n] > n/pow[n]; Select[Range[1000], aQ]

pful(f) = prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); \\ A057521
pfree(f) = for (k=1, #f~, if (f[k, 2] > 1, f[k, 2] = 0); ); factorback(f); \\ A055231
isok(n) = my(f=factor(n)); pful(f) > pfree(f); \\ Michel Marcus, Oct 02 2019

A349441 Dirichlet convolution of A057521 (powerful part of n) with A055615 (Dirichlet inverse of n).

Original entry on oeis.org

1, -1, -2, 2, -4, 2, -6, 0, 6, 4, -10, -4, -12, 6, 8, 0, -16, -6, -18, -8, 12, 10, -22, 0, 20, 12, 0, -12, -28, -8, -30, 0, 20, 16, 24, 12, -36, 18, 24, 0, -40, -12, -42, -20, -24, 22, -46, 0, 42, -20, 32, -24, -52, 0, 40, 0, 36, 28, -58, 16, -60, 30, -36, 0, 48, -20, -66, -32, 44, -24, -70, 0, -72, 36, -40, -36

Offset: 1

Antti Karttunen, Nov 18 2021

Multiplicative because A055615 and A057521 are.

Convolving this with Euler phi (A000010) produces A349379.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A055615, A057521, A349442 (Dirichlet inverse), A349443 (sum with it).

Cf. also A097945, A349379.

f[p_, e_] := Which[e > 2, 0, e == 2, p^2 - p, e == 1, 1 - p]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 19 2021 *)

A055615(n) = (n*moebius(n));
A057521(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); }; \\ From A057521
A349441(n) = sumdiv(n,d,A057521(n/d)*A055615(d));

a(n) = Sum_{d|n} A057521(n/d) * A055615(d).

Multiplicative with a(p^e) = 1 - p is e = 1, p^2 - p if e = 2, and 0 otherwise. - Amiram Eldar, Nov 19 2021

A381312 Numbers whose powerful part (A057521) is a power of a prime with an odd exponent >= 3 (A056824).

Original entry on oeis.org

8, 24, 27, 32, 40, 54, 56, 88, 96, 104, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 224, 232, 243, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 352, 375, 376, 378, 384, 408, 416, 424, 440, 456, 459, 472, 480, 486, 488, 512, 513, 520, 536, 544

Offset: 1

Amiram Eldar, Feb 19 2025

Subsequence of A301517 and A374459 and first differs from them at n = 21. A301517(21) = A374459(21) = 216 is not a term of this sequence.
Numbers having exactly one non-unitary prime factor and its multiplicity is odd.
Numbers whose prime signature (A118914) is of the form {1, 1, ..., 2*m+1} with m >= 1, i.e., any number (including zero) of 1's and then a single odd number > 1.
The asymptotic density of this sequence is (1/zeta(2)) * Sum_{p prime} 1/((p-1)*(p+1)^2) = 0.093382464285953613312...

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

Complement of A381311 within A190641.
Cf. A057521, A118914.
Subsequence of A268335, A301517 and A374459.
Subsequences: A030078, A050997, A056824, A065036, A079395, A092759, A138031, A178740, A179664, A179665, A179667, A179670, A179692, A179696, A179704, A189975, A189984, A190378, A190383, A190473.

q[n_] := Module[{e = ReverseSort[FactorInteger[n][[;; , 2]]]}, e[[1]] > 1 && OddQ[e[[1]]] && (Length[e] == 1 || e[[2]] == 1)]; Select[Range[1000], q]

isok(k) = if(k == 1, 0, my(e = vecsort(factor(k)[, 2], , 4)); e[1] % 2 && e[1] > 1 && (#e == 1 || e[2] == 1));

cp's OEIS Frontend

A295294 Sum of the divisors of the powerful part of n: a(n) = A000203(A057521(n)).

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A295295 Sum of squarefree divisors of the powerful part of n: a(n) = A048250(A057521(n)).

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A328013 Decimal expansion of the growth constant for the partial sums of powerful part of n (A057521).

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A370902 Partial sums of the powerful part function (A057521).

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A370903 Partial alternating sums of the powerful part function (A057521).

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A375142 Numbers whose powerful part (A057521) is a power of a squarefree number that is larger than 1 (A072777).

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A326185 a(n) = sigma(n) - A057521(n) - n.

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A328014 Numbers whose powerful part (A057521) is larger than their powerfree part (A055231).

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