A057521 -id:A057521 - cp's OEIS frontend

A386424 Numbers k such that sigma(k) has the same powerful part as k, where sigma is the sum of divisors function.

1, 2, 5, 12, 13, 26, 29, 37, 41, 44, 56, 61, 73, 74, 76, 90, 101, 109, 113, 122, 137, 146, 153, 157, 172, 173, 181, 193, 218, 229, 236, 257, 268, 277, 281, 312, 313, 314, 317, 353, 362, 373, 386, 389, 397, 401, 409, 421, 433, 457, 458, 461, 509, 522, 524, 528, 541, 554, 560, 569, 601, 613, 617, 626, 641, 652, 653

Offset: 1

Antti Karttunen, Aug 17 2025

nonn

Conjecture 1: the initial 1 is the only square in this sequence, and a(2) = 2 is the only term that is twice a square.

Conjecture 2: A323653 is a subsequence (which would follow from conjecture 1 (c) given there).

Cf. A000203, A003557, A057521, A387156.
Subsequences: A323653 (conjectured), A351549, A386425 (odd composites), A386426 (nondeficient terms).
Cf. also A006872, A351446, A387158.

rad[n_] := Times @@ First /@ FactorInteger[n];a057521[n_] := n/Denominator[n/rad[n]^2];Select[Range[653],a057521[DivisorSigma[1,#]]==a057521[#]&] (* James C. McMahon, Aug 18 2025 *)

A057521(n)=my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1))
isA386424(n) = (A057521(sigma(n))==A057521(n));

{k | A057521(A000203(k)) = A057521(k)}, or equally, {k | A387156(k) = A003557(k)}.

A386425 Odd composites k such that sigma(k) has the same powerful part as k, where sigma is the sum of divisors function.

Original entry on oeis.org

153, 801, 1773, 3725, 4689, 4753, 5013, 6957, 8577, 8725, 9549, 9873, 11493, 13437, 14409, 15381, 18621, 19269, 21213, 21537, 23481, 25101, 26073, 26225, 28989, 29161, 29313, 29961, 32229, 33849, 34173, 36117, 38061, 39033, 40653, 42597, 43893, 47457, 47781, 48725, 48753, 51669, 52317, 54261, 56953, 57177, 57501

Offset: 1

Antti Karttunen, Aug 17 2025

nonn

By definition, the sequence contains all odd perfect numbers, and also includes any hypothetical odd triperfect number that is not a multiple of 3 (see A005820 and A347391), and similarly, any odd term of A046060 that is not a multiple of 5, etc. If there are no squares in this sequence (see conjecture in A386424), then the latter categories of numbers certainly do not exist, and this is then a subsequence of A228058.

The first nondeficient term is a(32315) = 81022725. See A386426.

Intersection of A071904 and A386424.
Nonsquare terms form a subsequence of A228058.
Cf. A000203, A003557, A057521, A386426 (nondeficient terms).
Cf. also A324647, A349749.

rad[n_] := Times @@ First /@ FactorInteger[n];a057521[n_] := n/Denominator[n/rad[n]^2];Select[Range[9,57501,2],!PrimeQ[#]&&a057521[DivisorSigma[1,#]]==a057521[#]&] (* James C. McMahon, Aug 18 2025 *)

A057521(n)=my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1))
isA386425(n) = ((n>1) && (n%2) && !isprime(n) && (A057521(sigma(n))==A057521(n)));

{k | k is odd composite and A003557(A000203(k)) = A003557(k)}.

A191622 Decimal expansion of the growth constant for the partial sums of maximal unitary squarefree divisors.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jun 09 2011

The partial sums grow Sum_{n=1..N} A055231(n) = (this constant)*N^2/2 +O(N^(3/2)).

			0.64960669933734119473390488048021212670381089931988288391832103926132071...

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.
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004, Section 0.4. [Cached copy, with permission of the author]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 52 (constant beta).

Cf. A055231, A057521.

$MaxExtraPrecision = 1000; m = 1000; c = LinearRecurrence[{-2, 0, 2, 0, -1}, {0, -2, 0, 2, -5}, m]; RealDigits[Exp[NSum[Indexed[c, n]*PrimeZetaP[n]/n, {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]] (* Amiram Eldar, Jun 19 2019 *)

prodeulerrat(1 - (p^2+p-1)/(p^3*(p+1))) \\ Amiram Eldar, Mar 17 2021

Equals Product_{primes p=2,3,5,7,...} ( 1 - (p^2+p-1)/(p^3*(p+1)) ).

The constant d2 in the paper by Cloutier et al. such that Sum_{k=1..x} 1/A057521(x) = d2*x + O(x^(1/2)). - Amiram Eldar, Oct 01 2019

More terms from Amiram Eldar, Jun 19 2019

More terms from Vaclav Kotesovec, Jun 13 2021

A295879 Multiplicative with a(p) = 1, a(p^e) = prime(e-1) if e > 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 5, 5, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 7, 1, 2, 2, 4, 1, 1, 1, 3, 1, 1, 1, 6, 1, 1, 1, 5, 1, 1, 1, 2, 2, 1, 1, 3, 2, 1, 1, 2, 3, 2, 1, 13

Offset: 1

Antti Karttunen, Nov 29 2017

This sequence can be used as a filter. It matches at least to the following sequences related to the counting of various non-unitary prime divisors:
For all i, j:
  a(i) = a(j) => A056170(i) = A056170(j), as A056170(n) = A001222(a(n)).
  a(i) = a(j) => A162641(i) = A162641(j).
  a(i) = a(j) => A295659(i) = A295659(j).
  a(i) = a(j) => A295662(i) = A295662(j).
  a(i) = a(j) => A295883(i) = A295883(j), as A295883(n) = A007949(a(n)).
  a(i) = a(j) => A295884(i) = A295884(j).
An encoding of the prime signature of A057521(n), the powerful part of n. - Peter Munn, Apr 06 2024

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n.

Cf. A003557, A008578, A057521, A064989, A181819.
Cf. also A293515, A294875, A294895, A294897, A295878, A351944.
Differs from A000688 for the first time at n=128, where a(128) = 13, while A000688(128) = 15.

Array[Apply[Times, FactorInteger[#] /. {p_, e_} /; p > 0 :> Which[p == 1, 1, e == 1, 1, True, Prime[e - 1]]] &, 128] (* Michael De Vlieger, Nov 29 2017 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 1, 1, prime(f[i,2]-1)));} \\ Amiram Eldar, Nov 18 2022

a(1) = 1; for n>1, if n = Product prime(i)^e(i), then a(n) = Product A008578(e(i)).
a(n) = A064989(A181819(n)).
a(n) = A181819(A003557(n)). - Antti Karttunen, Apr 03 2022
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/p^2  + Sum_{k>=1} (prime(k+1)-prime(k))/p^(k+2)) = 2.208... . - Amiram Eldar, Nov 18 2022

A323332 The Dedekind psi function values of the powerful numbers, A001615(A001694(n)).

Original entry on oeis.org

1, 6, 12, 12, 24, 30, 36, 48, 72, 56, 96, 144, 108, 180, 216, 132, 150, 192, 288, 182, 336, 360, 432, 360, 324, 384, 576, 306, 648, 392, 380, 672, 720, 864, 672, 792, 900, 768, 552, 1152, 750, 1296, 1080, 1092, 972, 1344, 1440, 870, 1728, 2160, 992, 1584

Offset: 1

Amiram Eldar, Jan 11 2019

nonn

The sum of the reciprocals of all the terms of this sequence is Pi^2/6 (A013661).

The asymptotic density of a sequence S that possesses the property that an integer k is a term if and only if its powerful part, A057521(k) is a term, is (1/zeta(2)) * Sum_{n>=1, A001694(n) is a term of S} 1/a(n). Examples for such sequences are the e-perfect numbers (A054979), the exponential abundant numbers (A129575), and other sequences listed in the Crossrefs section. - Amiram Eldar, May 06 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, A property of Dedekind's psi-function, Proceedings of the American Mathematical Society, Vol. 12, No. 6 (1961), p. 996.

Cf. A001615, A001694, A013661, A082695, A112526.

Sequences whose density can be calculated using this sequence: A054979, A129575, A307958, A308053, A321147, A322858, A323310, A328135, A339936, A340109, A364990, A382061, A383693, A383695, A383697.

psi[1]=1; psi[n_] := n * Times@@(1+1/Transpose[FactorInteger[n]][[1]]); psi /@ Join[{1}, Select[Range@ 1200, Min@ FactorInteger[#][[All, 2]] > 1 &]] (* after T. D. Noe at A001615 and Harvey P. Dale at A001694 *)

from math import isqrt, prod
from sympy import mobius, integer_nthroot, primefactors
def A323332(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
    a = primefactors(m:=bisection(f,n,n))
    return m*prod(p+1 for p in a)//prod(a) # Chai Wah Wu, Sep 14 2024

A360902 Numbers with the same number of squarefree divisors and powerful divisors.

Original entry on oeis.org

1, 4, 9, 25, 36, 48, 49, 80, 100, 112, 121, 162, 169, 176, 196, 208, 225, 272, 289, 304, 361, 368, 405, 441, 464, 484, 496, 529, 567, 592, 656, 676, 688, 720, 752, 841, 848, 891, 900, 944, 961, 976, 1008, 1053, 1072, 1089, 1136, 1156, 1168, 1200, 1225, 1250, 1264

Offset: 1

Amiram Eldar, Feb 25 2023

nonn

Numbers k such that A034444(k) = A005361(k).
Numbers whose squarefree kernel (A007947) and powerful part (A057521) have the same number of divisors (A000005).
If k and m are coprime terms, then k*m is also a term.
All the terms are exponentially 2^n-numbers (A138302).
The characteristic function of this sequence depends only on prime signature.
Numbers whose canonical prime factorization has exponents whose geometric mean is 2.
Equivalently, numbers of the form Product_{i=1..m} p_i^(2^k_i), where p_i are distinct primes, and Sum_{i=1..m} k_i = m (i.e., the exponents k_i have an arithmetic mean 1).
1 is the only squarefree (A005117) term.
Includes the squares of squarefree numbers (A062503), which are the powerful (A001694) terms of this sequence.
The squares of primes (A001248) are the only terms that are prime powers (A246655).
Numbers of the for m*p^(2^k), where m is squarefree, p is prime, gcd(m, p) = 1 and omega(m) = k - 1, are all terms. In particular, this sequence includes numbers of the form p^4*q, where p != q are primes (A178739), and numbers of the form p^8*q*r where p, q, and r are distinct primes (A179747).
The corresponding numbers of squarefree (or powerful) divisors are 1, 2, 2, 2, 4, 4, 2, 4, 4, 4, 2, 4, ... . The least term with 2^k squarefree divisors is A360903(k).

			4 is a term since it has 2 squarefree divisors (1 and 2) and 2 powerful divisors (1 and 4).
36 is a term since it has 4 squarefree divisors (1, 2, 3 and 6) and 4 powerful divisors (1, 4, 9 and 36).

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

Cf. A000005, A001694, A005117, A005361, A007947, A034444, A057521, A246655.
Subsequence of A138302.
Subsequences: A001248, A062503, A178739, A179747, A360903, A360904, A360905.

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Times @@ e == 2^Length[e]]; q[1] = True; Select[Range[1300], q]

is(k) = {my(e = factor(k)[,2]); prod(i = 1, #e, e[i]) == 2^#e; }

A368040 The powerful part of the nonsquarefree numbers.

Original entry on oeis.org

4, 8, 9, 4, 16, 9, 4, 8, 25, 27, 4, 32, 36, 8, 4, 9, 16, 49, 25, 4, 27, 8, 4, 9, 64, 4, 72, 25, 4, 16, 81, 4, 8, 9, 4, 32, 49, 9, 100, 8, 108, 16, 4, 9, 8, 121, 4, 125, 9, 128, 4, 27, 8, 4, 144, 49, 4, 25, 8, 9, 4, 32, 81, 4, 8, 169, 9, 4, 25, 16, 36, 8, 4, 27

Offset: 1

Amiram Eldar, Dec 09 2023

nonn

The terms of A057521 that are larger than 1, since A057521(k) = 1 if and only if k is squarefree (A005117).

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

Cf. A005117, A013929, A057521.
Cf. A084936, A174961, A275699, A368038, A368039.
Cf. A013661, A229099, A328013.

f[p_, e_] := If[e > 1, p^e, 1]; powPart[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[powPart, 200], # > 1 &]

lista(kmax) = {my(p, f); for(k = 1, kmax, f = factor(k); p = prod(i=1, #f~, if(f[i, 2] > 1, f[i, 1]^f[i, 2], 1)); if(p > 1, print1(p, ", ")));}

a(n) = A057521(A013929(n)).

Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = d/(3*(1-1/zeta(2))^(3/2)) = 4.778771..., and d = A328013.

A368167 The largest unitary divisor of n that is a cubefull exponentially odd number (A335988).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 27, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 27, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Dec 14 2023

First differs from A056191 and A366126 at n = 32, and from A367513 at n = 64.
Also, the largest exponentially odd unitary divisor of the powerful part on n.
Also, the powerful part of the largest exponentially odd unitary divisor of n.

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

Cf. A057521, A077610, A335275, A335988, A350389, A368168, A368169.

Cf. A056191, A366126, A367513.

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

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1 || !(f[i, 2]%2), 1, f[i, 1]^f[i, 2]));}

Multiplicative with a(p^e) = p^e if e is odd that is larger than 1, and 1 otherwise.
a(n) = A350389(A057521(n)).
a(n) = A057521(A350389(n)).
a(n) >= 1, with equality if and only if n is in A335275.
a(n) <= n, with equality if and only if n is in A335988.

A372603 The maximal exponent in the prime factorization of the powerful part of n.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 3, 2, 0, 3, 2, 0, 0, 0, 5, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 2, 2, 0, 0, 4, 2, 2, 0, 2, 0, 3, 0, 3, 0, 0, 0, 2, 0, 0, 2, 6, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 2, 2, 0, 0, 0, 4, 4, 0, 0, 2, 0, 0, 0

Offset: 1

Amiram Eldar, May 07 2024

First differs from A275812 at n = 36, and from A212172 at n = 37.

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

Cf. A051903, A057521, A087156, A368710.
Cf. A013661, A033150, A059956.
Cf. A212172, A275812.
Similar sequences: A007424, A368781, A372601, A372602, A372604.

f[n_] := If[n == 1, 0, n]; a[n_] := f[Max[FactorInteger[n][[;; , 2]]]]; a[1] = 0; Array[a, 100]

s(n) = if(n == 1, 0, n);
a(n) = if(n>1, s(vecmax(factor(n)[,2])), 0);

a(n) = A051903(A057521(n)).
a(n) = A087156(A051903(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 - 1/zeta(2) + Sum_{i>=2} (1 - 1/zeta(i)) = A033150 - A059956 = 1.09728403825134113562... .

A382906 The powerful part of the n-th biquadratefree number.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 8, 9, 1, 1, 4, 1, 1, 1, 1, 9, 1, 4, 1, 1, 1, 8, 25, 1, 27, 4, 1, 1, 1, 1, 1, 1, 36, 1, 1, 1, 8, 1, 1, 1, 4, 9, 1, 1, 49, 25, 1, 4, 1, 27, 1, 8, 1, 1, 1, 4, 1, 1, 9, 1, 1, 1, 4, 1, 1, 1, 72, 1, 1, 25, 4, 1, 1, 1, 1, 1, 4, 1, 1, 1, 8, 1, 9, 1

Offset: 1

Amiram Eldar, Apr 08 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000000 terms)

Cf. A013662, A046100, A057521.

Similar sequences: A382902, A382903, A382904, A382905.

f[p_, e_] := p^If[e > 1, e, 0]; s[n_] := Module[{fct = FactorInteger[n]}, If[AllTrue[fct[[;; , 2]], # < 4 &], Times @@ f @@@ fct, Nothing]]; Array[s, 100]

list(lim) = {my(f); print1(1, ", "); for(k = 2, lim, f = factor(k); if(vecmax(f[, 2]) < 4, print1(prod(i = 1, #f~, f[i, 1]^if(f[i, 2] > 1, f[i, 2], 0)), ", ")));}

a(n) = A057521(A046100(n)).
a(n) = A046100(n)/A382905(n).
Sum_{k=1..n} a(k) ~ c * n^(3/2) / 3, where c = zeta(4)^(3/2) * Product_{p prime} (1 + 2/p^(3/2) - 1/p^2 - 2/p^(5/2)) = 2.21177275344948791706... .

cp's OEIS Frontend

A386424 Numbers k such that sigma(k) has the same powerful part as k, where sigma is the sum of divisors function.

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A386425 Odd composites k such that sigma(k) has the same powerful part as k, where sigma is the sum of divisors function.

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A191622 Decimal expansion of the growth constant for the partial sums of maximal unitary squarefree divisors.

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A295879 Multiplicative with a(p) = 1, a(p^e) = prime(e-1) if e > 1.

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A323332 The Dedekind psi function values of the powerful numbers, A001615(A001694(n)).

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A360902 Numbers with the same number of squarefree divisors and powerful divisors.

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A368040 The powerful part of the nonsquarefree numbers.

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A368167 The largest unitary divisor of n that is a cubefull exponentially odd number (A335988).