A183097 -id:A183097 - cp's OEIS frontend

A385005 The sum of the cubefull divisors of n.

1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 25, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 28, 1, 1, 1, 1, 57, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 25, 1, 1, 1, 1, 1, 28, 1, 9, 1, 1, 1, 1, 1, 1, 1, 121, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 25, 109, 1, 1, 1

Offset: 1

Amiram Eldar, Jun 15 2025

The sum of the terms in A036966 that divide n.

The number of these divisors is A190867(n), and the largest of them is A360540(n).

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

Cf. A036966, A190867, A183097, A360540.

The sum of divisors d of n such that d is: A000593 (odd), A033634 (exponentially odd), A035316 (square), A038712 (power of 2), A048250 (squarefree), A072079 (3-smooth), A073185 (cubefree), A113061 (cube), A162296 (nonsquarefree), A183097 (powerful), A186099 (5-rough), A353900 (exponentially 2^n), this sequence (cubefull), A385006 (biquadratefree).

f[p_, e_] := (p^(e+1)-1)/(p-1) - p - If[e == 1, 0, p^2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; (p^(e+1)-1)/(p-1) - p - if(e == 1, 0, p^2));}

Multiplicative with a(p^e) = 1 if e <= 2, and a(p^e) = ((p^(e+1)-1) / (p-1)) - p - p^2 if e >= 3.

Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - p^(s-1) + 1/p^(3*s-3)).

A349109 Powerful numbers (A001694) whose sum of powerful divisors (including 1) is also powerful.

Original entry on oeis.org

1, 64, 243, 441, 1764, 9800, 15552, 28224, 41616, 60516, 82369, 88200, 189728, 226576, 329476, 336200, 648675, 741321, 968256, 1317904, 1428025, 1707552, 1943236, 2039184, 2056356, 2381400, 2446227, 2798929, 2965284, 2986568, 4372281, 5189400, 5271616, 6508832

Offset: 1

Amiram Eldar, Nov 08 2021

nonn

Numbers k such that A112526(k) = A112526(A183097(k)) = 1.

			64 = 2^6 is a term since it is powerful and the sum of its powerful divisors, A183097(64) =  1 + 4 + 8 + 16 + 32 + 64 = 125 = 5^3 is also powerful.

Amiram Eldar, Table of n, a(n) for n = 1..12154 (terms below 10^19)

Cf. A001694, A112526, A180090, A183097, A337044, A337045, A349110.

powQ[n_] := n == 1 || AllTrue[FactorInteger[n][[;;,2]], # > 1 &]; f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - p; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := powQ[n] && powQ[s[n]]; Select[Range[7*10^6], q]

isok(n) = ispowerful(n) && ispowerful(sumdiv(n, d, d*ispowerful(d))); \\ Michel Marcus, Nov 08 2021

is(k) = {my(f = factor(k)); ispowerful(f) && ispowerful(prod(i = 1, #f~, (f[i,1]^(f[i,2]+1) - 1)/(f[i,1] - 1) - f[i,1]));} \\ Amiram Eldar, Sep 14 2024

A376217 Powerful numbers whose sum of powerful divisors (including 1) is even.

Original entry on oeis.org

9, 25, 36, 49, 72, 81, 100, 121, 144, 169, 196, 200, 225, 288, 289, 324, 361, 392, 400, 441, 484, 529, 576, 625, 648, 675, 676, 729, 784, 800, 841, 900, 961, 968, 1089, 1125, 1152, 1156, 1225, 1296, 1323, 1352, 1369, 1444, 1521, 1568, 1600, 1681, 1764, 1800, 1849, 1936

Offset: 1

Amiram Eldar, Sep 16 2024

The primitive terms of A376216: all the terms of A376216 are of the form k*m, where m is a term of this sequence and k is a squarefree number that is coprime to m.
Powerful numbers that have at least one odd prime factor in their prime factorization that has an even exponent.
Equivalently, powerful numbers whose odd part (A000265) is not an exponentially odd number (A268335).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to powerful numbers.

Intersection of A001694 and A376216.

Cf. A000265, A005117, A065487, A082695, A183097, A268335.

q[n_] := Module[{f = FactorInteger[n], i = 2 - Mod[n, 2]}, AllTrue[f[[;;, 2]], # > 1 &] && AnyTrue[f[[i;;-1, 2]], EvenQ]]; Select[Range[2000], q]

is(k) = {my(f = factor(k), i = 1 + !(k % 2)); ispowerful(f) && #select(x -> !(x%2), f[i..#f~,2]) > 0;}

Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - (9/7) * Product_{p prime} (1 + 1/(p*(p^2-1))) = A082695 - (9/7) * A065487 = 0.36050781682112605291... .

A183099 a(n) = sum of powerful divisors d (excluding 1) of n.

Original entry on oeis.org

0, 0, 0, 4, 0, 0, 0, 12, 9, 0, 0, 4, 0, 0, 0, 28, 0, 9, 0, 4, 0, 0, 0, 12, 25, 0, 36, 4, 0, 0, 0, 60, 0, 0, 0, 49, 0, 0, 0, 12, 0, 0, 0, 4, 9, 0, 0, 28, 49, 25, 0, 4, 0, 36, 0, 12, 0, 0, 0, 4, 0, 0, 9, 124, 0, 0, 0, 4, 0, 0, 0, 129, 0, 0, 25, 4, 0, 0, 0, 28, 117, 0, 0, 4, 0, 0, 0, 12, 0, 9, 0, 4, 0, 0, 0, 60, 0, 49, 9, 129

Offset: 1

Jaroslav Krizek, Dec 25 2010

a(n) = sum of divisors d of n from set A001694(m) - powerful numbers for m >=2.

			For n = 12, set of such divisors is {4}; a(12) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..16385
Index entries for sequences related to sums of divisors.

Cf. A000203, A001694, A183097, A183100.

f[p_, e_] := (p^(e+1)-1)/(p-1) - p; a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - 1; Array[a, 100] (* Amiram Eldar, Aug 29 2023 *)

A183099(n) = (sumdiv(n, d, ispowerful(d)*d) - 1); \\ Antti Karttunen, Oct 07 2017

a(n) = A000203(n) - A183100(n) = A183097(n) - 1.

a(1) = 0, a(p) = 0, a(pq) = 0, a(pq...z) = 0, a(p^k) = ((p^(k+1)-1) / (p-1))-p-1, for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q, ..., z.

A360722 a(n) is the sum of infinitary divisors of n that are powerful (A001694).

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 13, 10, 1, 1, 5, 1, 1, 1, 17, 1, 10, 1, 5, 1, 1, 1, 13, 26, 1, 37, 5, 1, 1, 1, 49, 1, 1, 1, 50, 1, 1, 1, 13, 1, 1, 1, 5, 10, 1, 1, 17, 50, 26, 1, 5, 1, 37, 1, 13, 1, 1, 1, 5, 1, 1, 10, 85, 1, 1, 1, 5, 1, 1, 1, 130, 1, 1, 26, 5, 1, 1, 1, 17

Offset: 1

Amiram Eldar, Feb 18 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to divisors of numbers.

Cf. A001694, A049417, A077609, A077610, A360723.

Similar sequences: A183097, A360721.

f[p_, e_] := Times @@ (p^(2^(-1 + Flatten @ Position[Reverse@IntegerDigits[e, 2], ?(# == 1 &)])) + 1) - If[OddQ[e], p, 0]; a[1] = 1; a[n] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), b); prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], f[i, 1]^(2^(#b-k))+1, 1)) - if(f[i, 2]%2, f[i, 1], 0));}

Multiplicative with a(p^e) = f(p, e) if e is even, and f(p, e) - p is e is odd, where f(p, e) = Product{k>=1, e_k=1} (p^(2^k) + 1), where e = Sum_{k} e_k * 2^k is the binary representation of e, i.e., e_k is bit k of e.
a(n) <= A049417(n), with equality if and only if n is a square.
a(n) <= A183097(n), with equality if and only if n is not in A360723.

A376216 Numbers whose sum of powerful divisors (including 1) is even.

Original entry on oeis.org

9, 18, 25, 36, 45, 49, 50, 63, 72, 75, 81, 90, 98, 99, 100, 117, 121, 126, 144, 147, 150, 153, 162, 169, 171, 175, 180, 196, 198, 200, 207, 225, 234, 242, 245, 252, 261, 275, 279, 288, 289, 294, 300, 306, 315, 324, 325, 333, 338, 342, 350, 360, 361, 363, 369, 387, 392, 396, 400

Offset: 1

Amiram Eldar, Sep 15 2024

The sequence of numbers whose number of powerful divisors (including 1, A005361) is even is A072587, which is the sequence of numbers that are not exponentially odd (A268335).
The primitive terms of this sequence are the powerful terms (A376217). If m is a powerful term then k*m is a term of this sequence for all squarefree numbers k that are coprime to m.
Numbers that have at least one odd prime factor in their prime factorization that has an even exponent.
Numbers whose odd part (A000265) is not an exponentially odd number (A268335).
Also, numbers k such that A335341(k) is even.
The asymptotic density of this sequence is 1 - (6/5) * A065463 = 0.15466935880100128871... .

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

Subsequence of A013929.

Cf. A000265, A005117, A005361, A065463, A072587, A183097, A268335, A335341, A376217 (subsequence).

q[n_] := Module[{f = Select[FactorInteger[n], First[#] == 2 || Last[#] > 1 &], i = 2 - Mod[n, 2]}, Length[f] > 0 && AnyTrue[f[[i;;-1, 2]], EvenQ]]; Select[Range[400], q]

is(k) = {my(f = factor(k), i = 1 + !(k % 2)); #select(x -> !(x%2), f[i..#f~,2]) > 0;}

A349110 Powerful numbers (A001694) whose sum of aliquot powerful divisors (including 1) is larger than 1 and is also powerful.

Original entry on oeis.org

128, 729, 900, 4900, 10404, 17424, 24336, 52900, 78400, 79524, 81796, 297025, 304175, 304200, 313600, 346921, 417316, 532900, 1612900, 1656200, 1960000, 2238016, 2464900, 3129361, 3232804, 3334276, 3496900, 3534400, 3992004, 6056521, 6974881, 9245000, 10672200

Offset: 1

Amiram Eldar, Nov 08 2021

nonn

Numbers k such that A112526(k) = A112526(A183097(k) - k) = 1.

			128 = 2^7 is a term since it is powerful and the sum of its aliquot powerful divisors, A183097(128) - 128 =  1 + 4 + 8 + 16 + 32 + 64 = 125 = 5^3 is also powerful.

Cf. A001694, A183097, A349109.

powQ[n_] := AllTrue[FactorInteger[n][[;;,2]], # > 1 &]; f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - p; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := powQ[n] && powQ[s[n] - n]; Select[Range[1.1*10^7], q]

isok(n) = my(s); ispowerful(n) && (s=sumdiv(n, d, if (d1) && ispowerful(s); \\ Michel Marcus, Nov 08 2021

A376204 Numbers whose sum of powerful divisors (including 1) is a powerful number that is larger than 1.

Original entry on oeis.org

64, 192, 243, 320, 441, 448, 486, 704, 832, 882, 960, 1088, 1215, 1216, 1344, 1472, 1701, 1764, 1856, 1984, 2112, 2205, 2240, 2368, 2430, 2496, 2624, 2673, 2752, 3008, 3159, 3264, 3392, 3402, 3520, 3648, 3776, 3904, 4131, 4160, 4288, 4410, 4416, 4544, 4617, 4672, 4851, 4928

Offset: 1

Amiram Eldar, Sep 15 2024

Numbers k such that A112526(A183097(k)) = 1.
The primitive terms of this sequence are the powerful terms (A349109 \ {1}). If m > 1 is a powerful term then k*m is a term of this sequence for all squarefree numbers k that are coprime to m.
The asymptotic density of this sequence is Sum_{i>=2} f(A349109(i))/A349109(i) = 0.00935344863979..., where f(k) = (6/Pi^2) * Product_{p|k} (p/(p+1)).

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

Subsequence of A013929.
A349109 \ {1} is a subsequence.
Cf. A005117, A059956, A112526, A183097.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - p; s[1] = 1; s[k_] := Times @@ f @@@ FactorInteger[k]; q[k_] := AllTrue[FactorInteger[k][[;; , 2]], # > 1 &]; Select[Range[5000], q[s[#]] &]

s(k) = {my(f = factor(k)); prod(i = 1, #f~, (f[i,1]^(f[i,2]+1) - 1)/(f[i,1] - 1) - f[i,1]);}
is(k) = {my(s1 = s(k)); s1 > 1 && ispowerful(s1);}

A385542 The sum of the aliquot divisors of n that are powerful.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 13, 1, 10, 1, 5, 1, 1, 1, 13, 1, 1, 10, 5, 1, 1, 1, 29, 1, 1, 1, 14, 1, 1, 1, 13, 1, 1, 1, 5, 10, 1, 1, 29, 1, 26, 1, 5, 1, 37, 1, 13, 1, 1, 1, 5, 1, 1, 10, 61, 1, 1, 1, 5, 1, 1, 1, 58, 1, 1, 26, 5, 1, 1, 1, 29, 37

Offset: 1

Amiram Eldar, Jul 03 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)
Index entries for sequences related to powerful numbers.

Cf. A001248, A001694, A005117, A112526, A167207, A183097.

Cf. A002117, A078434.

f[p_, e_] := (p^(e+1)-1)/(p-1) - p; a[1] = 0; a[n_] := Times @@ f @@@ (fct = FactorInteger[n]) - If[AllTrue[fct[[;;, 2]], # > 1 &], n, 0]; Array[a, 100]

a(n) = {my(f = factor(n), s); s = prod(i=1, #f~, (f[i,1]^(f[i,2]+1)-1)/(f[i,1]-1) - f[i,1]); if(n==1 || vecmin(f[,2]) > 1, s -= n); s};

a(n) = Sum_{d|n, d < n} A112526(d) * d.
a(n) = A183097(n) - A112526(n) * n.
a(n) = 1 if and only if n is either a squarefree number (A005117) > 1 or a square of a prime (A001248), i.e., if and only if n is in A167207 \ {1}.
Dirichlet g.f.: (zeta(s) - 1)* zeta(2*s-2) * zeta(3*s-3) / zeta(6*s-6).	
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = zeta(3/2)*(zeta(3/2)-1)/(3*zeta(3)) = 1.168033893310319119603... .
More precise asymptotics: Sum_{k=1..n} a(k) ~ (zeta(3/2) - 1)*zeta(3/2)*n^(3/2) / (3*zeta(3)) + 3*zeta(2/3)*(zeta(4/3) - 1)*n^(4/3) / (2*Pi^2) - n/2. - Vaclav Kotesovec, Jul 03 2025

cp's OEIS Frontend

A385005 The sum of the cubefull divisors of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A349109 Powerful numbers (A001694) whose sum of powerful divisors (including 1) is also powerful.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

A376217 Powerful numbers whose sum of powerful divisors (including 1) is even.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A183099 a(n) = sum of powerful divisors d (excluding 1) of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A360722 a(n) is the sum of infinitary divisors of n that are powerful (A001694).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A376216 Numbers whose sum of powerful divisors (including 1) is even.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A349110 Powerful numbers (A001694) whose sum of aliquot powerful divisors (including 1) is larger than 1 and is also powerful.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

A376204 Numbers whose sum of powerful divisors (including 1) is a powerful number that is larger than 1.

Views

Author