A065487 -id:A065487 - cp's OEIS frontend

A377818 Powerful numbers that have a single even exponent in their prime factorization.

4, 9, 16, 25, 49, 64, 72, 81, 108, 121, 169, 200, 256, 288, 289, 361, 392, 432, 500, 529, 625, 648, 675, 729, 800, 841, 961, 968, 972, 1024, 1125, 1152, 1323, 1352, 1369, 1372, 1568, 1681, 1728, 1849, 2000, 2209, 2312, 2401, 2592, 2809, 2888, 3087, 3200, 3267, 3481

Offset: 1

Amiram Eldar, Nov 09 2024

Each term can be represented in a unique way as m * p^(2*k), k >= 1, where m is a cubefull exponentially odd number (A335988) and p is a prime that does not divide m.

Powerful numbers k such that A350388(k) is a prime power with an even positive exponent (A056798 \ {1}).

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

Intersection of A001694 and A377816.
Subsequence of A377819.
Cf. A056798, A065487, A335988, A350388.

With[{max = 3500}, Select[Union@ Flatten@ Table[i^2 * j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}], Count[FactorInteger[#][[;; , 2]], _?EvenQ] == 1 &]]

is(k) = if(k == 1, 0, my(e = factor(k)[, 2]); vecmin(e) > 1 && #select(x -> !(x%2), e) == 1);

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p*(p^2-1))) * Sum_{p prime} (p/(p^3-p+1)) = 0.61399274770712398109... .

A366764 The sum of divisors of n that have no exponent 2 in their prime factorization.

Original entry on oeis.org

1, 3, 4, 3, 6, 12, 8, 11, 4, 18, 12, 12, 14, 24, 24, 27, 18, 12, 20, 18, 32, 36, 24, 44, 6, 42, 31, 24, 30, 72, 32, 59, 48, 54, 48, 12, 38, 60, 56, 66, 42, 96, 44, 36, 24, 72, 48, 108, 8, 18, 72, 42, 54, 93, 72, 88, 80, 90, 60, 72, 62, 96, 32, 123, 84, 144, 68

Offset: 1

Amiram Eldar, Oct 21 2023

The sum of terms of A337050 that divide n.

The number of these divisors is A366763(n), and the largest of them is A366765(n).

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

Cf. A000203, A004709, A005117, A034448, A065487, A337050, A366763, A366765.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - p^2; f[p_, 1] := p + 1; 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, 1] + 1, (f[i, 1]^(f[i,2] + 1) - 1)/(f[i, 1] - 1) - f[i, 1]^2));}

Multiplicative with a(p) = p + 1, and a(p^e) = (p^(e+1) - 1)/(p - 1) - p^2 for e >= 2.
a(n) <= A000203(n), with equality if and only if n is squarefree (A005117).
a(n) >= A034448(n), with equality if and only if n is cubefree (A004709).
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - 1/p^(2*s-2) + 1/p^(3*s-3)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + 1/(p^3-p)) = 1.231291... (A065487).

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... .

A381825 Odd cubefull exponentially odd numbers: numbers whose prime factorization has only odd primes and odd exponents that are larger than 1 (except for 1 whose prime factorization is empty).

Original entry on oeis.org

1, 27, 125, 243, 343, 1331, 2187, 2197, 3125, 3375, 4913, 6859, 9261, 12167, 16807, 19683, 24389, 29791, 30375, 35937, 42875, 50653, 59319, 68921, 78125, 79507, 83349, 84375, 103823, 132651, 148877, 161051, 166375, 177147, 185193, 205379, 226981, 273375, 274625

Offset: 1

Amiram Eldar, Mar 08 2025

Differs from its subsequence A369118 by having the terms 1, 19683 = 3^9, 1953125 = 5^9, 2460375 = 3^9 * 5^3, 6751269 = 3^9 * 7^3, 14348907 = 3^15, ... .

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

Intersection of A005408 and A335988.
Intersection A036966 and A376218.
Subsequence of A381824.
A369118 is a subsequence.
Cf. A065487.

Join[{1}, Select[Range[3, 300000, 2], AllTrue[FactorInteger[#][[;;, 2]],  #1 > 1 && OddQ[#1] &] &]]

isok(k) = k == 1 || (k % 2 && #select(x -> (x == 1) || !(x % 2), factor(k)[, 2]) == 0);

Sum_{n>=1} 1/a(n) = Product_{prime p >= 3} (1 + 1/(p*(p^2-1))) = (6/7) * A065487 = 1.05539241333308876809... .

A377819 Powerful numbers that have no more than one even exponent in their prime factorization.

Original entry on oeis.org

1, 4, 8, 9, 16, 25, 27, 32, 49, 64, 72, 81, 108, 121, 125, 128, 169, 200, 216, 243, 256, 288, 289, 343, 361, 392, 432, 500, 512, 529, 625, 648, 675, 729, 800, 841, 864, 961, 968, 972, 1000, 1024, 1125, 1152, 1323, 1331, 1352, 1369, 1372, 1568, 1681, 1728, 1849, 1944, 2000

Offset: 1

Amiram Eldar, Nov 09 2024

Powerful numbers k such that A350388(k) is either 1 or a prime power with an even positive exponent (A056798 \ {1}).

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

Disjoint union of A335988 and A377818.
Intersection of A001694 and the complement of A377817.
Cf. A056798, A065487, A350388.

With[{max = 2000}, Select[Union@ Flatten@Table[i^2 * j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}], Count[FactorInteger[#][[;; , 2]], _?EvenQ] <= 1 &]]

is(k) = if(k == 1, 1, my(e = factor(k)[, 2]); vecmin(e) > 1 && #select(x -> !(x%2), e) <= 1);

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p*(p^2-1))) * (1 + Sum_{p prime} (p/(p^3-p+1))) = 1.84528389659572754387... .

A072802 Continued fraction expansion of Product_{p prime} (1 + 1/(p*(p^2-1))).

Original entry on oeis.org

1, 4, 3, 11, 33, 1, 2, 1, 4, 103, 1, 1, 1, 2, 2, 12, 1, 1, 3, 2, 3, 1, 1, 15, 1, 5, 11, 1, 1, 1, 5, 1, 1, 14, 1, 1, 1, 2, 20, 2, 1, 2, 2, 29, 1, 8, 1, 1, 4, 2, 2, 6, 4, 1, 1, 1, 2, 1, 12, 1, 1, 3, 2, 2, 2, 4, 2, 7, 1, 11, 2, 4, 2, 1, 1, 1, 2, 2, 1, 3, 1, 2, 7, 2, 3, 1, 11, 3, 15, 1, 1, 3, 2, 1, 3, 1, 1

Offset: 0

Rick L. Shepherd, Jul 11 2002

G. Niklasch, Some number-theoretical constants: 1000-digit values [Cached copy]

Cf. A065487 (decimal expansion).

\p 1002 x=1.2312911488886035627747876512720337... (cut-and-paste all 1002 digits from link) contfrac(x)

contfrac(prodeulerrat(1 + 1/(p*(p^2-1)))) \\ Amiram Eldar, Jun 13 2021

A365337 The sum of divisors of the largest exponentially odd number dividing n.

Original entry on oeis.org

1, 3, 4, 3, 6, 12, 8, 15, 4, 18, 12, 12, 14, 24, 24, 15, 18, 12, 20, 18, 32, 36, 24, 60, 6, 42, 40, 24, 30, 72, 32, 63, 48, 54, 48, 12, 38, 60, 56, 90, 42, 96, 44, 36, 24, 72, 48, 60, 8, 18, 72, 42, 54, 120, 72, 120, 80, 90, 60, 72, 62, 96, 32, 63, 84, 144, 68

Offset: 1

Amiram Eldar, Sep 01 2023

The number of divisors of the largest exponentially odd number dividing n is A286324(n).

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

Cf. A000203, A065487, A286324, A350390.

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

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

a(n) = A000203(A350390(n)).
Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if e is odd and (p^e-1)/(p-1) if e is even.
Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-1) - 1/p^(2*s-2) + 1/p^(3*s-2)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + 1/(p*(p^2-1))) = 1.2312911488886... (A065487). - Amiram Eldar, Sep 01 2023

A373058 The sum of the aliquot coreful divisors of the nonsquarefree numbers.

Original entry on oeis.org

2, 6, 3, 6, 14, 6, 10, 18, 5, 12, 14, 30, 36, 30, 22, 15, 42, 7, 10, 26, 24, 42, 30, 21, 62, 34, 96, 15, 38, 70, 39, 42, 66, 30, 46, 90, 14, 33, 80, 78, 126, 98, 58, 39, 90, 11, 62, 30, 42, 126, 66, 60, 102, 70, 216, 21, 74, 30, 114, 51, 78, 150, 78, 82, 126, 13

Offset: 1

Amiram Eldar, May 21 2024

A coreful divisor d of n is a divisor that is divisible by every prime that divides n (see also A307958).
The positive terms of A336563: if k is a squarefree number (A005117) then the only coreful divisor of k is k itself, so k has no aliquot coreful divisors.
The number of the aliquot coreful divisors of the n-th nonsquarefree number is A368039(n).

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

Cf. A005117, A013661, A013929, A065487, A229099, A307958, A336563.

Cf. A084936, A174961, A275699, A368038, A368039, A368040, A368541, A368713.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; Select[Array[s, 300], # > 0 &]

lista(kmax) = {my(f); for(k = 1, kmax, f = factor(k); if(!issquarefree(f), print1(prod(i = 1, #f~, (f[i, 1]^(f[i, 2]+1) - 1)/(f[i, 1] - 1) - 1) - k, ", "))); }

from math import prod, isqrt
from sympy import mobius, factorint
def A373058(n):
    def f(x): return n+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return prod((p**(e+1)-1)//(p-1)-1 for p, e in factorint(m).items())-m # Chai Wah Wu, Jul 22 2024

a(n) = A336563(A013929(n)).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = (A065487 - 1)/(1-1/zeta(2))^2 = 1.50461493205911656114... .

cp's OEIS Frontend

A377818 Powerful numbers that have a single even exponent in their prime factorization.

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A366764 The sum of divisors of n that have no exponent 2 in their prime factorization.

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

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A381825 Odd cubefull exponentially odd numbers: numbers whose prime factorization has only odd primes and odd exponents that are larger than 1 (except for 1 whose prime factorization is empty).

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A377819 Powerful numbers that have no more than one even exponent in their prime factorization.

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A072802 Continued fraction expansion of Product_{p prime} (1 + 1/(p*(p^2-1))).

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A365337 The sum of divisors of the largest exponentially odd number dividing n.

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A373058 The sum of the aliquot coreful divisors of the nonsquarefree numbers.

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