A166590 -id:A166590 - cp's OEIS frontend

A247213 Numbers n = Product_(p_i^e_i) such that nn = Product_((p_i + 2)^e_i) is divisible by n.

1, 2, 4, 8, 16, 32, 64, 105, 128, 210, 256, 315, 420, 512, 630, 840, 1024, 1260, 1575, 1680, 2048, 2520, 3150, 3360, 4096, 5040, 6300, 6720, 8192, 10080, 11025, 12600, 13440, 16384, 20160, 22050, 25200, 26880, 32768, 33075, 40320, 44100, 50400, 53760, 65536

Offset: 1

Michel Marcus, after a suggestion from Thomas Ordowski, Nov 26 2014

nonn

That is, numbers n, such that A166590(n) is divisible by n.
A000079, powers of 2, is a subsequence.
Thomas Ordowski remarks that the only squarefrees of this sequence are: 1, 2, 105, and 210.

			A166590(2)=4 is divisible by 2, so 2 is in the sequence.
A166590(105) = A166590(3*5*7) = 5*7*9 = 3*(3*5*7), so 105 is in the sequence.

Chai Wah Wu, Table of n, a(n) for n = 1..164

Cf. A000079, A166590.

a247213[n_] := Select[Range@n, Mod[Times @@ Power @@@ Transpose[{Plus[First /@ FactorInteger@#, 2], Last /@ FactorInteger@#}], #] == 0 &]; a247213[2^16] (* Michael De Vlieger, Jan 07 2015 *)

isok(n) = { f = factor(n); for (i=1, #f~, f[i,1] += 2); newn = factorback(f);  newn % n == 0;}

from operator import mul
from functools import reduce
from sympy import factorint
A247213_list = [n for n in range(1,10**4) if n <= 1 or not reduce(mul,[(p+2)**e for p,e in factorint(n).items()]) % n]
# Chai Wah Wu, Jan 05 2015

A359530 Multiplicative with a(p^e) = (p + 4)^e.

Original entry on oeis.org

1, 6, 7, 36, 9, 42, 11, 216, 49, 54, 15, 252, 17, 66, 63, 1296, 21, 294, 23, 324, 77, 90, 27, 1512, 81, 102, 343, 396, 33, 378, 35, 7776, 105, 126, 99, 1764, 41, 138, 119, 1944, 45, 462, 47, 540, 441, 162, 51, 9072, 121, 486, 147, 612, 57, 2058, 135, 2376, 161

Offset: 1

Vaclav Kotesovec, Feb 26 2023

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A166589 (multiplicative with a(p^e) = (p-3)^e), A166586 (p-2), A003958 (p-1), A000027 (p), A003959 (p+1), A166590 (p+2), A166591 (p+3).

g[p_, e_] := (p + 4)^e; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]

for(n=1, 100, print1(direuler(p=2, n, 1/(1-p*X-4*X))[n], ", "))

from math import prod
from sympy import factorint
def A359530(n): return prod((p+4)**e for p, e in factorint(n).items()) # Chai Wah Wu, Feb 26 2023

Dirichlet g.f.: Product_{primes p} 1 / (1 - p^(1-s) - 4*p^(-s)).
Dirichlet g.f.: zeta(s-1) * (1 + 4/(2^s - 6)) * Product_{primes p, p>2} (1 + 4/(p^s - p - 4)).
Sum_{k=1..n} a(k) has an average value 2*c*zeta(r-1) * n^r / (3*log(6)), where r = 1 + log(3)/log(2) = 2.5849625007211561814537389439478165... and c = Product_{primes p, p>2} (1 + 4/(p^r - p - 4)) = 1.5747380964592139...

cp's OEIS Frontend

A247213 Numbers n = Product_(p_i^e_i) such that nn = Product_((p_i + 2)^e_i) is divisible by n.

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