A174932 - OEIS

%I #12 Sep 08 2022 08:45:51

%S 1,4,6,16,10,54,14,96,45,130,22,1860,26,238,270,1216,34,6048,38,8300,

%T 504,550,46,335688,175,754,864,22484,58,811050,62,35200,1188,1258,

%U 1330,10095048,74,1558,1638,2576920,82,3113586,86,86372,92070,2254,94,255478416

%N a(n) = Sum_{d|n} A007955(d) * A000027(n/d) = Sum_{d|n} A007955(d) * (n/d), where A007955(m) = product of divisors of m.

%H Andrew Howroyd, <a href="/A174932/b174932.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = n*A322671(n). - _Andrew Howroyd_, Jan 05 2020

%e For n = 4, A007955(n) = b(n): a(4) = b(1)*(4/1) + b(2)*(4/2) + b(4)*(4/4) = 1*4 + 2*2 + 8*1 = 16.

%o (PARI) a(n)={n*sumdiv(n, d, vecprod(divisors(d))/d)} \\ _Andrew Howroyd_, Jan 05 2020

%o (Magma) [&+[&*Divisors(d)*(n div d):d in Divisors(n)]:n in [1..50]]; // _Marius A. Burtea_, Jan 05 2020

%o (Python)

%o from math import isqrt

%o from sympy import divisor_count, divisors

%o def A174932(n): return n*sum(isqrt(d)**(c-2) if (c:=divisor_count(d)) & 1 else d**(c//2-1) for d in divisors(n,generator=True)) # _Chai Wah Wu_, Jun 25 2022

%Y Cf. A007955 (product of divisors), A322671.

%K nonn

%O 1,2

%A _Jaroslav Krizek_, Apr 02 2010