A341885 - OEIS

%I #21 Aug 14 2022 10:16:00

%S 0,3,6,6,15,9,28,9,12,18,66,12,91,31,21,12,153,15,190,21,34,69,276,15,

%T 30,94,18,34,435,24,496,15,72,156,43,18,703,193,97,24,861,37,946,72,

%U 27,279,1128,18,56,33,159,97,1431,21,81,37,196,438,1770,27,1891,499,40,18,106,75,2278,159,282

%N a(n) is the sum of A000217(p) over the prime factors p of n, counted with multiplicity.

%C By definition, this sequence is completely additive. - _Peter Munn_, Aug 14 2022

%H Robert Israel, <a href="/A341885/b341885.txt">Table of n, a(n) for n = 1..10000</a>

%e 18 = 2*3*3 so a(18) = 2*3/2 + 3*4/2 + 3*4/2 = 15.

%p f:= proc(n) local t; add(t[1]*(t[1]+1)/2*t[2], t = ifactors(n)[2]) end proc:

%p map(f, [$1..100]);

%t Prepend[Array[Total@ PolygonalNumber@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[#]] &, 68, 2], 0] (* _Michael De Vlieger_, Feb 22 2021 *)

%o (Python3)

%o from sympy import factorint

%o def A341885(n): return sum(k*m*(m+1)//2 for m,k in factorint(n).items()) # _Chai Wah Wu_, Feb 25 2021

%o (PARI) a(n) = my(f=factor(n), p); sum(k=1, #f~, p=f[k,1]; f[k,2]*p*(p+1)/2); \\ _Michel Marcus_, Aug 14 2022

%Y Cf. A000217, A340834.

%Y For other completely additive sequences with primes p mapped to a function of p, see A001414.

%K nonn,easy

%O 1,2

%A _J. M. Bergot_ and _Robert Israel_, Feb 22 2021