A347124 - OEIS

%I #19 Dec 24 2023 01:08:17

%S 1,7,17,44,39,119,83,268,261,273,153,748,233,581,663,1616,339,1827,

%T 455,1716,1411,1071,689,4556,1385,1631,3933,3652,927,4641,1177,9712,

%U 2601,2373,3237,11484,1553,3185,3961,10452,1803,9877,2063,6732,10179,4823,2537,27472,6433,9695,5763,10252,3179,27531,5967,22244

%N Möbius transform of A341528, n * sigma(A003961(n)).

%C Multiplicative because A341528 is.

%H Antti Karttunen, <a href="/A347124/b347124.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>.

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.

%F a(n) = Sum_{d|n} A008683(n/d) * A341528(d).

%F a(n) = A347125(n) - A346239(n).

%F Multiplicative with a(p^e) = p^(e-1)*(q^e*(p*q-1)-p+1)/(q-1), where q = A151800(p). - _Sebastian Karlsson_, Sep 02 2021

%F Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = (1/zeta(3)) / Product_{p prime} (1 - (q(p)-p)/p^2 - q(p)/p^3) = 5.6488805... , and q(p) = A151800(p). - _Amiram Eldar_, Dec 24 2023

%t f[p_, e_] := Module[{q = NextPrime[p]}, p^(e-1) * (q^e * (p*q-1) - p + 1)/(q-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Dec 24 2023 *)

%o (PARI)

%o A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };

%o A003973(n) = sigma(A003961(n));

%o A341528(n) = (n*A003973(n));

%o A347124(n) = sumdiv(n,d,moebius(n/d)*A341528(d));

%Y Cf. A002117, A003961, A003973, A008683, A151800, A341528, A341512, A346239, A347125.

%K nonn,mult

%O 1,2

%A _Antti Karttunen_, Aug 24 2021