cp's OEIS Frontend

A336840 Inverse Möbius transform of A048673.

%I A336840 #26 Nov 30 2024 12:55:00
%S A336840 1,3,4,8,5,14,7,22,17,18,8,42,10,26,26,63,11,65,13,55,38,30,16,124,30,
%T A336840 38,80,81,17,100,20,185,44,42,50,206,22,50,56,164,23,148,25,94,127,62,
%U A336840 28,368,68,117,62,120,31,316,58,244,74,66,32,318,35,78,189,550,74,172,37,133,92,196,38,626,41,86,174,159
%N A336840 Inverse Möbius transform of A048673.
%C A336840 Arithmetic mean of the number of divisors (A000005) and prime-shifted sigma (A003973), thus a(n) is the average between the number of and the sum of divisors of A003961(n).
%C A336840 The local minima occur on primes p, where p/2 < a(p) <= (p+1).
%H A336840 Antti Karttunen, <a href="/A336840/b336840.txt">Table of n, a(n) for n = 1..16384</a>
%H A336840 Antti Karttunen, <a href="/A336840/a336840.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%H A336840 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A336840 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A336840 a(n) = Sum_{d|n} A048673(d).
%F A336840 a(n) = (1/2) * (A000005(n) + A003973(n)).
%F A336840 a(n) = A113415(A003961(n)). - _Antti Karttunen_, Jun 01 2022
%F A336840 a(n) = A349371(A003961(n)) = A364063(A048673(n)). - _Antti Karttunen_, Nov 30 2024
%o A336840 (PARI)
%o A336840 A048673(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)+1)/2; };
%o A336840 A336840(n) = sumdiv(n,d,A048673(d));
%o A336840 (PARI) A336840(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (1/2)*(numdiv(n)+sigma(factorback(f))); };
%Y A336840 Cf. A000005, A003961, A003973, A007503, A048673, A113415, A336838, A336839, A336841, A349371, A349384, A349385, A349910 [= a(n)-A048673(n)], A364063, A378520 (Dirichlet inverse).
%K A336840 nonn
%O A336840 1,2
%A A336840 _Antti Karttunen_, Aug 07 2020