A336840 - OEIS

%I #21 Jun 01 2022 17:44:07

%S 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 38,80,81,17,100,20,185,44,42,50,206,22,50,56,164,23,148,25,94,127,62,

%U 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 Inverse Möbius transform of A048673.

%C 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 The local minima occur on primes p, where p/2 < a(p) <= (p+1).

%H Antti Karttunen, <a href="/A336840/b336840.txt">Table of n, a(n) for n = 1..16384</a>

%H Antti Karttunen, <a href="/A336840/a336840.txt">Data supplement: n, a(n) computed for n = 1..65537</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} A048673(d).

%F a(n) = (1/2) * (A000005(n) + A003973(n)).

%F a(n) = A113415(A003961(n)). - _Antti Karttunen_, Jun 01 2022

%o (PARI)

%o A048673(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)+1)/2; };

%o A336840(n) = sumdiv(n,d,A048673(d));

%o (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 Cf. A000005, A003961, A003973, A048673, A113415, A336838, A336839, A336841, A349384, A349385.

%K nonn

%O 1,2

%A _Antti Karttunen_, Aug 07 2020