login
Inverse Möbius transform of A003961; a(n) = sigma(A003961(n)), where A003961 shifts the prime factorization of n one step towards the larger primes.
86

%I #60 Dec 08 2022 07:36:26

%S 1,4,6,13,8,24,12,40,31,32,14,78,18,48,48,121,20,124,24,104,72,56,30,

%T 240,57,72,156,156,32,192,38,364,84,80,96,403,42,96,108,320,44,288,48,

%U 182,248,120,54,726,133,228,120,234,60,624,112,480,144,128,62,624,68

%N Inverse Möbius transform of A003961; a(n) = sigma(A003961(n)), where A003961 shifts the prime factorization of n one step towards the larger primes.

%C Sum of the divisors of the prime shifted n, or equally, sum of the prime shifted divisors of n. - _Antti Karttunen_, Aug 17 2020

%H Charles R Greathouse IV, <a href="/A003973/b003973.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 Multiplicative with a(p^e) = (q^(e+1)-1)/(q-1) where q = nextPrime(p). - _David W. Wilson_, Sep 01 2001

%F From _Antti Karttunen_, Aug 06-12 2020: (Start)

%F a(n) = Sum_{d|n} A003961(d) = Sum_{d|A003961(n)} d.

%F a(n) = A000203(A003961(n)) = A000593(A003961(n)).

%F a(n) = 2*A336840(n) - A000005(n) = 2*Sum_{d|n} (A048673(d) - (1/2)).

%F a(n) = A008438(A108228(n)) = A008438(A048673(n)-1).

%F a(n) = A336838(n) * A336856(n).

%F a(n) is odd if and only if n is a square.

%F (End)

%F Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/2) * Product_{p prime} p^3/((p+1)*(p^2-nextprime(p))) = 3.39513795..., where nextprime is A151800. - _Amiram Eldar_, Dec 08 2022

%t b[1] = 1; b[p_?PrimeQ] := b[p] = Prime[ PrimePi[p] + 1]; b[n_] := b[n] = Times @@ (b[First[#]]^Last[#] &) /@ FactorInteger[n]; a[n_] := Sum[ b[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 70}] (* _Jean-François Alcover_, Jul 18 2013 *)

%o (PARI) aPrime(p,e)=my(q=nextprime(p+1));(q^(e+1)-1)/(q-1)

%o a(n)=my(f=factor(n));prod(i=1,#f~,aPrime(f[i,1],f[i,2])) \\ _Charles R Greathouse IV_, Jul 18 2013

%o (PARI) A003973(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); sigma(factorback(f)); }; \\ _Antti Karttunen_, Aug 06 2020

%o (Python)

%o from math import prod

%o from sympy import factorint, nextprime

%o def A003973(n): return prod(((q:=nextprime(p))**(e+1)-1)//(q-1) for p,e in factorint(n).items()) # _Chai Wah Wu_, Jul 05 2022

%Y Cf. A000203, A000290 (positions of odd terms), A003961, A007814, A048673, A108228, A151800, A295664, A336840.

%Y Permutation of A008438.

%Y Used in the definitions of the following sequences: A326042, A336838, A336841, A336844, A336846, A336847, A336848, A336849, A336850, A336851, A336852, A336856, A336931, A336932.

%Y Cf. also A003972.

%K nonn,easy,mult

%O 1,2

%A _Marc LeBrun_

%E More terms from _David W. Wilson_, Aug 29 2001

%E Secondary name added by _Antti Karttunen_, Aug 06 2020