A348993 - OEIS

%I #15 Nov 28 2021 12:53:23

%S 1,1,1,5,2,1,1,3,11,2,2,5,5,1,2,29,4,11,3,1,1,2,2,1,29,5,1,5,6,2,1,5,

%T 2,4,2,55,17,3,5,3,10,1,7,5,22,2,2,29,34,29,4,25,8,1,4,3,1,6,6,1,29,1,

%U 11,113,2,2,13,5,2,2,4,11,31,17,29,15,2,5,3,29,49,10,10,5,8,7,2,3,12,22,5,5,1,2,6,5

%N a(n) = A064989(sigma(n) / gcd(sigma(n), A003961(n))), where A003961 shifts the prime factorization of n one step towards larger primes, while A064989 shifts it back towards smaller primes, and sigma is the sum of divisors function.

%H Antti Karttunen, <a href="/A348993/b348993.txt">Table of n, a(n) for n = 1..22968</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) = A064989(A349162(n)) = A064989(A348992(n)).

%t Array[Times @@ Map[If[#1 <= 2, 1, NextPrime[#1, -1]]^#2 & @@ # &, FactorInteger[#1/GCD[##]]] & @@ {DivisorSigma[1, #], Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]} &, 96] (* _Michael De Vlieger_, Nov 11 2021 *)

%o (PARI)

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

%o A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};

%o A349162(n) = { my(s=sigma(n)); (s/gcd(s,A003961(n))); };

%o A348993(n) = A064989(A349162(n));

%Y Cf. A000203, A000265, A003961, A064989, A161942, A342671, A348992, A349162, A349169 (gives odd k for which a(k) = A319627(k)).

%K nonn,look

%O 1,4

%A _Antti Karttunen_, Nov 10 2021