cp's OEIS Frontend

A349163 a(n) = A064989(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.

%I A349163 #22 Nov 28 2021 12:53:56
%S A349163 1,2,1,1,1,2,1,2,1,2,1,1,1,2,1,1,1,2,1,10,1,2,1,6,1,2,3,1,1,2,1,4,1,2,
%T A349163 1,1,1,2,1,4,1,2,1,2,1,2,1,1,1,2,1,1,1,6,1,2,3,2,1,10,1,2,1,1,5,2,1,4,
%U A349163 1,2,1,6,1,2,1,1,1,2,1,2,1,2,1,1,1,2,3,4,1,2,1,2,1,2,1,4,1,2,11,5,1,2,1,2,1
%N A349163 a(n) = A064989(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 A349163 Antti Karttunen, <a href="/A349163/b349163.txt">Table of n, a(n) for n = 1..22968</a>
%H A349163 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A349163 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A349163 a(n) = A064989(A342671(n)).
%F A349163 a(n) = n / A349164(n).
%t A349163 Array[Times @@ Map[If[#1 <= 2, 1, NextPrime[#1, -1]]^#2 & @@ # &, FactorInteger@ GCD[##]] & @@ {DivisorSigma[1, #], Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]} &, 105] (* _Michael De Vlieger_, Nov 11 2021 *)
%o A349163 (PARI)
%o A349163 A003961(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A349163 A064989(n) = { my(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 A349163 A349163(n) = A064989(gcd(sigma(n),A003961(n)));
%Y A349163 Cf. A000203, A003961, A342671, A349161, A349162, A349165 (positions of 1's), A349166 (of terms > 1).
%Y A349163 Cf. A349144 and A349168 [positions where a(n) is / is not relatively prime with A349164(n) = n/a(n)].
%K A349163 nonn
%O A349163 1,2
%A A349163 _Antti Karttunen_, Nov 09 2021