A349163 - OEIS

%I #22 Nov 28 2021 12:53:56

%S 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 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 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 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 Antti Karttunen, <a href="/A349163/b349163.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(A342671(n)).

%F a(n) = n / A349164(n).

%t 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 (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=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(n) = A064989(gcd(sigma(n),A003961(n)));

%Y Cf. A000203, A003961, A342671, A349161, A349162, A349165 (positions of 1's), A349166 (of terms > 1).

%Y Cf. A349144 and A349168 [positions where a(n) is / is not relatively prime with A349164(n) = n/a(n)].

%K nonn

%O 1,2

%A _Antti Karttunen_, Nov 09 2021