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a(n) = A064989(A003961(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.
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%I #28 Nov 28 2021 12:54:03

%S 1,1,3,4,5,3,7,4,9,5,11,12,13,7,15,16,17,9,19,2,21,11,23,4,25,13,9,28,

%T 29,15,31,8,33,17,35,36,37,19,39,10,41,21,43,22,45,23,47,48,49,25,51,

%U 52,53,9,55,28,19,29,59,6,61,31,63,64,13,33,67,17,69,35,71,12,73,37,75,76,77,39,79,40,81,41,83,84

%N a(n) = A064989(A003961(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="/A349164/b349164.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(A349161(n)).

%F a(n) = n / A349163(n).

%t Array[Times @@ Map[If[#1 <= 2, 1, NextPrime[#1, -1]]^#2 & @@ # &, FactorInteger[#2/GCD[##]]] & @@ {DivisorSigma[1, #], Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]} &, 84] (* _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 A349164(n) = { my(u=A003961(n)); A064989(u/gcd(u,sigma(n))); };

%Y Cf. A000203, A003961, A319630, A326042, A336702, A342671, A348993, A349161, A349162, A349169, A349174.

%Y Cf. A349144 and A349168 [positions where a(n) is / is not relatively prime with A349163(n) = n/a(n)].

%K nonn

%O 1,3

%A _Antti Karttunen_, Nov 09 2021