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A349161
a(n) = A003961(n) / gcd(sigma(n), A003961(n)), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of divisors function.
20
1, 1, 5, 9, 7, 5, 11, 9, 25, 7, 13, 45, 17, 11, 35, 81, 19, 25, 23, 3, 55, 13, 29, 9, 49, 17, 25, 99, 31, 35, 37, 27, 65, 19, 77, 225, 41, 23, 85, 21, 43, 55, 47, 39, 175, 29, 53, 405, 121, 49, 95, 153, 59, 25, 91, 99, 23, 31, 61, 15, 67, 37, 275, 729, 17, 65, 71, 19, 145, 77, 73, 45, 79, 41, 245, 207, 143, 85, 83
OFFSET
1,3
COMMENTS
Numerator of ratio A003961(n) / A000203(n). Sequence A349162 gives the denominators.
Numerator of ratio A003961(n) / A161942(n). Sequence A348992 gives the denominators.
Both ratios are multiplicative because the constituent sequences are.
No 1's occur as terms after a(2), because for n > 2, sigma(n) < A003961(n). (See A286385).
FORMULA
a(n) = A003961(n) / A342671(n) = A003961(n) / gcd(A000203(n), A003961(n)).
a(n) = A003961(A349164(n)).
MATHEMATICA
Array[#2/GCD[##] & @@ {DivisorSigma[1, #], If[# == 1, 1, Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]]} &, 79] (* Michael De Vlieger, Nov 11 2021 *)
PROG
(PARI)
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A349161(n) = { my(u=A003961(n)); (u/gcd(u, sigma(n))); };
(Python)
from math import prod, gcd
from sympy import nextprime, factorint
def A349161(n):
f = factorint(n).items()
a = prod(nextprime(p)**e for p, e in f)
b = prod((p**(e+1)-1)//(p-1) for p, e in f)
return a//gcd(a, b) # Chai Wah Wu, Mar 17 2023
KEYWORD
nonn,frac
AUTHOR
Antti Karttunen, Nov 09 2021
STATUS
approved