A306694 a(n) is the denominator of log(A014963(n))/log(n) if n > 1 and a(1) = 1.

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset: 1

Author

Mats Granvik, Mar 05 2019

Keywords

• nonn

Comments

Log(A112624(n)) is the inverse Möbius transform of log(a(n)).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..100000
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A014963, A112624, A246655.

Programs

Maple

with(numtheory): pexp := n -> ifactors(n)[2][1][2]:
a := n -> if nops(factorset(n)) = 1 then pexp(n) else 1 fi:
seq(a(n), n=1..101); # Peter Luschny, Mar 19 2019

Mathematica

Table[Denominator[FullSimplify[MangoldtLambda[n]/Log[n]]], {n, 1, 101}]

PARI

A306694(n) = if((n=isprimepower(n))>0,n,1); \\ Antti Karttunen, Nov 17 2019

Sage

def a(n):
    F = n.factor()
    return 1 if len(F) != 1 else F[0][1]
print([a(n) for n in (1..101)]) # Peter Luschny, Mar 18 2019

Formula

If n is a prime power (in the sense of A246655) then a(n) is the exponent of this prime and otherwise a(n) is 1. - Peter Luschny, Mar 18 2019

Dirichlet generating function: zeta(s) + Sum_{n>=1} n*primezeta((n + 1)*s). - Mats Granvik, Mar 24 2019

Extensions

Data section extended up to term a(121) by Antti Karttunen, Nov 17 2019