A095874 a(n) = k if n = A000961(k) (powers of primes), a(n) = 0 if n is not in A000961.

1, 2, 3, 4, 5, 0, 6, 7, 8, 0, 9, 0, 10, 0, 0, 11, 12, 0, 13, 0, 0, 0, 14, 0, 15, 0, 16, 0, 17, 0, 18, 19, 0, 0, 0, 0, 20, 0, 0, 0, 21, 0, 22, 0, 0, 0, 23, 0, 24, 0, 0, 0, 25, 0, 0, 0, 0, 0, 26, 0, 27, 0, 0, 28, 0, 0, 29, 0, 0, 0, 30, 0, 31, 0, 0, 0, 0, 0, 32, 0, 33, 0, 34, 0, 0, 0, 0, 0, 35, 0

Offset: 1

Author

Reinhard Zumkeller, Jun 10 2004

Keywords

• nonn

Comments

The name has been edited to clarify that the indices k refer to A000961 ("powers of primes" = {1} U A246655) and not to the list A246655 of proper prime powers. - M. F. Hasler, Jun 16 2021

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000961 (right inverse), A049084, A097621.

Programs

Haskell

a095874 n | y == n    = length xs + 1
          | otherwise = 0
          where (xs, y:ys) = span (< n) a000961_list
-- Reinhard Zumkeller, Feb 16 2012, Jun 26 2011

Mathematica

Join[{1},Module[{k=2},Table[If[PrimePowerQ[n],k;k++,0],{n,2,100}]]] (* Harvey P. Dale, Aug 15 2020 *)

PARI

a(n)=if(isprimepower(n), sum(i=1,logint(n,2), primepi(sqrtnint(n,i)))+1, n==1) \\ Charles R Greathouse IV, Apr 29 2015

PARI

{M95874=Map(); A095874(n,k)=if(mapisdefined(M95874,n,&k),k, isprimepower(n), mapput(M95874,n, k=sum(i=1,exponent(n), primepi(sqrtnint(n,i)))+1); k,n==1)} \\ Variant with memoization, possibly useful to compute A097621, A344826 and related. One may omit "isprimepower(n)," (possibly requiring factorization) and ",n==1" if n is known to be a power of a prime, i.e., to get a left inverse for A000961. - M. F. Hasler, Jun 15 2021

Python

from sympy import primepi, integer_nthroot, primefactors
def A095874(n): return 1+int(primepi(n)+sum(primepi(integer_nthroot(n,k)[0]) for k in range(2,n.bit_length()))) if n==1 or len(primefactors(n))==1 else 0 # Chai Wah Wu, Jan 19 2025

Formula

a(n) = Sum_{1 <= k <= n} A010055(k); [corrected by M. F. Hasler, Jun 15 2021]

a(n) = A065515(n)*(A065515(n)-A065515(n-1)).

a(n) = A065515(n)*A069513(n). - M. F. Hasler, Jun 16 2021

Extensions

Edited by M. F. Hasler, Jun 15 2021