A025474 -id:A025474 - cp's OEIS frontend

A330669 The prime indices of the prime powers (A000961).

0, 1, 2, 1, 3, 4, 1, 2, 5, 6, 1, 7, 8, 9, 3, 2, 10, 11, 1, 12, 13, 14, 15, 4, 16, 17, 18, 1, 19, 20, 21, 22, 2, 23, 24, 25, 26, 27, 28, 29, 30, 5, 3, 31, 1, 32, 33, 34, 35, 36, 37, 38, 39, 6, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49

Offset: 1

Grant E. Martin and Robert G. Wilson v, Dec 23 2019

			a(16) is 2 since A000961(16) is 27 which is 3^3 = (p_2)^3, i.e., the prime index of 3 is 2.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000040, A000961, A000720, A025473, A025474, A065515.

b:= proc(n) option remember; local k; for k from
      1+b(n-1) while nops(ifactors(k)[2])>1 do od; k
    end: b(1):=1:
a:= n-> `if`(n=1, 0, numtheory[pi](ifactors(b(n))[2, 1$2])):
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 20 2020

mxn = 500; Join[{0}, Transpose[ Sort@ Flatten[ Table[ {Prime@n^ex, n}, {n, PrimePi@ mxn}, {ex, Log[Prime@n, mxn]}], 1]][[2]]]

lista(nn) = {print1(0); for(n=2, nn, if(isprimepower(n, &p), print1(", ", primepi(p)))); } \\ Jinyuan Wang, Feb 19 2020

from sympy import primepi, integer_nthroot, primefactors
def A330669(n):
    if n == 1: return 0
    def f(x): return int(n-2+x-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return int(primepi(primefactors(kmax)[0])) # Chai Wah Wu, Aug 20 2024

a(n) = A000720(A025473(n)). - Michel Marcus, Dec 24 2019

A000040(a(n))^A025474(n) = A000961(n) for n > 0. - Alois P. Heinz, Feb 20 2020

A082950 Power base and exponent of n-th prime power exchanged.

Original entry on oeis.org

0, 1, 1, 4, 1, 1, 9, 8, 1, 1, 16, 1, 1, 1, 32, 27, 1, 1, 25, 1, 1, 1, 1, 128, 1, 1, 1, 36, 1, 1, 1, 1, 64, 1, 1, 1, 1, 1, 1, 1, 1, 2048, 243, 1, 49, 1, 1, 1, 1, 1, 1, 1, 1, 8192, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 125, 1, 64, 1, 1, 1, 1, 1, 1, 1, 131072, 1, 1, 1, 1, 1, 1, 1, 2187, 1, 1, 1, 1

Offset: 1

Reinhard Zumkeller, May 26 2003

nonn

Cf. A000961, A025473, A025474.

s[n_] := Module[{f = FactorInteger[n]}, If[Length[f] == 1, f[[1, 2]]^f[[1, 1]], Nothing]]; s[1] = 0; Array[s, 250] (* Amiram Eldar, May 16 2025 *)

a(n) = A025474(n)^A025473(n) while A025473(n)^A025474(n) = A000961(n).

a(n) = 1 iff A000961(n) is prime.

a(71) and following corrected by Georg Fischer, Dec 09 2022

A271394 1 and the prime powers (p^k, p prime, k >= 1) such that p^k - k and p^k + k are prime powers.

Original entry on oeis.org

1, 2, 3, 8, 9, 25, 32, 512, 6561, 36703368217294125441230211032033660188801

Offset: 1

Juri-Stepan Gerasimov, Apr 06 2016

a(10) = 7^48. If it exists, a(11) > 10^100. - Giovanni Resta, Apr 12 2016

			6561 is in this sequence because 6561 = 3^8, 6561 - 8 = 6553 is prime and 6561 + 8 = 6569 is prime.

Cf. A000961, A025474.

nn = 10^6; {1}~Join~Sort@ Apply[Power, Select[Flatten[Function[k, {#, k}] /@ Range[Floor@ Log[#, nn]] & /@ Prime@ Range@ PrimePi@ nn, 1], With[{p = First@ #, k = Last@ #}, And[Or[PrimePowerQ@ #, # == 1] &[p^k - k], Or[PrimePowerQ@ #, # == 1] &[p^k + k]]] &], 1] (* Michael De Vlieger, Apr 06 2016 *)

ispp(n) = (n==1) || isprimepower(n);
isok(n) = (n==1) || ((k=isprimepower(n)) && ispp(n+k) && ispp(n-k));
\\ Michel Marcus, Apr 07 2016

a(10) from Giovanni Resta, Apr 12 2016

A335089 Decimal expansion of log(Pi^2/6).

Original entry on oeis.org

4, 9, 7, 7, 0, 0, 3, 0, 2, 4, 7, 0, 7, 4, 5, 3, 4, 7, 4, 7, 4, 3, 7, 7, 3, 4, 4, 3, 2, 5, 4, 1, 5, 1, 5, 0, 5, 7, 1, 5, 9, 8, 9, 3, 3, 6, 4, 7, 6, 1, 8, 4, 3, 7, 1, 7, 1, 8, 7, 1, 7, 9, 9, 8, 1, 3, 3, 8, 7, 6, 2, 4, 5, 8, 1, 3, 4, 8, 4, 7, 7, 0, 8, 7, 7, 6, 7, 4, 5, 8, 7, 4, 0, 8, 2, 8, 6, 3, 9, 0, 7, 4, 0, 4, 8, 1

Offset: 0

Terry D. Grant, Sep 11 2020

			Equals 1/(2^2) + 1/(3^2) + (1/(4^2))*(1/2) + 1/(5^2) + + 1/(7^2) + (1/(8^2))*(1/3) + ... = 0.4977003024707...

Mathematics Stack Exchange, Infinite series involving Von Mangoldt's function.
Grant Sanderson, What makes the natural log "natural"?, 3Blue1Brown video (2020).
Eric Weisstein's World of Mathematics, Mangold Function.
Eric Weisstein's World of Mathematics, Prime Zeta Function.

Cf. A000796, A013661, A053510, A131659, A016629.

Cf. A100995, A014963, A025474, A246655.

RealDigits[Log[Pi^2/6], 10, 120][[1]]
RealDigits[Sum[PrimeZetaP[2 k]/k, {k, 1, inf}], 10, 120][[1]]

log(Pi^2/6) \\ Michel Marcus, Sep 15 2020

Equals Sum_{k>=2} MangoldtLambda(k) / ((k^2)*log(k)).
Equals Sum_{k>=1} (1/k)*(1/(A246655(n)^2)) where k is the exponent of the prime power, A025474(n+1).
Equals Sum_{k>=1} primezeta(2*k)/k.
Equals 2*log(Pi) - log(6).
Equals log(zeta(2)) = log(A013661).

cp's OEIS Frontend

A330669 The prime indices of the prime powers (A000961).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A082950 Power base and exponent of n-th prime power exchanged.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

Extensions

A271394 1 and the prime powers (p^k, p prime, k >= 1) such that p^k - k and p^k + k are prime powers.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A335089 Decimal expansion of log(Pi^2/6).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula