A138929 Twice the prime powers A000961.

2, 4, 6, 8, 10, 14, 16, 18, 22, 26, 32, 34, 38, 46, 50, 54, 58, 62, 64, 74, 82, 86, 94, 98, 106, 118, 122, 128, 134, 142, 146, 158, 162, 166, 178, 194, 202, 206, 214, 218, 226, 242, 250, 254, 256, 262, 274, 278, 298, 302, 314, 326, 334, 338, 346, 358, 362, 382

Offset: 1

Author

M. F. Hasler, Apr 04 2008

Keywords

• nonn

Comments

Except for the initial term a(1)=2, indices k such that A020513(k)=Phi[k](-1) is prime, where Phi is a cyclotomic polynomial.

This is illustrated by the PARI code, although it is probably more efficient to calculate a(n) as 2*A000961(n).

{ a(n)/2 ; n>1 } are also the indices for which A020500(k)=Phi[k](1) is prime.

A188666(k) = A000961(k+1) for k: a(k) <= k < a(k+1), k > 0;

A188666(a(n)) = A000961(n+1). [Reinhard Zumkeller, Apr 25 2011]

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000
• Index entries for cyclotomic polynomials, values at X

Crossrefs

Cf. A000961, A020513, A138920-A138940, A230078 (complement).

Programs

Maple

a := n -> `if`(1>=nops(numtheory[factorset](n)),2*n,NULL):
seq(a(i),i=1..192); # Peter Luschny, Aug 12 2009

Mathematica

Join[{2}, Select[ Range[3, 1000], PrimeQ[ Cyclotomic[#, -1]] &]] (* Robert G. Wilson v, Mar 25 2012 - modified by Paolo Xausa, Aug 30 2024 to include a(1) *)
2*Join[{1}, Select[Range[500], PrimePowerQ]] (* Paolo Xausa, Aug 30 2024 *)

PARI

print1(2);for( i=1,999, isprime( polcyclo(i,-1)) & print1(",",i)) /* use ...subst(polcyclo(i),x,-2)... in PARI < 2.4.2. It should be more efficient to calculate a(n) as 2*A000961(n) ! */

Python

from sympy import primepi, integer_nthroot
def A138929(n):
    def f(x): return int(n-1+x-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    kmin, kmax = 0,1
    while f(kmax) > kmax:
        kmax <<= 1
    while kmax-kmin > 1:
        kmid = kmax+kmin>>1
        if f(kmid) <= kmid:
            kmax = kmid
        else:
            kmin = kmid
    return kmax<<1 # Chai Wah Wu, Aug 29 2024

Formula

a(n) = 2*A000961(n).

Equals {2} union { k | Phi[k](-1)=A020513(k) is prime } = {2} union { 2k | Phi[k](1)=A020500(k) is prime }.