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

1,1

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

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 }.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A225241 A087370 A364158 * A334169 A180081 A340854

Adjacent sequences: A138926 A138927 A138928 * A138930 A138931 A138932

Keyword

nonn

Author

M. F. Hasler, Apr 04 2008

Status

approved