A085156 Powers of primes or powers of semiprimes.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 67, 69, 71, 73, 74, 77, 79, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 100, 101

Offset

1,2

Comments

m is a term iff A067029(m) = A071178(m) and A001221(m) <= 2.

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Mathematica

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Length[e] == 1 || (Length[e] == 2 && Equal @@ e)]; Select[Range[100], q] (* Amiram Eldar, Apr 19 2025 *)

Prog

(PARI) is(n)=my(f=factor(n)[, 2]); #f<2 || (#f==2 && f[1]==f[2]) \\ Charles R Greathouse IV, Oct 19 2015
(Python)
from math import isqrt
from sympy import primepi, primerange, integer_nthroot
from oeis_sequences.OEISsequences import bisection
def A085156(n):
    def f(x): return int(n+x-1+sum(((t:=primepi(s:=isqrt(y:=integer_nthroot(x, k)[0])))*(t-1)>>1)-sum(primepi(y//p) for p in primerange(s+1))-(primepi(y) if k&1 else 0) for k in range(1, x.bit_length())))
    return bisection(f, n, n) # Chai Wah Wu, Jan 02 2026

Crossrefs

Union of A000961 and A085155.

Cf. A001221, A067029, A071178.

Sequence in context: A303554 A325328 A316521 * A342522 A102466 A354144

Adjacent sequences: A085153 A085154 A085155 * A085157 A085158 A085159

Keyword

nonn

Author

Reinhard Zumkeller, Jun 21 2003

Status

approved