A360204 Primitive prime powers. p is a primitive prime power iff it is an odd prime power that exceeds the preceding odd prime power by more than any smaller odd prime power does. ('Prime power' defined in the sense of A246655.)

5, 17, 37, 97, 149, 211, 307, 907, 1151, 1361, 5623, 8501, 9587, 15727, 19661, 31469, 156007, 360749, 370373, 492227, 1349651, 1357333, 2010881, 4652507, 17051887, 20831533, 47326913, 122164969, 189695893, 191913031, 387096383, 436273291, 1294268779

Offset: 1

Author

Peter Luschny, Feb 01 2023

Keywords

• nonn

Comments

Conjecture: Primitive prime powers are primes.

Examples

			The first few terms of the sequence are the minuends in the following differences. The differences are strictly increasing by definition.
  5 -   3 = 2;
 17 -  13 = 4;
 37 -  31 = 6;
 97 -  89 = 8;
149 - 139 = 10;
211 - 199 = 12;
307 - 293 = 14;
907 - 887 = 20.
The example above might suggest the subtrahends are also prime. In general they are not, as the example a(10) shows, where 1361 - 1331 = 30, but 1331 is not prime. - _Ivan N. Ianakiev_, Feb 02 2023
		

Crossrefs

Cf. A246655, A061345.

Programs

Mathematica

a[1] = 5; candidates[n_] := Select[Range[NextPrime[n, -1], n], OddQ[#] && PrimePowerQ[#]&];
difference[n_] := candidates[n][[-1]] - candidates[n][[-2]];
a[n_] := a[n] = Module[{k = a[n-1]+2}, While[OddQ[k] && !PrimePowerQ[k] || difference[k] <= difference[a[n-1]], k = k+2]; k];
a/@Range[16] (* Ivan N. Ianakiev, Feb 02 2023 *)

SageMath

def A360204_list(n):
    R = []; L = 3; MAX = 1
    for C in xsrange(3, n, 2):
        if C.is_prime_power():
            if C - L > MAX:
                MAX = C - L
                R.append(C)
            L = C
    return R
print(A360204_list(6000))