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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.)
0
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
1,1
COMMENTS
Conjecture: Primitive prime powers are primes.
EXAMPLE
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
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 *)
PROG
(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))
CROSSREFS
Sequence in context: A181423 A146874 A145012 * A147204 A168024 A121326
KEYWORD
nonn
AUTHOR
Peter Luschny, Feb 01 2023
STATUS
approved