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%I #63 Jan 20 2024 03:16:09
%S 4,16,27,64,256,729,1024,3125,4096,16384,19683,65536,262144,531441,
%T 823543,1048576,4194304,9765625,14348907,16777216,67108864,268435456,
%U 387420489,1073741824,4294967296,10460353203,17179869184,30517578125,68719476736,274877906944,282429536481
%N Prime powers p^m such that p | m.
%C Proper subset of A072873, which in turn is a proper subset of A342090.
%C This sequence represents the prime power block in A072873 and A342090.
%C A342090 \ {a(n)} is in A126706.
%C A072873 \ {{1} U {a(n)}} is in A286708, in turn a proper subset of A001694.
%C Contains A051674.
%H Michael De Vlieger, <a href="/A368107/b368107.txt">Table of n, a(n) for n = 1..3351</a>
%F Sum_{n>=1} 1/a(n) = Sum_{n>=1} 1/A088730(n) = 0.372116188498... . - _Amiram Eldar_, Jan 20 2024
%e This sequence contains prime powers of the following form:
%e 2^2, 2^4, i.e., 2^k such that k is even.
%e 3^3, 3^6, 3^9, i.e., 3^k such that 3 | k.
%e 5^5, 5^10, 5^15, i.e., 5^k such that 5 | k, etc.
%p N:= 10^13: # for terms <= N
%p R:= NULL:
%p for i from 1 do
%p p:= ithprime(i);
%p if p^p > N then break fi;
%p R:= R, seq(p^k,k=p..floor(log[p](N)), p);
%p od:
%p sort([R]); # _Robert Israel_, Jan 16 2024
%t nn = 10^12; i = 1; p = 2; While[p^p <= nn, p = NextPrime[p] ];
%t MapIndexed[Set[S[First[#2]], #1] &, Prime@ Range@ PrimePi[p] ];
%t Union@ Reap[
%t While[j = S[i];
%t While[S[i]^j < nn,
%t Sow[S[i]^j]; j += S[i] ]; j > 2,
%t i++] ][[-1, 1]]
%o (Python)
%o import heapq
%o from itertools import islice
%o from sympy import nextprime
%o def agen(): # generator of terms
%o v, h, m, nextp = 4, [(4, 2)], 4, 3
%o while True:
%o v, p = heapq.heappop(h)
%o yield v
%o if v >= m:
%o m = nextp**nextp
%o heapq.heappush(h, (m, nextp))
%o nextp = nextprime(nextp)
%o heapq.heappush(h, (v*p**p, p))
%o print(list(islice(agen(), 31))) # _Michael S. Branicky_, Jan 16 2024
%Y Cf. A001694, A051674, A072873, A088730, A126706, A246547, A286708, A342090.
%K nonn,easy
%O 1,1
%A _Michael De Vlieger_, Jan 15 2024
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