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A368107 Prime powers p^m such that p | m.

%I A368107 #67 Sep 13 2024 08:03:49
%S A368107 4,16,27,64,256,729,1024,3125,4096,16384,19683,65536,262144,531441,
%T A368107 823543,1048576,4194304,9765625,14348907,16777216,67108864,268435456,
%U A368107 387420489,1073741824,4294967296,10460353203,17179869184,30517578125,68719476736,274877906944,282429536481
%N A368107 Prime powers p^m such that p | m.
%C A368107 Proper subset of A072873, which in turn is a proper subset of A342090.
%C A368107 This sequence represents the prime power block in A072873 and A342090.
%C A368107 A342090 \ {a(n)} is in A126706.
%C A368107 A072873 \ {{1} U {a(n)}} is in A286708, in turn a proper subset of A001694.
%C A368107 Contains A051674.
%H A368107 Michael De Vlieger, <a href="/A368107/b368107.txt">Table of n, a(n) for n = 1..3351</a>
%F A368107 Sum_{n>=1} 1/a(n) = Sum_{n>=1} 1/A088730(n) = 0.372116188498... . - _Amiram Eldar_, Jan 20 2024
%e A368107 This sequence contains prime powers of the following form:
%e A368107   2^2, 2^4, i.e., 2^k such that k is even.
%e A368107   3^3, 3^6, 3^9, i.e., 3^k such that 3 | k.
%e A368107   5^5, 5^10, 5^15, i.e., 5^k such that 5 | k, etc.
%p A368107 N:= 10^13: # for terms <= N
%p A368107 R:= NULL:
%p A368107 for i from 1 do
%p A368107   p:= ithprime(i);
%p A368107   if p^p > N then break fi;
%p A368107   R:= R, seq(p^k,k=p..floor(log[p](N)), p);
%p A368107 od:
%p A368107 sort([R]); # _Robert Israel_, Jan 16 2024
%t A368107 nn = 10^12; i = 1; p = 2; While[p^p <= nn, p = NextPrime[p] ];
%t A368107 MapIndexed[Set[S[First[#2]], #1] &, Prime@ Range@ PrimePi[p] ];
%t A368107 Union@ Reap[
%t A368107     While[j = S[i];
%t A368107     While[S[i]^j < nn,
%t A368107       Sow[S[i]^j]; j += S[i] ]; j > 2,
%t A368107     i++] ][[-1, 1]]
%o A368107 (Python)
%o A368107 import heapq
%o A368107 from itertools import islice
%o A368107 from sympy import nextprime
%o A368107 def agen(): # generator of terms
%o A368107     v, h, m, nextp = 4, [(4, 2)], 4, 3
%o A368107     while True:
%o A368107         v, p = heapq.heappop(h)
%o A368107         yield v
%o A368107         if v >= m:
%o A368107             m = nextp**nextp
%o A368107             heapq.heappush(h, (m, nextp))
%o A368107             nextp = nextprime(nextp)
%o A368107         heapq.heappush(h, (v*p**p, p))
%o A368107 print(list(islice(agen(), 31))) # _Michael S. Branicky_, Jan 16 2024
%o A368107 (Python)
%o A368107 from sympy import integer_nthroot, primefactors
%o A368107 def A368107(n):
%o A368107     def f(x):
%o A368107         c = n+x
%o A368107         for k in range(1,x.bit_length()):
%o A368107             m = integer_nthroot(x,k)[0]
%o A368107             c -= sum(1 for p in primefactors(k) if p<=m)
%o A368107         return c
%o A368107     def bisection(f,kmin=0,kmax=1):
%o A368107         while f(kmax) > kmax: kmax <<= 1
%o A368107         while kmax-kmin > 1:
%o A368107             kmid = kmax+kmin>>1
%o A368107             if f(kmid) <= kmid:
%o A368107                 kmax = kmid
%o A368107             else:
%o A368107                 kmin = kmid
%o A368107         return kmax
%o A368107     return bisection(f,n,n) # _Chai Wah Wu_, Sep 12 2024
%Y A368107 Cf. A001694, A051674, A072873, A088730, A126706, A246547, A286708, A342090.
%K A368107 nonn,easy
%O A368107 1,1
%A A368107 _Michael De Vlieger_, Jan 15 2024