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%I #22 Jul 11 2024 01:35:06
%S 8,27,125,128,343,1331,2187,2197,4913,6859,12167,24389,29791,32768,
%T 50653,68921,78125,79507,103823,148877,205379,226981,300763,357911,
%U 389017,493039,571787,704969,823543,912673,1030301,1092727,1225043,1295029,1442897,2048383,2248091
%N Composite numbers m for which A064380(m) = A000010(m).
%C Theorem. A064380(m) = A000010(m) iff m has the form m=p^(2^k-1), k>=1, p a prime. Eliminating the primes (k=1), the terms of the sequence have this form for k>1. All terms of A030078 (k=2) and A092759 (k=3) and prime powers of A010803 (k=4) are in the sequence, for example.
%H Amiram Eldar, <a href="/A176509/b176509.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) ~ n^3 log^3 n. - _Charles R Greathouse IV_, Feb 19 2013
%F Sum_{n>=1} 1/a(n) = Sum_{k>=2} 1/P(2^k-1) = 0.183077059924063305405..., where P(s) is the prime zeta function. - _Amiram Eldar_, Jul 11 2024
%t seq[max_] := Module[{ps = Select[Range[Floor[Surd[max, 3]]], PrimeQ], e, k, s = {}}, Do[e = Floor[Log[ps[[i]], max]]; k = Floor[Log2[e + 1]]; s = Join[s, ps[[i]]^(2^Range[2, k] - 1)], {i, 1, Length[ps]}]; Sort[s]]; seq[3*10^6] (* _Amiram Eldar_, Mar 26 2023 *)
%o (PARI) is(n)=my(e=isprimepower(n));e>2 && 2^valuation(e+1,2)==e+1 \\ _Charles R Greathouse IV_, Feb 19 2013
%Y Cf. A000010, A010803, A030078, A050376, A056824, A064380, A092759, A176472.
%K nonn
%O 1,1
%A _Vladimir Shevelev_, Apr 19 2010
%E 128 inserted, 1024 deleted, 2187 inserted, 32768 inserted, etc. - _R. J. Mathar_, Nov 21 2010
%E More terms from _Amiram Eldar_, Mar 26 2023
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