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A375270 Numbers of the form p^Fibonacci(2*k), where p is a prime and k >= 0.

%I A375270 #12 Mar 23 2025 07:26:47
%S A375270 1,2,3,5,7,8,11,13,17,19,23,27,29,31,37,41,43,47,53,59,61,67,71,73,79,
%T A375270 83,89,97,101,103,107,109,113,125,127,131,137,139,149,151,157,163,167,
%U A375270 173,179,181,191,193,197,199,211,223,227,229,233,239,241,251,256
%N A375270 Numbers of the form p^Fibonacci(2*k), where p is a prime and k >= 0.
%C A375270 Differs from A186285 by having the terms 1, 2^8 = 256, 3^8 = 6561, ..., and not having the terms 2^9 = 512, 3^9 = 19683, ... .
%C A375270 The partial products of this sequence (A375271) are the sequence of numbers with record numbers of Zeckendorf-infinitary divisors (A318465).
%H A375270 Amiram Eldar, <a href="/A375270/b375270.txt">Table of n, a(n) for n = 1..10000</a>
%F A375270 a(n) = A375271(n)/A375271(n-1) for n >= 2.
%e A375270 The positive even-indexed Fibonacci numbers are 1, 3, 8, 21, ..., so the sequence includes 2^1 = 2, 2^3 = 8, 2^8 = 256, ..., 3^1 = 3, 3^3 = 27, 3^8 = 6561, ... .
%t A375270 fib[lim_] := Module[{s = {}, f = 1, k = 2}, While[f <= lim, AppendTo[s, f]; k += 2; f = Fibonacci[k]]; s];
%t A375270 seq[max_] := Module[{s = {1}, p = 2, e = 1, f = {}}, While[e > 0, e = Floor[Log[p, max]]; If[f == {}, f = fib[e], f = Select[f, # <= e &]]; s = Join[s, p^f]; p = NextPrime[p]]; Sort[s]]; seq[256]
%o A375270 (PARI) fib(lim) = {my(s = List(), f = 1, k = 2); while(f <= lim, listput(s, f); k += 2; f = fibonacci(k)); Vec(s);}
%o A375270 lista(pmax) = {my(s = [1], p = 2, e = 1, f = []); while(e > 0, e = logint(pmax, p); if(#f == 0, f = fib(e), f = select(x -> x <= e, f)); s = concat(s, apply(x -> p^x, f)); p = nextprime(p+1)); vecsort(s);}
%Y A375270 Subsequence of A115975.
%Y A375270 Subsequences: A000040, A030078, A179645.
%Y A375270 Cf. A000045, A001906, A050376, A186285, A318465, A375271 (partial products).
%K A375270 nonn
%O A375270 1,2
%A A375270 _Amiram Eldar_, Aug 09 2024