A375271 -id:A375271 - cp's OEIS frontend

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

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, 83, 89, 97, 101, 103, 107, 109, 113, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 256

Offset: 1

Amiram Eldar, Aug 09 2024

nonn

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, ... .

The partial products of this sequence (A375271) are the sequence of numbers with record numbers of Zeckendorf-infinitary divisors (A318465).

			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, ... .

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A115975.
Subsequences: A000040, A030078, A179645.
Cf. A000045, A001906, A050376, A186285, A318465, A375271 (partial products).

fib[lim_] := Module[{s = {}, f = 1, k = 2}, While[f <= lim, AppendTo[s, f]; k += 2; f = Fibonacci[k]]; s];
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]

fib(lim) = {my(s = List(), f = 1, k = 2); while(f <= lim, listput(s, f); k += 2; f = fibonacci(k)); Vec(s);}
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);}

a(n) = A375271(n)/A375271(n-1) for n >= 2.

A375269 Partial products of A115975.

Original entry on oeis.org

1, 2, 6, 24, 120, 840, 6720, 60480, 665280, 8648640, 147026880, 2793510720, 64250746560, 1606268664000, 43369253928000, 1257708363912000, 38988959281272000, 1247646697000704000, 46162927789026048000, 1892680039350067968000, 81385241692052922624000, 3825106359526487363328000

Offset: 1

Amiram Eldar, Aug 09 2024

nonn

First differs from A334395 at n = 42.
Numbers with a record number of dual-Zeckendorf-infinitary divisors (A331109). Also, indices of records in A375272.
a(n) is the least number k such that A375272(k) = n-1 and A331109(k) = 2^(n-1).

			A115975 begins with 1, 2, 3, 4, 5, 7, ..., so, a(1) = 1, a(2) = 1 * 2 = 2, a(3) = 1 * 2 * 3 = 6, ..., a(6) = 1 * 2 * 3 * 4 * 5 * 7 = 840.

Amiram Eldar, Table of n, a(n) for n = 1..358

Cf. A037992 (analogous with "Fermi-Dirac primes", A050376), A115975, A331109, A334395, A375271, A375272.

Subsequence of A025487.

fib[lim_] := Module[{s = {}, f = 1, k = 2}, While[f <= lim, AppendTo[s, f]; k++; f = Fibonacci[k]]; s];
seq[max_] := Module[{s = {}, 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]]; FoldList[Times, 1, Sort[s]]]; seq[250]

fib(lim) = {my(s = List(), f = 1, k = 2); while(f <= lim, listput(s, f); k++; f = fibonacci(k)); Vec(s);}
lista(pmax) = {my(s = [1], p = 2, e = 1, f = [], r = 1); 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)); s = vecsort(s); for(i = 1, #s, r *= s[i]; print1(r, ", "));}

a(n) = Product_{k=1..n} A115975(k).

cp's OEIS Frontend

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

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