A096165 -id:A096165 - cp's OEIS frontend

A164336 a(1)=1. Thereafter, all terms are primes raised to the values of earlier terms of the sequence.

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227

Offset: 1

Leroy Quet, Aug 13 2009

nonn

These are the values of exponent towers consisting completely of primes coefficients. (For example, p^(q^(r^(s^..))), all variables being primes.) This sequence first differs from the terms of A096165, after the initial 1 in this sequence, when 18446744073709551616 = 2^64 occurs in A096165 but not in this sequence.

A064372(a(n)) = 1. [Reinhard Zumkeller, Aug 27 2011]

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Power Tower
Wikipedia, Tetration

Cf. A096165, A164337.

q:= n-> is(n=1 or (l-> nops(l)=1 and q(l[1, 2]))(ifactors(n)[2])):
select(q, [$1..350])[];  # Alois P. Heinz, Dec 30 2020

Block[{a = {1}}, Do[If[Length@ # == 1 && MemberQ[a, First@ #], AppendTo[a, i]] &[FactorInteger[i][[All, -1]]], {i, 2, 227}]; a] (* Michael De Vlieger, Aug 31 2017 *)

L=1000;S=[1];SS=[];while(#S!=#SS, SS=S;S=[];for(i=1,#SS,forprime(p=2,floor(L^(1/SS[i])),S=concat(S,p^SS[i])));S=eval(setunion(S,SS)));vecsort(S) \\ Hagen von Eitzen, Oct 03 2009

More terms from Hagen von Eitzen, Oct 03 2009

A164345 Powers of primes where the exponents are not powers of primes.

Original entry on oeis.org

1, 64, 729, 1024, 4096, 15625, 16384, 32768, 59049, 117649, 262144, 531441, 1048576, 1771561, 2097152, 4194304, 4782969, 4826809, 9765625, 14348907, 16777216, 24137569, 47045881, 67108864, 148035889, 244140625, 268435456

Offset: 1

Leroy Quet, Aug 13 2009

nonn

First differs from A164337, after the initial 1 in this sequence: 2^64 = 18446744073709551616 is in sequence A164337, but is not in this sequence.

This sequence contains those powers of primes that are not in sequence A096165.

			2^12 = 4096. Since 2 is prime, and since 12 is not a power of a prime, then 4096 is in this sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A024619, A096165 (complement with respect to A000961), A164337.

isok(k) = if(k==1, return(1)); my(q=isprimepower(k)); (q>1) && !isprimepower(q); \\ Michel Marcus, Nov 26 2020

Sum_{n>=1} 1/a(n) = 1 + Sum_{k in A024619} P(k) = 1.018407114609068368636..., where P is the prime zeta function. - Amiram Eldar, Nov 26 2020

Extended beyond 16384 by R. J. Mathar, Sep 27 2009

cp's OEIS Frontend

A164336 a(1)=1. Thereafter, all terms are primes raised to the values of earlier terms of the sequence.

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