A192135 Prime powers p^e with p < e.

8, 16, 32, 64, 81, 128, 243, 256, 512, 729, 1024, 2048, 2187, 4096, 6561, 8192, 15625, 16384, 19683, 32768, 59049, 65536, 78125, 131072, 177147, 262144, 390625, 524288, 531441, 1048576, 1594323, 1953125, 2097152, 4194304, 4782969, 5764801, 8388608, 9765625, 14348907

Offset: 1

Author

Reinhard Zumkeller, Jun 26 2011

Keywords

• nonn

Links

• Donovan Johnson, Table of n, a(n) for n = 1..10000

Crossrefs

Complement to A074583 with respect to A000961.

Cf. A095874, A192187, A257278.

Programs

Maple

A192135 := proc(nmax) local s ,i,p,e ; s := {} ; for i from 1 do p := ithprime(i) ; if p^(p+1) > nmax then break; end if; for e from p+1 do if p^e > nmax then break; end if; s := s union {p^e} ; end do: end do: sort(s) ; end proc:
A192135(20000000) ; # R. J. Mathar, Jul 09 2011

Mathematica

seq[lim_] := Module[{s = {}, p = 2}, While[p^p <= lim, AppendTo[s, p^Range[p+1, Log[p, lim]]]; p = NextPrime[p]]; Sort[Flatten[s]]]; seq[10^7] (* Amiram Eldar, Apr 14 2025 *)

Formula

a(n) = A000961(A192187(n)).

A095874(a(n)) = A192187(n).

Sum_{n>=1} 1/a(n) = Sum_{p prime} 1/(p^p*(p-1)) = 0.26859872089648243789... . - Amiram Eldar, Apr 14 2025