A076704 Numbers k of the form p^q where both p and q are prime and all digits of k are odd.

9, 1331, 357911, 5177717, 5735339, 9393931, 17171515157399, 335571975137771, 7979737131773191, 13337513771953951, 13137917533317175739371379, 33159599371999557199755557, 1593395573971551557179777111133, 131755773357537951113179771515713, 315113377779977515359339551539771

Offset: 1

Author

Zak Seidov, Oct 26 2002

Keywords

• nonn
• base

Comments

Up to 10^17, there are only 10 odd-digit prime powers of prime numbers. a(1) = 3^2, a(2) = 11^3, a(3) = 71^3, a(4) = 173^3, a(5) = 179^3, a(6) = 211^3, a(7) = 25799^3, a(8) = 69491^3, a(9) = 199831^3, and a(10) = 237151^3.

The only candidates for even-digit prime powers of prime numbers are of the form 2^n, and below 2^10000 there are only 2, 4, 8, 64, and 2048, two of which are not raised to prime powers.

a(11) <= 13137917533317175739371379 and a(12) <= 33159599371999557199755557. - Jinyuan Wang, Mar 02 2020

Links

• Giovanni Resta, Table of n, a(n) for n = 1..36 (terms < 10^57)

Crossrefs

Cf. A014261, A053810, A075308.

Programs

Mathematica

pp = Sort[ Flatten[ Table[ Prime[n]^Prime[i], {n, 1, PrimePi[ Sqrt[10^17]]}, {i, 1, PrimePi[ Floor[ Log[ Prime[n], 10^17]]]}]]]; Do[ If[ Union[ OddQ[ IntegerDigits[ pp[[n]]]]] == {True}, Print[ pp[[n]]]], {n, 1, Length[pp]}]

PARI

lista(nn) = {my(k, v=List([])); forprime(p=2, nn, forprime(q=2, logint(nn, p), if(Set(digits(k=p^q)%2)==[1], listput(v, k)))); Set(v); } \\ Jinyuan Wang, Mar 02 2020

Extensions

Edited and extended by Robert G. Wilson v, Oct 31 2002

Corrected and edited by Elliott Line, Jul 11 2013

Better definition from Jon E. Schoenfield, Nov 19 2018

Terms a(11) and beyond from Giovanni Resta, Mar 03 2020