A257279 Zeroless prime powers p^m with p <= m.

4, 8, 16, 27, 32, 64, 81, 128, 243, 256, 512, 729, 2187, 3125, 6561, 8192, 15625, 16384, 19683, 32768, 65536, 78125, 177147, 262144, 524288, 531441, 823543, 1594323, 1953125, 4782969, 9765625, 16777216, 33554432, 48828125, 134217728, 268435456, 282475249, 1162261467

Offset: 1

Author

M. F. Hasler, Apr 28 2015

Keywords

• nonn
• base

Comments

A few years ago, challenges had been launched to find a prime power p^n, n>1 as large as possible, cf. links. I have remarked that it is easy to find arbitrarily large examples by taking the square of very large primes, rather than high powers of smaller primes, and suggested a merit function to take into account and penalize such "trivial" solutions. This led to a new challenge including the condition n > p. This sequence lists such numbers with the last condition relaxed to n >= p, which is sufficient to make the search nontrivial but includes a few more terms, namely the zeroless powers p^p (A051674 intersect A052382).

Possibly is a(80) = 19^44 the largest term; there are no greater ones in the first 500000 terms of A257278. - Reinhard Zumkeller, May 01 2015

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..80
• S. S. Gupta, Can you find?, CYF n° 410 (Jan. 26, 2010) and CYF n° 412 (Nov. 13, 2012).
• C. Rivera, Puzzle 607. A zeroless Prime power, primepuzzles.net, 2011

Crossrefs

Cf. A052382, A000961, A025475; A122494.

Equals A257278 \ A011540 = intersection of A052382 and A257278.

Subsequence of A025475 \ A011540 and of A195943, see also A168046.

Programs

Haskell

a257279 n = a257279_list !! (n-1)
a257279_list = filter ((== 1) . a168046) a257278_list
-- Reinhard Zumkeller, May 01 2015

PARI

is(n)=vecmin(digits(n)) && isprimepower(n,&n)>=n

PARI

L=List();lim=10;forprime(p=1,lim,for(n=p,lim*log(lim)\log(p),listput(L,p^n)));listsort(select(n->vecmin(digits(n)),L));