A307458 Composites c where an integer b with 1 < b < c exists such that when the k digits in the base-b expansion of c are considered as exponents in an ordered list of primes prime(1), prime(2), ..., prime(k), then Product_{i=1..k} prime(i)^d[i] = c, where d[h] gives the h-th most significant digit in the expansion.

6, 10, 18, 36, 54, 96, 100, 162, 200, 216, 256, 324, 486, 1296, 1458, 2916, 4374, 5832, 13122, 26244, 39366, 46656, 47250, 49000, 65536, 82944, 104976, 118098, 157464, 181500, 236196, 354294, 746496, 1062882, 1492992, 1679616, 1990656, 2125764, 3188646, 3538944

Offset: 1

Author

Felix Fröhlich, Apr 09 2019

Keywords

• nonn
• base

Comments

In other words, integers k where an integer b with 1 < b < k exists such that row k of A067255 gives the digits of the base-b expansion of k.

Clearly, all terms are even, since all expansions start with a nonzero digit and thus the factorization of each term contains the prime 2.

Examples

			The base-4 expansion of 200 is 3020. 2^3 * 3^0 * 5^2 * 7^0 = 200, so 200 is a term of the sequence.
		

Crossrefs

Cf. A067255.

Programs

Mathematica

base[n_] := Block[{e, t=0, m, b=0, s=False, p, x, pp}, pp = PrimePi@ FactorInteger[n][[-1, 1]]; If[2^(pp - 1) > n, 0, e = IntegerExponent[n, Prime@ Range@ pp]; m = Max[e] + 1; p = Total[Reverse[e] x^Range[0, Length[e] - 1]]; While[((p x^t) /. x -> m ) <= n, s = Reduce[p x^t == n && m <= x < n, x, Integers]; If[s === False, t++, b = x /. List[ToRules@ s][[1]]; Break[], t++]]; b]]; Select[Range[4, 10^5, 2], base[#] > 0 &] (* Giovanni Resta, Apr 10 2019 *)

PARI

is(n) = for(b=2, n-1, my(d=digits(n, b), k=#d, x=1); while(k > 0, x=x*prime(k)^d[k]; k--); if(x==n, return(1))); 0
for(t=1, oo, if(is(2*t), print1(2*t, ", ")))

Extensions

a(21)-a(40) from Giovanni Resta, Apr 10 2019