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A307458
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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.
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0
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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, ", ")))
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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