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A186000 Consider the list s(1), s(2), ... of numbers that are products of exactly n primes; a(n) is the smallest s(j) whose decimal expansion ends in j.

%I A186000 #34 Aug 24 2025 05:50:18
%S A186000 1,17,291,12,56,78645,1350,192,896,7936,36096,3072,14336,250880,
%T A186000 1247232,49152,229376,4014080,6718464,786432,3670016,64225280,
%U A186000 45203456000,12582912,58720256,622854144,219792015360,201326592,939524096,8321499136,37849399296,3221225472,15032385536,263066746880,2924872728576,51539607552,240518168576,4209067950080,7044820107264,824633720832,3848290697216
%N A186000 Consider the list s(1), s(2), ... of numbers that are products of exactly n primes; a(n) is the smallest s(j) whose decimal expansion ends in j.
%C A186000 For n = 0, 1, 2, ..., the index j corresponding to a(n) is 1, 7, 91, 2, 6, 8645, 50, 2, 6, 36, 96, 2, 6, 80, 232, 2, 6, 80, 64, 2, 6, 80, >6136, 2, 6, 44, > 564, 2, 6, 36, 96, 2, 6, 80, >286, 2, 6, 80, 64, 2, 6, ..., .
%H A186000 <a href="/index/Ar#automorphic">Index entries for sequences related to automorphic numbers</a>
%e A186000 a(0) = 1 because 1 is the first and only positive integer (A000027) which is not a prime, a semiprime, a triprime, etc.;
%e A186000 a(1) = 17 because 17 is the seventh term of A000040 (it is also the first term of A046883);
%e A186000 a(2) = 291 because 291 is the 91st term of A001358;
%e A186000 a(3) = 12 because 12 is the second term of A014612;
%e A186000 a(4) = 56 because 56 is the sixth term of A014613; etc.
%t A186000 nextKthAlmostPrime[n_, k_] := Block[{m = n + 1}, While[ Plus @@ Last /@ FactorInteger@ m != k, m++]; m] (* _Eric W. Weisstein_, Feb 07 2006 *); f[n_] := Block[{c = 1, kp = 2^n}, While[ Mod[kp, 10^Floor[1 + Log10@ c]] != c, c++; kp = nextKthAlmostPrime[kp, n]]; kp]
%t A186000 (* These terms can be crosschecked by: *)
%t A186000 AlmostPrimePi[k_Integer, n_] := Module[{a, i}, a[0] = 1; If[k == 1, PrimePi[n], Sum[PrimePi[n/Times @@ Prime[Array[a, k - 1]]] - a[k - 1] + 1, Evaluate[ Sequence @@ Table[{a[i], a[i - 1], PrimePi[(n/Times @@ Prime[Array[a, i - 1]])^(1/(k - i + 1))]}, {i, k - 1}]]]]] (* _Eric W. Weisstein_, Feb 07 2006 *)
%Y A186000 Cf. A000040, A001358, A046883, A014612, A014613, A078840, A185999.
%K A186000 nonn,base,changed
%O A186000 0,2
%A A186000 _Jonathan Vos Post_ and _Robert G. Wilson v_, Feb 24 2011
%E A186000 Edited by _N. J. A. Sloane_, Mar 04 2011
%E A186000 The missing values, a(22), a(26) & a(34), were supplied to me via email dtd Mar 03 2011 from _Donovan Johnson_. - _Robert G. Wilson v_, Mar 22 2011