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A360844 a(n) is the least k-full number that is sandwiched between twin primes.

%I A360844 #15 Feb 16 2025 08:34:04
%S A360844 4,432,2592,139968,139968,174960000000,56358560858112,84537841287168,
%T A360844 578415690713088,578415690713088,1141260857376768,61628086298345472,
%U A360844 61628086298345472,61628086298345472,322850407500000000000000000000,322850407500000000000000000000,62518864539857068333550694039552
%N A360844 a(n) is the least k-full number that is sandwiched between twin primes.
%C A360844 k-full number is a number m such that if a prime p divides m then so does p^k. All the exponents in the canonical prime factorization of a k-full number are not smaller than k.
%C A360844 a(2)-a(15) are the terms below 3*10^19. Except for a(7) = 174960000000, they are all 3-smooth numbers (A003586, and thus they are terms of A027856). Are there other terms that are not 3-smooth?
%C A360844 a(168) = 2^176 * 3^173 * 7^168 is the first term that is not 5-smooth. - _Bert Dobbelaere_, Feb 24 2023
%H A360844 Bert Dobbelaere, <a href="/A360844/b360844.txt">Table of n, a(n) for n = 2..200</a>
%H A360844 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TwinPrimes.html">Twin Primes</a>.
%H A360844 Wikipedia, <a href="https://en.wikipedia.org/wiki/Powerful_number#Generalization">Powerful number: Generalization</a>.
%H A360844 <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>.
%e A360844 The first 3 terms, their factorizations and the corresponding twin primes are:
%e A360844   n |   a(n)  prime factorization  A051904(a(n))  {a(n)-1, a(n)+1}
%e A360844   ----------------------------------------------------------------
%e A360844   2 |     4                  2^2              2             {3, 5}
%e A360844   3 |   432            2^4 * 3^3              3         {431, 433}
%e A360844   4 |  2592            2^5 * 3^4              4       {2591, 2593}
%Y A360844 Cf. A001097, A001694, A014574, A036966, A036967, A069492, A069493.
%Y A360844 Cf. A113839, A360840, A360841, A360842, A360843.
%Y A360844 Cf. A027856, A003586, A051904.
%K A360844 nonn
%O A360844 2,1
%A A360844 _Amiram Eldar_, Feb 23 2023
%E A360844 More terms from _Bert Dobbelaere_, Feb 24 2023