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Numbers in whose prime factorization the exponent of prime(k) is greater than k for all prime indices k.
7

%I #12 Oct 24 2020 04:12:41

%S 1,4,8,16,27,32,64,81,108,128,216,243,256,324,432,512,625,648,729,864,

%T 972,1024,1296,1728,1944,2048,2187,2500,2592,2916,3125,3456,3888,4096,

%U 5000,5184,5832,6561,6912,7776,8192,8748,10000,10368,11664,12500,13824,15552

%N Numbers in whose prime factorization the exponent of prime(k) is greater than k for all prime indices k.

%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions where each part k appears more than k times. Such partitions are counted by A115584.

%H Amiram Eldar, <a href="/A325127/b325127.txt">Table of n, a(n) for n = 1..2847</a> (terms up to 10^12)

%F Sum_{n>=1} 1/a(n) = Product_{k>=1} 1 + 1/(prime(k)^k * (prime(k)-1)) = 1.58661114052385082598.... - _Amiram Eldar_, Oct 24 2020

%e The sequence of terms together with their prime indices begins:

%e 1: {}

%e 4: {1,1}

%e 8: {1,1,1}

%e 16: {1,1,1,1}

%e 27: {2,2,2}

%e 32: {1,1,1,1,1}

%e 64: {1,1,1,1,1,1}

%e 81: {2,2,2,2}

%e 108: {1,1,2,2,2}

%e 128: {1,1,1,1,1,1,1}

%e 216: {1,1,1,2,2,2}

%e 243: {2,2,2,2,2}

%e 256: {1,1,1,1,1,1,1,1}

%e 324: {1,1,2,2,2,2}

%e 432: {1,1,1,1,2,2,2}

%e 512: {1,1,1,1,1,1,1,1,1}

%e 625: {3,3,3,3}

%e 648: {1,1,1,2,2,2,2}

%e 729: {2,2,2,2,2,2}

%e 864: {1,1,1,1,1,2,2,2}

%e 972: {1,1,2,2,2,2,2}

%t Select[Range[1000],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>k>PrimePi[p]]&]

%t With[{k = 4}, m = Prime[k]^(k + 1); s = {}; Do[p = Prime[i]; AppendTo[s, Join[{1}, p^Range[i + 1, Floor[Log[p, m]]]]], {i, 1, k}]; Union @ Select[Times @@@ Tuples[s], # <= m &]] (* _Amiram Eldar_, Oct 24 2020 *)

%Y Cf. A000720, A056239, A062457, A109298, A112798, A115584, A118914, A276078.

%Y Cf. A324524, A324525, A324571, A325128, A325130.

%K nonn

%O 1,2

%A _Gus Wiseman_, Apr 01 2019