A325127 Numbers in whose prime factorization the exponent of prime(k) is greater than k for all prime indices k.

1, 4, 8, 16, 27, 32, 64, 81, 108, 128, 216, 243, 256, 324, 432, 512, 625, 648, 729, 864, 972, 1024, 1296, 1728, 1944, 2048, 2187, 2500, 2592, 2916, 3125, 3456, 3888, 4096, 5000, 5184, 5832, 6561, 6912, 7776, 8192, 8748, 10000, 10368, 11664, 12500, 13824, 15552

Offset: 1

Author

Gus Wiseman, Apr 01 2019

Keywords

• nonn

Comments

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.

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.

Examples

			The sequence of terms together with their prime indices begins:
    1: {}
    4: {1,1}
    8: {1,1,1}
   16: {1,1,1,1}
   27: {2,2,2}
   32: {1,1,1,1,1}
   64: {1,1,1,1,1,1}
   81: {2,2,2,2}
  108: {1,1,2,2,2}
  128: {1,1,1,1,1,1,1}
  216: {1,1,1,2,2,2}
  243: {2,2,2,2,2}
  256: {1,1,1,1,1,1,1,1}
  324: {1,1,2,2,2,2}
  432: {1,1,1,1,2,2,2}
  512: {1,1,1,1,1,1,1,1,1}
  625: {3,3,3,3}
  648: {1,1,1,2,2,2,2}
  729: {2,2,2,2,2,2}
  864: {1,1,1,1,1,2,2,2}
  972: {1,1,2,2,2,2,2}
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..2847 (terms up to 10^12)

Crossrefs

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

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

Programs

Mathematica

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

Formula

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