A352492 Powerful numbers whose prime indices are all prime numbers.

1, 9, 25, 27, 81, 121, 125, 225, 243, 289, 625, 675, 729, 961, 1089, 1125, 1331, 1681, 2025, 2187, 2601, 3025, 3125, 3267, 3375, 3481, 4489, 4913, 5625, 6075, 6561, 6889, 7225, 7803, 8649, 9801, 10125, 11881, 11979, 14641, 15125, 15129, 15625, 16129, 16875

Offset: 1

Author

Gus Wiseman, Mar 24 2022

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.

Examples

			The terms together with their prime indices (not prime factors) begin:
    1: {}
    9: {2,2}
   25: {3,3}
   27: {2,2,2}
   81: {2,2,2,2}
  121: {5,5}
  125: {3,3,3}
  225: {2,2,3,3}
  243: {2,2,2,2,2}
  289: {7,7}
  625: {3,3,3,3}
  675: {2,2,2,3,3}
  729: {2,2,2,2,2,2}
  961: {11,11}
For example, 675 = prime(2)^3 prime(3)^2 = 3^3 * 5^2.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Powerful numbers are A001694, counted by A007690.

The version for prime exponents instead of indices is A056166, counted by A055923.

This is the powerful case of A076610 (products of A006450), counted by A000607.

The partitions with these Heinz numbers are counted by A339218.

A000040 lists primes.

A031368 lists primes of odd index, products A066208.

A101436 counts exponents in prime factorization that are themselves prime.

A112798 lists prime indices, reverse A296150, sum A056239.

A124010 gives prime signature, sorted A118914, length A001221, sum A001222.

A053810 lists all numbers p^q with p and q prime, counted by A230595.

A257994 counts prime indices that are themselves prime, complement A330944.

Cf. A000720, A000961, A001597, A007821, A007916, A052485, A109297, A140319, A164336, A181819, A325131, A330945, A346068.

Programs

Mathematica

Select[Range[1000],#==1||And@@PrimeQ/@PrimePi/@First/@FactorInteger[#]&&Min@@Last/@FactorInteger[#]>1&]

Formula

Intersection of A001694 and A076610.

Sum_{n>=1} 1/a(n) = Product_{p in A006450} (1 + 1/(p*(p-1))) = 1.24410463... - Amiram Eldar, May 04 2022