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A352492
Powerful numbers whose prime indices are all prime numbers.
7
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
1,2
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.
LINKS
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
EXAMPLE
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.
MATHEMATICA
Select[Range[1000], #==1||And@@PrimeQ/@PrimePi/@First/@FactorInteger[#]&&Min@@Last/@FactorInteger[#]>1&]
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.
Sequence in context: A340238 A020308 A108989 * A068583 A074852 A322177
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 24 2022
STATUS
approved