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A352492
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Powerful numbers whose prime indices are all prime numbers.
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7
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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Sum_{n>=1} 1/a(n) = Product_{p in A006450} (1 + 1/(p*(p-1))) = 1.24410463... - Amiram Eldar, May 04 2022
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EXAMPLE
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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.
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MATHEMATICA
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Select[Range[1000], #==1||And@@PrimeQ/@PrimePi/@First/@FactorInteger[#]&&Min@@Last/@FactorInteger[#]>1&]
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CROSSREFS
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The version for prime exponents instead of indices is A056166, counted by A055923.
The partitions with these Heinz numbers are counted by A339218.
A101436 counts exponents in prime factorization that are themselves prime.
A257994 counts prime indices that are themselves prime, complement A330944.
Cf. A000720, A000961, A001597, A007821, A007916, A052485, A109297, A140319, A164336, A181819, A325131, A330945, A346068.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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