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A325130
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Numbers in whose prime factorization the exponent of prime(k) is not equal to k for any prime index k.
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10
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1, 3, 4, 5, 7, 8, 11, 12, 13, 15, 16, 17, 19, 20, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 35, 37, 39, 40, 41, 43, 44, 47, 48, 49, 51, 52, 53, 55, 56, 57, 59, 60, 61, 64, 65, 67, 68, 69, 71, 73, 75, 76, 77, 79, 80, 81, 83, 84, 85, 87, 88, 89, 91, 92, 93, 95, 96
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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.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of the integer partitions counted by A276429.
The asymptotic density of this sequence is Product_{k>=1} (1 - 1/prime(k)^k + 1/prime(k)^(k+1)) = 0.68974964705635552968... - Amiram Eldar, Jan 09 2021
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LINKS
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EXAMPLE
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The sequence of terms together with their prime indices begins:
1: {}
3: {2}
4: {1,1}
5: {3}
7: {4}
8: {1,1,1}
11: {5}
12: {1,1,2}
13: {6}
15: {2,3}
16: {1,1,1,1}
17: {7}
19: {8}
20: {1,1,3}
21: {2,4}
23: {9}
24: {1,1,1,2}
25: {3,3}
27: {2,2,2}
28: {1,1,4}
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MAPLE
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q:= n-> andmap(i-> numtheory[pi](i[1])<>i[2], ifactors(n)[2]):
a:= proc(n) option remember; local k; for k from 1+
`if`(n=1, 0, a(n-1)) while not q(k) do od; k
end:
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MATHEMATICA
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Select[Range[100], And@@Cases[If[#==1, {}, FactorInteger[#]], {p_, k_}:>k!=PrimePi[p]]&]
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CROSSREFS
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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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