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A325127
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Numbers in whose prime factorization the exponent of prime(k) is greater than k for all prime indices k.
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7
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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
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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 integer partitions where each part k appears more than k times. Such partitions are counted by A115584.
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LINKS
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FORMULA
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Sum_{n>=1} 1/a(n) = Product_{k>=1} 1 + 1/(prime(k)^k * (prime(k)-1)) = 1.58661114052385082598.... - Amiram Eldar, Oct 24 2020
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EXAMPLE
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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}
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MATHEMATICA
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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 *)
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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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