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A325800
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Numbers whose sum of prime indices is equal to the number of distinct subset-sums of their prime indices.
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3
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3, 10, 28, 66, 88, 156, 208, 306, 340, 408, 544, 570, 684, 760, 912, 966, 1216, 1242, 1288, 1380, 1656, 1840, 2208, 2436, 2610, 2900, 2944, 3132, 3248, 3480, 3906, 4092, 4176, 4340, 4640, 4650, 5022, 5208, 5456, 5568, 5580, 6200, 6696, 6944, 7326, 7424, 7440
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OFFSET
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1,1
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COMMENTS
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First differs from A325793 in lacking 70.
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, with sum A056239(n). A subset-sum of an integer partition is any sum of a submultiset of it.
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 whose sum is equal to their number of distinct subset-sums. The enumeration of these partitions by sum is given by A126796 interlaced with zeros.
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LINKS
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FORMULA
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EXAMPLE
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340 has prime indices {1,1,3,7} which sum to 12 and have 12 distinct subset-sums: {0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12}, so 340 is in the sequence.
The sequence of terms together with their prime indices begins:
3: {2}
10: {1,3}
28: {1,1,4}
66: {1,2,5}
88: {1,1,1,5}
156: {1,1,2,6}
208: {1,1,1,1,6}
306: {1,2,2,7}
340: {1,1,3,7}
408: {1,1,1,2,7}
544: {1,1,1,1,1,7}
570: {1,2,3,8}
684: {1,1,2,2,8}
760: {1,1,1,3,8}
912: {1,1,1,1,2,8}
966: {1,2,4,9}
1216: {1,1,1,1,1,1,8}
1242: {1,2,2,2,9}
1288: {1,1,1,4,9}
1380: {1,1,2,3,9}
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MATHEMATICA
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hwt[n_]:=Total[Cases[FactorInteger[n], {p_, k_}:>PrimePi[p]*k]];
Select[Range[1000], hwt[#]==Length[Union[hwt/@Divisors[#]]]&]
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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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