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A342192
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Heinz numbers of partitions of crank 0.
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3
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6, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 100, 106, 118, 122, 134, 140, 142, 146, 158, 166, 178, 194, 196, 202, 206, 214, 218, 220, 226, 254, 260, 262, 274, 278, 298, 300, 302, 308, 314, 326, 334, 340, 346, 358, 362, 364, 380, 382, 386, 394, 398
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OFFSET
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1,1
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COMMENTS
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The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
See A257989 or the program for a definition of crank of a partition.
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LINKS
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Table of n, a(n) for n=1..55.
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EXAMPLE
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The sequence of terms together with their prime indices begins:
6: {1,2} 106: {1,16} 218: {1,29}
10: {1,3} 118: {1,17} 220: {1,1,3,5}
14: {1,4} 122: {1,18} 226: {1,30}
22: {1,5} 134: {1,19} 254: {1,31}
26: {1,6} 140: {1,1,3,4} 260: {1,1,3,6}
34: {1,7} 142: {1,20} 262: {1,32}
38: {1,8} 146: {1,21} 274: {1,33}
46: {1,9} 158: {1,22} 278: {1,34}
58: {1,10} 166: {1,23} 298: {1,35}
62: {1,11} 178: {1,24} 300: {1,1,2,3,3}
74: {1,12} 194: {1,25} 302: {1,36}
82: {1,13} 196: {1,1,4,4} 308: {1,1,4,5}
86: {1,14} 202: {1,26} 314: {1,37}
94: {1,15} 206: {1,27} 326: {1,38}
100: {1,1,3,3} 214: {1,28} 334: {1,39}
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
ck[y_]:=With[{w=Count[y, 1]}, If[w==0, Max@@y, Count[y, _?(#>w&)]-w]];
Select[Range[100], ck[primeMS[#]]==0&]
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CROSSREFS
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Indices of zeros in A257989.
A000005 counts constant partitions.
A000041 counts partitions (strict: A000009).
A001522 counts partitions of positive crank.
A003242 counts anti-run compositions.
A064391 counts partitions by crank.
A064428 counts partitions of nonnegative crank.
A224958 counts compositions with alternating parts unequal.
A257989 gives the crank of the partition with Heinz number n.
Cf. A000726, A008965, A056239, A112798, A124010, A130091, A325351, A325352.
Sequence in context: A080784 A072978 A055163 * A119431 A207574 A278972
Adjacent sequences: A342189 A342190 A342191 * A342193 A342194 A342195
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KEYWORD
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nonn
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AUTHOR
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Gus Wiseman, Apr 05 2021
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STATUS
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approved
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