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A338552
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Non-powers of primes whose prime indices have a common divisor > 1.
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3
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21, 39, 57, 63, 65, 87, 91, 111, 115, 117, 129, 133, 147, 159, 171, 183, 185, 189, 203, 213, 235, 237, 247, 259, 261, 267, 273, 299, 301, 303, 305, 319, 321, 325, 333, 339, 351, 365, 371, 377, 387, 393, 399, 417, 427, 441, 445, 453, 477, 481, 489, 497, 507
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OFFSET
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1,1
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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.
Also Heinz numbers of non-constant, non-relatively prime partitions. The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.
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LINKS
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FORMULA
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EXAMPLE
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The sequence of terms together with their prime indices begins:
21: {2,4} 183: {2,18} 305: {3,18}
39: {2,6} 185: {3,12} 319: {5,10}
57: {2,8} 189: {2,2,2,4} 321: {2,28}
63: {2,2,4} 203: {4,10} 325: {3,3,6}
65: {3,6} 213: {2,20} 333: {2,2,12}
87: {2,10} 235: {3,15} 339: {2,30}
91: {4,6} 237: {2,22} 351: {2,2,2,6}
111: {2,12} 247: {6,8} 365: {3,21}
115: {3,9} 259: {4,12} 371: {4,16}
117: {2,2,6} 261: {2,2,10} 377: {6,10}
129: {2,14} 267: {2,24} 387: {2,2,14}
133: {4,8} 273: {2,4,6} 393: {2,32}
147: {2,4,4} 299: {6,9} 399: {2,4,8}
159: {2,16} 301: {4,14} 417: {2,34}
171: {2,2,8} 303: {2,26} 427: {4,18}
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MATHEMATICA
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Select[Range[100], !(#==1||PrimePowerQ[#]||GCD@@PrimePi/@First/@FactorInteger[#]==1)&]
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CROSSREFS
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A327685 allows nonprime prime powers.
A338330 is the coprime instead of relatively prime version.
A338554 counts the partitions with these Heinz numbers.
A000740 counts relatively prime compositions.
A051424 counts pairwise coprime or singleton partitions.
A108572 counts nontrivial periodic partitions, with Heinz numbers A001597.
A302696 gives the Heinz numbers of pairwise coprime partitions.
A327516 counts pairwise coprime partitions, with Heinz numbers A302696.
Cf. A000005, A000837, A007916, A056239, A112798, A289508, A302569, A302796, A318716, A327658, A328867, A328677, A338331, A338553.
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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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