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A325373
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Composite totally abnormal numbers. Heinz numbers of non-singleton totally abnormal integer partitions.
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1
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9, 25, 27, 49, 81, 100, 121, 125, 169, 196, 225, 243, 289, 343, 361, 441, 484, 529, 625, 676, 729, 841, 961, 1000, 1089, 1156, 1225, 1331, 1369, 1444, 1521, 1681, 1764, 1849, 2116, 2187, 2197, 2209, 2401, 2601, 2744, 2809, 3025, 3125, 3249, 3364, 3375, 3481
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OFFSET
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1,1
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COMMENTS
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The first term that is not a perfect power (A001597) is 11880, with prime indices {1,1,1,2,2,2,3,5} and prime signature {1,1,3,3}.
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. A number n is totally abnormal iff (1) the prime indices of n do not cover an initial interval of positive integers, and either (2a) n is prime, or (2b) the prime exponents (or prime signature) of n forms a totally abnormal integer partition, or, equivalently to (2b), A181819(n) is totally abnormal.
The enumeration of totally abnormal integer partitions by sum is given by A325332.
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LINKS
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EXAMPLE
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The sequence of terms together with their prime indices begins:
9: {2,2}
25: {3,3}
27: {2,2,2}
49: {4,4}
81: {2,2,2,2}
100: {1,1,3,3}
121: {5,5}
125: {3,3,3}
169: {6,6}
196: {1,1,4,4}
225: {2,2,3,3}
243: {2,2,2,2,2}
289: {7,7}
343: {4,4,4}
361: {8,8}
441: {2,2,4,4}
484: {1,1,5,5}
529: {9,9}
625: {3,3,3,3}
676: {1,1,6,6}
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MATHEMATICA
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normQ[n_Integer]:=Or[n==1, PrimePi/@First/@FactorInteger[n]==Range[PrimeNu[n]]];
totabnQ[n_]:=And[!normQ[n], PrimeQ[n]||totabnQ[Times@@Prime/@Last/@If[n==1, {}, FactorInteger[n]]]];
Select[Range[10000], !PrimeQ[#]&&totabnQ[#]&]
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CROSSREFS
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Cf. A001597, A055932, A056239, A112798, A181819, A317089, A317090, A317246, A319152, A319810, A325332, A325370, A325372.
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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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