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%I #6 May 02 2019 16:05:31
%S 9,25,27,49,81,100,121,125,169,196,225,243,289,343,361,441,484,529,
%T 625,676,729,841,961,1000,1089,1156,1225,1331,1369,1444,1521,1681,
%U 1764,1849,2116,2187,2197,2209,2401,2601,2744,2809,3025,3125,3249,3364,3375,3481
%N Composite totally abnormal numbers. Heinz numbers of non-singleton totally abnormal integer partitions.
%C 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}.
%C 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.
%C The enumeration of totally abnormal integer partitions by sum is given by A325332.
%e The sequence of terms together with their prime indices begins:
%e 9: {2,2}
%e 25: {3,3}
%e 27: {2,2,2}
%e 49: {4,4}
%e 81: {2,2,2,2}
%e 100: {1,1,3,3}
%e 121: {5,5}
%e 125: {3,3,3}
%e 169: {6,6}
%e 196: {1,1,4,4}
%e 225: {2,2,3,3}
%e 243: {2,2,2,2,2}
%e 289: {7,7}
%e 343: {4,4,4}
%e 361: {8,8}
%e 441: {2,2,4,4}
%e 484: {1,1,5,5}
%e 529: {9,9}
%e 625: {3,3,3,3}
%e 676: {1,1,6,6}
%t normQ[n_Integer]:=Or[n==1,PrimePi/@First/@FactorInteger[n]==Range[PrimeNu[n]]];
%t totabnQ[n_]:=And[!normQ[n],PrimeQ[n]||totabnQ[Times@@Prime/@Last/@If[n==1,{},FactorInteger[n]]]];
%t Select[Range[10000],!PrimeQ[#]&&totabnQ[#]&]
%Y Cf. A001597, A055932, A056239, A112798, A181819, A317089, A317090, A317246, A319152, A319810, A325332, A325370, A325372.
%K nonn
%O 1,1
%A _Gus Wiseman_, May 02 2019
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