cp's OEIS Frontend

A325373 Composite totally abnormal numbers. Heinz numbers of non-singleton totally abnormal integer partitions.

%I A325373 #6 May 02 2019 16:05:31
%S A325373 9,25,27,49,81,100,121,125,169,196,225,243,289,343,361,441,484,529,
%T A325373 625,676,729,841,961,1000,1089,1156,1225,1331,1369,1444,1521,1681,
%U A325373 1764,1849,2116,2187,2197,2209,2401,2601,2744,2809,3025,3125,3249,3364,3375,3481
%N A325373 Composite totally abnormal numbers. Heinz numbers of non-singleton totally abnormal integer partitions.
%C A325373 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 A325373 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 A325373 The enumeration of totally abnormal integer partitions by sum is given by A325332.
%e A325373 The sequence of terms together with their prime indices begins:
%e A325373      9: {2,2}
%e A325373     25: {3,3}
%e A325373     27: {2,2,2}
%e A325373     49: {4,4}
%e A325373     81: {2,2,2,2}
%e A325373    100: {1,1,3,3}
%e A325373    121: {5,5}
%e A325373    125: {3,3,3}
%e A325373    169: {6,6}
%e A325373    196: {1,1,4,4}
%e A325373    225: {2,2,3,3}
%e A325373    243: {2,2,2,2,2}
%e A325373    289: {7,7}
%e A325373    343: {4,4,4}
%e A325373    361: {8,8}
%e A325373    441: {2,2,4,4}
%e A325373    484: {1,1,5,5}
%e A325373    529: {9,9}
%e A325373    625: {3,3,3,3}
%e A325373    676: {1,1,6,6}
%t A325373 normQ[n_Integer]:=Or[n==1,PrimePi/@First/@FactorInteger[n]==Range[PrimeNu[n]]];
%t A325373 totabnQ[n_]:=And[!normQ[n],PrimeQ[n]||totabnQ[Times@@Prime/@Last/@If[n==1,{},FactorInteger[n]]]];
%t A325373 Select[Range[10000],!PrimeQ[#]&&totabnQ[#]&]
%Y A325373 Cf. A001597, A055932, A056239, A112798, A181819, A317089, A317090, A317246, A319152, A319810, A325332, A325370, A325372.
%K A325373 nonn
%O A325373 1,1
%A A325373 _Gus Wiseman_, May 02 2019