cp's OEIS Frontend

A325372 Totally abnormal numbers. Heinz numbers of totally abnormal integer partitions.

%I A325372 #7 May 02 2019 16:05:24
%S A325372 3,5,7,9,11,13,17,19,23,25,27,29,31,37,41,43,47,49,53,59,61,67,71,73,
%T A325372 79,81,83,89,97,100,101,103,107,109,113,121,125,127,131,137,139,149,
%U A325372 151,157,163,167,169,173,179,181,191,193,196,197,199,211,223,225,227
%N A325372 Totally abnormal numbers. Heinz numbers of totally abnormal integer partitions.
%C A325372  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 A325372 The enumeration of totally abnormal integer partitions by sum is given by A325332.
%e A325372 The sequence of terms together with their prime indices are the following. See also the example at A325373.
%e A325372     3: {2}
%e A325372     5: {3}
%e A325372     7: {4}
%e A325372     9: {2,2}
%e A325372    11: {5}
%e A325372    13: {6}
%e A325372    17: {7}
%e A325372    19: {8}
%e A325372    23: {9}
%e A325372    25: {3,3}
%e A325372    27: {2,2,2}
%e A325372    29: {10}
%e A325372    31: {11}
%e A325372    37: {12}
%e A325372    41: {13}
%e A325372    43: {14}
%e A325372    47: {15}
%e A325372    49: {4,4}
%e A325372    53: {16}
%e A325372    59: {17}
%t A325372 normQ[n_Integer]:=Or[n==1,PrimePi/@First/@FactorInteger[n]==Range[PrimeNu[n]]];
%t A325372 totabnQ[n_]:=And[!normQ[n],PrimeQ[n]||totabnQ[Times@@Prime/@Last/@If[n==1,{},FactorInteger[n]]]];
%t A325372 Select[Range[100],totabnQ]
%Y A325372 Cf. A055932, A056239, A112798, A181819, A317089, A317090, A317246 (supernormal), A317492 (fully normal), A317589 (uniformly normal), A319151, A325332, A325373.
%K A325372 nonn
%O A325372 1,1
%A A325372 _Gus Wiseman_, May 02 2019