OFFSET
1,1
COMMENTS
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.
EXAMPLE
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}
MATHEMATICA
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[#]&]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 02 2019
STATUS
approved