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

3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 100, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 196, 197, 199, 211, 223, 225, 227

Offset

1,1

Comments

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.

Links

Table of n, a(n) for n=1..59.

Example

The sequence of terms together with their prime indices are the following. See also the example at A325373.
    3: {2}
    5: {3}
    7: {4}
    9: {2,2}
   11: {5}
   13: {6}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   37: {12}
   41: {13}
   43: {14}
   47: {15}
   49: {4,4}
   53: {16}
   59: {17}

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[100], totabnQ]

Crossrefs

Cf. A055932, A056239, A112798, A181819, A317089, A317090, A317246 (supernormal), A317492 (fully normal), A317589 (uniformly normal), A319151, A325332, A325373.

Sequence in context: A371533 A112372 A082916 * A098903 A061345 A238266

Adjacent sequences: A325369 A325370 A325371 * A325373 A325374 A325375

Keyword

nonn

Author

Gus Wiseman, May 02 2019

Status

approved