A325332 -id:A325332 - cp's OEIS frontend

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

Gus Wiseman, May 02 2019

nonn

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.

			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}

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

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]

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

Original entry on oeis.org

9, 25, 27, 49, 81, 100, 121, 125, 169, 196, 225, 243, 289, 343, 361, 441, 484, 529, 625, 676, 729, 841, 961, 1000, 1089, 1156, 1225, 1331, 1369, 1444, 1521, 1681, 1764, 1849, 2116, 2187, 2197, 2209, 2401, 2601, 2744, 2809, 3025, 3125, 3249, 3364, 3375, 3481

Offset: 1

Gus Wiseman, May 02 2019

nonn

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.

			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}

Cf. A001597, A055932, A056239, A112798, A181819, A317089, A317090, A317246, A319152, A319810, A325332, A325370, A325372.

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[#]&]

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

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