A319152 -id:A319152 - cp's OEIS frontend

A319149 Number of superperiodic integer partitions of n.

1, 1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 6, 1, 3, 3, 5, 1, 7, 1, 7, 3, 3, 1, 13, 2, 3, 4, 9, 1, 13, 1, 11, 3, 3, 3, 23, 1, 3, 3, 20, 1, 17, 1, 16, 9, 3, 1, 38, 2, 9, 3, 23, 1, 25, 3, 36, 3, 3, 1, 71, 1, 3, 11, 49, 3, 31, 1, 52, 3, 19

Offset: 1

Gus Wiseman, Sep 12 2018

nonn

An integer partition is superperiodic if either it consists of a single part equal to 1 or its parts have a common divisor > 1 and its multiset of multiplicities is itself superperiodic. For example, (8,8,6,6,4,4,4,4,2,2,2,2) has multiplicities (4,4,2,2) with multiplicities (2,2) with multiplicities (2) with multiplicities (1). The first four of these partitions are periodic and the last is (1), so (8,8,6,6,4,4,4,4,2,2,2,2) is superperiodic.

			The a(24) = 11 superperiodic partitions:
  (24)
  (12,12)
  (8,8,8)
  (9,9,3,3)
  (8,8,4,4)
  (6,6,6,6)
  (10,10,2,2)
  (6,6,6,2,2,2)
  (6,6,4,4,2,2)
  (4,4,4,4,4,4)
  (4,4,4,4,2,2,2,2)
  (3,3,3,3,3,3,3,3)
  (2,2,2,2,2,2,2,2,2,2,2,2)

Cf. A000041, A000837, A018783, A024994, A047966, A098859, A100953, A181819, A182857, A304660, A305563, A317245, A317256, A319151, A319152, A319157.

wotperQ[m_]:=Or[m=={1},And[GCD@@m>1,wotperQ[Sort[Length/@Split[Sort[m]]]]]];
Table[Length[Select[IntegerPartitions[n],wotperQ]],{n,30}]

A319151 Heinz numbers of superperiodic integer partitions.

Original entry on oeis.org

2, 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, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233

Offset: 1

Gus Wiseman, Sep 12 2018

nonn

First differs from A061345 at a(1) = 2 and next at a(98) = 441.
A number n is in the sequence iff n = 2 or the prime indices of n have a common divisor > 1 and the Heinz number of the multiset of prime multiplicities of n, namely A181819(n), is already in the sequence.
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The sequence of partitions whose Heinz numbers belong to the sequence begins: (1), (2), (3), (4), (2,2), (5), (6), (7), (8), (9), (3,3), (2,2,2), (10), (11), (12), (13), (14), (15), (4,4), (16), (17), (18), (19), (20), (21), (22), (2,2,2,2).

Vincenzo Librandi, Table of n, a(n) for n = 1..2326

Cf. A001597, A056239, A072774, A098859, A181819, A182850, A289509, A296150, A298748, A304464, A305732, A317246, A317257, A319149, A319152, A319157.

supperQ[n_]:=Or[n==2,And[GCD@@PrimePi/@FactorInteger[n][[All,1]]>1,supperQ[Times@@Prime/@FactorInteger[n][[All,2]]]]];
Select[Range[500],supperQ]

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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A319149 Number of superperiodic integer partitions of n.

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A319151 Heinz numbers of superperiodic integer partitions.

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

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