A358169 Row n lists the first differences plus one of the prime indices of n with 1 prepended.

1, 2, 1, 1, 3, 1, 2, 4, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 2, 6, 1, 4, 2, 2, 1, 1, 1, 1, 7, 1, 2, 1, 8, 1, 1, 3, 2, 3, 1, 5, 9, 1, 1, 1, 2, 3, 1, 1, 6, 2, 1, 1, 1, 1, 4, 10, 1, 2, 2, 11, 1, 1, 1, 1, 1, 2, 4, 1, 7, 3, 2, 1, 1, 2, 1, 12, 1, 8, 2, 5, 1, 1, 1, 3

Offset

2,2

Comments

Every nonempty composition appears as a row exactly once.

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. Here this multiset is regarded as a sequence in weakly increasing order.

Also the reversed augmented differences of the integer partition with Heinz number n, where the augmented differences aug(q) of a sequence q of length k are given by aug(q)_i = q_i - q_{i+1} + 1 if i < k and aug(q)_k = q_k, and the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The non-reversed version is A355534.

Links

Table of n, a(n) for n=2..85.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Example

Triangle begins:
   2: 1
   3: 2
   4: 1 1
   5: 3
   6: 1 2
   7: 4
   8: 1 1 1
   9: 2 1
  10: 1 3
  11: 5
  12: 1 1 2
  13: 6
  14: 1 4
  15: 2 2
  16: 1 1 1 1
  17: 7
  18: 1 2 1
  19: 8
  20: 1 1 3

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Differences[Prepend[primeMS[n], 1]]+1, {n, 30}]

Crossrefs

Row-lengths are A001222.

The first term of each row is A055396.

Row-sums are A252464.

The rows appear to be ranked by A253566.

Another variation is A287352.

Constant rows have indices A307824.

The Heinz numbers of the rows are A325351.

Strict rows have indices A325366.

Row-minima are A355531, also A355524 and A355525.

Row-maxima are A355532, non-augmented A286470, also A355526.

Reversing rows gives A355534.

The non-augmented version A355536, also A355533.

A112798 lists prime indices, sum A056239.

Cf. A124010, A243055, A243056, A325352, A325390, A355523, A355528.

Sequence in context: A198789 A134521 A131375 * A327392 A112798 A187846

Adjacent sequences: A358166 A358167 A358168 * A358170 A358171 A358172

Keyword

nonn,tabf

Author

Gus Wiseman, Nov 01 2022

Status

approved