A355536 Irregular triangle read by rows where row n lists the differences between adjacent prime indices of n; if n is prime, row n is empty.

0, 1, 0, 0, 0, 2, 0, 1, 3, 1, 0, 0, 0, 1, 0, 0, 2, 2, 4, 0, 0, 1, 0, 5, 0, 0, 0, 3, 1, 1, 0, 0, 0, 0, 3, 6, 1, 0, 1, 0, 7, 4, 0, 0, 2, 1, 2, 0, 4, 0, 1, 8, 0, 0, 0, 1, 0, 2, 0, 5, 0, 5, 1, 0, 0, 2, 0, 0, 3, 6, 9, 0, 1, 1, 10, 0, 2, 0, 0, 0, 0, 0, 3, 1, 3, 0, 6

Offset

2,6

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.

The version where zero is prepended to the prime indices is A287352.

One could argue that row n = 1 is empty, but adding it changes only the offset, not the data.

Links

Table of n, a(n) for n=2..88.

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

Example

Triangle begins (showing n, prime indices, differences*):
   2:    (1)       .
   3:    (2)       .
   4:   (1,1)      0
   5:    (3)       .
   6:   (1,2)      1
   7:    (4)       .
   8:  (1,1,1)    0 0
   9:   (2,2)      0
  10:   (1,3)      2
  11:    (5)       .
  12:  (1,1,2)    0 1
  13:    (6)       .
  14:   (1,4)      3
  15:   (2,3)      1
  16: (1,1,1,1)  0 0 0

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Differences[primeMS[n]], {n, 2, 100}]

Crossrefs

Row-lengths are A001222 minus one.

The prime indices are A112798, sum A056239.

Row-sums are A243055.

Constant rows have indices A325328.

The Heinz numbers of the rows plus one are A325352.

Strict rows have indices A325368.

Row minima are A355524.

Row maxima are A286470, also A355526.

An adjusted version is A358169, reverse A355534.

Cf. A066312, A124010, A129654, A243056, A287352, A325394, A355523, A355528, A355531.

Sequence in context: A089994 A178107 A272472 * A100260 A165317 A174067

Adjacent sequences: A355533 A355534 A355535 * A355537 A355538 A355539

Keyword

nonn,tabf

Author

Gus Wiseman, Jul 12 2022

Status

approved