A355533 Irregular triangle read by rows where row n lists the differences between adjacent prime indices of n; if n is prime(k), then row n is just (k).

1, 2, 0, 3, 1, 4, 0, 0, 0, 2, 5, 0, 1, 6, 3, 1, 0, 0, 0, 7, 1, 0, 8, 0, 2, 2, 4, 9, 0, 0, 1, 0, 5, 0, 0, 0, 3, 10, 1, 1, 11, 0, 0, 0, 0, 3, 6, 1, 0, 1, 0, 12, 7, 4, 0, 0, 2, 13, 1, 2, 14, 0, 4, 0, 1, 8, 15, 0, 0, 0, 1, 0, 2, 0

Offset: 2

Author

Gus Wiseman, Jul 12 2022

Keywords

• nonn
• tabf

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 before taking differences is A287352.

One could argue that row n = 1 is empty, but adding it changes only the offset, with no effect on the data.

Examples

			Triangle begins (showing n, prime indices, differences*):
   2:    (1)       1
   3:    (2)       2
   4:   (1,1)      0
   5:    (3)       3
   6:   (1,2)      1
   7:    (4)       4
   8:  (1,1,1)    0 0
   9:   (2,2)      0
  10:   (1,3)      2
  11:    (5)       5
  12:  (1,1,2)    0 1
  13:    (6)       6
  14:   (1,4)      3
  15:   (2,3)      1
  16: (1,1,1,1)  0 0 0
For example, the prime indices of 24 are (1,1,1,2), with differences (0,0,1).
		

Links

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

Crossrefs

Crossrefs found in the link are not repeated here.

Row sums are A243056.

The version for prime indices prepended by 0 is A287352.

Constant rows have indices A325328.

Strict rows have indices A325368.

Number of distinct terms in each row are 1 if prime, otherwise A355523.

Row minima are A355525, augmented A355531.

Row maxima are A355526, augmented A355535.

The augmented version is A355534, Heinz number A325351.

The version with prime-indexed rows empty is A355536, Heinz number A325352.

A112798 lists prime indices, sum A056239.

Cf. A001222, A066312, A124010, A286469, A286470, A325160, A325390, A355524.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[PrimeQ[n],{PrimePi[n]},Differences[primeMS[n]]],{n,2,30}]

Formula

Row lengths are 1 or A001222(n) - 1 depending on whether n is prime.