%I #13 Jul 14 2022 17:23:27
%S 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,
%T 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,
%U 15,0,0,0,1,0,2,0
%N 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).
%C 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.
%C The version where zero is prepended to the prime indices before taking differences is A287352.
%C One could argue that row n = 1 is empty, but adding it changes only the offset, with no effect on the data.
%H Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts</a>.
%F Row lengths are 1 or A001222(n) - 1 depending on whether n is prime.
%e Triangle begins (showing n, prime indices, differences*):
%e 2: (1) 1
%e 3: (2) 2
%e 4: (1,1) 0
%e 5: (3) 3
%e 6: (1,2) 1
%e 7: (4) 4
%e 8: (1,1,1) 0 0
%e 9: (2,2) 0
%e 10: (1,3) 2
%e 11: (5) 5
%e 12: (1,1,2) 0 1
%e 13: (6) 6
%e 14: (1,4) 3
%e 15: (2,3) 1
%e 16: (1,1,1,1) 0 0 0
%e For example, the prime indices of 24 are (1,1,1,2), with differences (0,0,1).
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Table[If[PrimeQ[n],{PrimePi[n]},Differences[primeMS[n]]],{n,2,30}]
%Y Crossrefs found in the link are not repeated here.
%Y Row sums are A243056.
%Y The version for prime indices prepended by 0 is A287352.
%Y Constant rows have indices A325328.
%Y Strict rows have indices A325368.
%Y Number of distinct terms in each row are 1 if prime, otherwise A355523.
%Y Row minima are A355525, augmented A355531.
%Y Row maxima are A355526, augmented A355535.
%Y The augmented version is A355534, Heinz number A325351.
%Y The version with prime-indexed rows empty is A355536, Heinz number A325352.
%Y A112798 lists prime indices, sum A056239.
%Y Cf. A001222, A066312, A124010, A286469, A286470, A325160, A325390, A355524.
%K nonn,tabf
%O 2,2
%A _Gus Wiseman_, Jul 12 2022