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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).

%I A355533 #13 Jul 14 2022 17:23:27
%S A355533 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 A355533 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 A355533 15,0,0,0,1,0,2,0
%N 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).
%C A355533 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 A355533 The version where zero is prepended to the prime indices before taking differences is A287352.
%C A355533 One could argue that row n = 1 is empty, but adding it changes only the offset, with no effect on the data.
%H A355533 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts</a>.
%F A355533 Row lengths are 1 or A001222(n) - 1 depending on whether n is prime.
%e A355533 Triangle begins (showing n, prime indices, differences*):
%e A355533    2:    (1)       1
%e A355533    3:    (2)       2
%e A355533    4:   (1,1)      0
%e A355533    5:    (3)       3
%e A355533    6:   (1,2)      1
%e A355533    7:    (4)       4
%e A355533    8:  (1,1,1)    0 0
%e A355533    9:   (2,2)      0
%e A355533   10:   (1,3)      2
%e A355533   11:    (5)       5
%e A355533   12:  (1,1,2)    0 1
%e A355533   13:    (6)       6
%e A355533   14:   (1,4)      3
%e A355533   15:   (2,3)      1
%e A355533   16: (1,1,1,1)  0 0 0
%e A355533 For example, the prime indices of 24 are (1,1,1,2), with differences (0,0,1).
%t A355533 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A355533 Table[If[PrimeQ[n],{PrimePi[n]},Differences[primeMS[n]]],{n,2,30}]
%Y A355533 Crossrefs found in the link are not repeated here.
%Y A355533 Row sums are A243056.
%Y A355533 The version for prime indices prepended by 0 is A287352.
%Y A355533 Constant rows have indices A325328.
%Y A355533 Strict rows have indices A325368.
%Y A355533 Number of distinct terms in each row are 1 if prime, otherwise A355523.
%Y A355533 Row minima are A355525, augmented A355531.
%Y A355533 Row maxima are A355526, augmented A355535.
%Y A355533 The augmented version is A355534, Heinz number A325351.
%Y A355533 The version with prime-indexed rows empty is A355536, Heinz number A325352.
%Y A355533 A112798 lists prime indices, sum A056239.
%Y A355533 Cf. A001222, A066312, A124010, A286469, A286470, A325160, A325390, A355524.
%K A355533 nonn,tabf
%O A355533 2,2
%A A355533 _Gus Wiseman_, Jul 12 2022