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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.

%I A355536 #15 Nov 04 2022 19:24:27
%S A355536 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,
%T A355536 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,
%U A355536 3,6,9,0,1,1,10,0,2,0,0,0,0,0,3,1,3,0,6
%N 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.
%C A355536 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 A355536 The version where zero is prepended to the prime indices is A287352.
%C A355536 One could argue that row n = 1 is empty, but adding it changes only the offset, not the data.
%H A355536 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%e A355536 Triangle begins (showing n, prime indices, differences*):
%e A355536    2:    (1)       .
%e A355536    3:    (2)       .
%e A355536    4:   (1,1)      0
%e A355536    5:    (3)       .
%e A355536    6:   (1,2)      1
%e A355536    7:    (4)       .
%e A355536    8:  (1,1,1)    0 0
%e A355536    9:   (2,2)      0
%e A355536   10:   (1,3)      2
%e A355536   11:    (5)       .
%e A355536   12:  (1,1,2)    0 1
%e A355536   13:    (6)       .
%e A355536   14:   (1,4)      3
%e A355536   15:   (2,3)      1
%e A355536   16: (1,1,1,1)  0 0 0
%t A355536 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A355536 Table[Differences[primeMS[n]],{n,2,100}]
%Y A355536 Row-lengths are A001222 minus one.
%Y A355536 The prime indices are A112798, sum A056239.
%Y A355536 Row-sums are A243055.
%Y A355536 Constant rows have indices A325328.
%Y A355536 The Heinz numbers of the rows plus one are A325352.
%Y A355536 Strict rows have indices A325368.
%Y A355536 Row minima are A355524.
%Y A355536 Row maxima are A286470, also A355526.
%Y A355536 An adjusted version is A358169, reverse A355534.
%Y A355536 Cf. A066312, A124010, A129654, A243056, A287352, A325394, A355523, A355528, A355531.
%K A355536 nonn,tabf
%O A355536 2,6
%A A355536 _Gus Wiseman_, Jul 12 2022