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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.
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%I #15 Nov 04 2022 19:24:27

%S 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 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 3,6,9,0,1,1,10,0,2,0,0,0,0,0,3,1,3,0,6

%N 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 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 is A287352.

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

%H Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>

%e Triangle begins (showing n, prime indices, differences*):

%e 2: (1) .

%e 3: (2) .

%e 4: (1,1) 0

%e 5: (3) .

%e 6: (1,2) 1

%e 7: (4) .

%e 8: (1,1,1) 0 0

%e 9: (2,2) 0

%e 10: (1,3) 2

%e 11: (5) .

%e 12: (1,1,2) 0 1

%e 13: (6) .

%e 14: (1,4) 3

%e 15: (2,3) 1

%e 16: (1,1,1,1) 0 0 0

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

%t Table[Differences[primeMS[n]],{n,2,100}]

%Y Row-lengths are A001222 minus one.

%Y The prime indices are A112798, sum A056239.

%Y Row-sums are A243055.

%Y Constant rows have indices A325328.

%Y The Heinz numbers of the rows plus one are A325352.

%Y Strict rows have indices A325368.

%Y Row minima are A355524.

%Y Row maxima are A286470, also A355526.

%Y An adjusted version is A358169, reverse A355534.

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

%K nonn,tabf

%O 2,6

%A _Gus Wiseman_, Jul 12 2022