A355536 - OEIS

%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