cp's OEIS Frontend

A358169 Row n lists the first differences plus one of the prime indices of n with 1 prepended.

%I A358169 #8 Nov 02 2022 11:08:01
%S A358169 1,2,1,1,3,1,2,4,1,1,1,2,1,1,3,5,1,1,2,6,1,4,2,2,1,1,1,1,7,1,2,1,8,1,
%T A358169 1,3,2,3,1,5,9,1,1,1,2,3,1,1,6,2,1,1,1,1,4,10,1,2,2,11,1,1,1,1,1,2,4,
%U A358169 1,7,3,2,1,1,2,1,12,1,8,2,5,1,1,1,3
%N A358169 Row n lists the first differences plus one of the prime indices of n with 1 prepended.
%C A358169 Every nonempty composition appears as a row exactly once.
%C A358169 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. Here this multiset is regarded as a sequence in weakly increasing order.
%C A358169 Also the reversed augmented differences of the integer partition with Heinz number n, where the augmented differences aug(q) of a sequence q of length k are given by aug(q)_i = q_i - q_{i+1} + 1 if i < k and aug(q)_k = q_k, and the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The non-reversed version is A355534.
%H A358169 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%e A358169 Triangle begins:
%e A358169    2: 1
%e A358169    3: 2
%e A358169    4: 1 1
%e A358169    5: 3
%e A358169    6: 1 2
%e A358169    7: 4
%e A358169    8: 1 1 1
%e A358169    9: 2 1
%e A358169   10: 1 3
%e A358169   11: 5
%e A358169   12: 1 1 2
%e A358169   13: 6
%e A358169   14: 1 4
%e A358169   15: 2 2
%e A358169   16: 1 1 1 1
%e A358169   17: 7
%e A358169   18: 1 2 1
%e A358169   19: 8
%e A358169   20: 1 1 3
%t A358169 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A358169 Table[Differences[Prepend[primeMS[n],1]]+1,{n,30}]
%Y A358169 Row-lengths are A001222.
%Y A358169 The first term of each row is A055396.
%Y A358169 Row-sums are A252464.
%Y A358169 The rows appear to be ranked by A253566.
%Y A358169 Another variation is A287352.
%Y A358169 Constant rows have indices A307824.
%Y A358169 The Heinz numbers of the rows are A325351.
%Y A358169 Strict rows have indices A325366.
%Y A358169 Row-minima are A355531, also A355524 and A355525.
%Y A358169 Row-maxima are A355532, non-augmented A286470, also A355526.
%Y A358169 Reversing rows gives A355534.
%Y A358169 The non-augmented version A355536, also A355533.
%Y A358169 A112798 lists prime indices, sum A056239.
%Y A358169 Cf. A124010, A243055, A243056, A325352, A325390, A355523, A355528.
%K A358169 nonn,tabf
%O A358169 2,2
%A A358169 _Gus Wiseman_, Nov 01 2022