cp's OEIS Frontend

A359361 Irregular triangle read by rows whose n-th row lists the partial sums of the integer partition with Heinz number n.

%I A359361 #13 Mar 31 2023 05:51:08
%S A359361 1,2,1,2,3,2,3,4,1,2,3,2,4,3,4,5,2,3,4,6,4,5,3,5,1,2,3,4,7,2,4,5,8,3,
%T A359361 4,5,4,6,5,6,9,2,3,4,5,3,6,6,7,2,4,6,4,5,6,10,3,5,6,11,1,2,3,4,5,5,7,
%U A359361 7,8,4,7,2,4,5,6,12,8,9,6,8,3,4,5,6,13
%N A359361 Irregular triangle read by rows whose n-th row lists the partial sums of the integer partition with Heinz number n.
%C A359361 The partial sums of a sequence (a, b, c, ...) are (a, a+b, a+b+c, ...).
%C A359361 The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The partition with Heinz number n is the reversed n-th row of A112798.
%e A359361 Triangle begins:
%e A359361    2: 1
%e A359361    3: 2
%e A359361    4: 1 2
%e A359361    5: 3
%e A359361    6: 2 3
%e A359361    7: 4
%e A359361    8: 1 2 3
%e A359361    9: 2 4
%e A359361   10: 3 4
%e A359361   11: 5
%e A359361   12: 2 3 4
%e A359361   13: 6
%e A359361   14: 4 5
%e A359361   15: 3 5
%e A359361   16: 1 2 3 4
%e A359361 For example, the integer partition with Heinz number 90 is (3,2,2,1), so row n = 90 is (3,5,7,8).
%p A359361 T:= n-> ListTools[PartialSums](sort([seq(numtheory
%p A359361        [pi](i[1])$i[2], i=ifactors(n)[2])], `>`))[]:
%p A359361 seq(T(n), n=2..50);  # _Alois P. Heinz_, Jan 01 2023
%t A359361 Table[Accumulate[Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],{n,2,30}]
%Y A359361 Row-lengths are A001222.
%Y A359361 The version for standard compositions is A048793, non-reversed A358134.
%Y A359361 Last element in each row is A056239.
%Y A359361 First element in each row is A061395
%Y A359361 Rows are the partial sums of rows of A296150.
%Y A359361 Row-sums are A304818.
%Y A359361 A reverse version is A358136, row sums A318283, Heinz numbers A358137.
%Y A359361 The sorted Heinz numbers of rows are A359397.
%Y A359361 A000041 counts partitions, strict A000009.
%Y A359361 A112798 lists prime indices, product A003963.
%Y A359361 A355536 lists differences of prime indices.
%Y A359361 Cf. A000720, A001221, A055396, A261079, A325362.
%K A359361 nonn,tabf
%O A359361 2,2
%A A359361 _Gus Wiseman_, Dec 30 2022