A339258 - OEIS

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A339258

Triangle read by rows T(n,k), (n >= 1, k > = 1), in which row n has length A000070(n-1) and every column gives A000005, the number of divisors function.

1, 2, 1, 2, 2, 1, 1, 3, 2, 2, 2, 1, 1, 1, 2, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 4, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 4, 2, 2, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 2, 4, 4, 2, 2, 2, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Conjecture: the sum of row n equals A006128(n), the total number of parts in all partitions of n.

LINKS

Paolo Xausa, Table of n, a(n) for n = 1..10980 (rows 1..21 of the triangle, flattened)

FORMULA

T(n,k) = A000005(A176206(n,k)).

EXAMPLE

Triangle begins:

2, 1;

2, 2, 1, 1;

3, 2, 2, 2, 1, 1, 1;

2, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1;

4, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1;

2, 4, 2, 2, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, ...

...

MATHEMATICA

A339258row[n_]:=Flatten[Table[ConstantArray[DivisorSigma[0, n-m], PartitionsP[m]], {m, 0, n-1}]]; Array[A339258row, 10] (* Paolo Xausa, Sep 02 2023 *)

PROG

(PARI) f(n) = sum(k=0, n-1, numbpart(k));

T(n, k) = {if (k > f(n), error("invalid k")); if (k==1, return (numdiv(n))); my(s=0); while (k <= f(n-1), s++; n--; ); numdiv(1+s); }

tabf(nn) = {for (n=1, nn, for (k=1, f(n), print1(T(n, k), ", "); ); print; ); } \\ Michel Marcus, Jan 13 2021

CROSSREFS

Row sums give A006128 (conjectured).

Cf. A000005, A000041, A176206, A221530, A221531, A337209.

Sequence in context: A304093 A238580 A035176 * A360326 A322976 A011793

Adjacent sequences: A339255 A339256 A339257 * A339259 A339260 A339261

KEYWORD

nonn,tabf

AUTHOR

Omar E. Pol, Nov 29 2020

STATUS

approved