A259154 -id:A259154

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A259153

Triangle T(n,k) in which n-th row lists in increasing order the values v whose pi-based arithmetic derivative equals n; n>=0, 1<=k<=A259154(n).

+20
4

0, 1, 2, 3, 5, 4, 7, 11, 13, 6, 17, 19, 23, 29, 10, 31, 8, 9, 37, 41, 43, 14, 47, 53, 59, 61, 15, 67, 12, 71, 22, 73, 79, 83, 89, 26, 97, 21, 101, 103, 107, 109, 25, 113, 34, 127, 16, 20, 131, 18, 137, 139, 38, 149, 151, 33, 157, 163, 167, 173, 35, 46, 179

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

OFFSET

0,3

LINKS

Alois P. Heinz, Rows n = 0..12000, flattened

FORMULA

A258851(T(n,k)) = n.

T(n,1) = A258861(n).

T(n,A259154(n)) = A008578(n+1).

T(n,A259154(n)) = A000040(n) for n>0.

EXAMPLE

Triangle T(n,k) begins:

0, 1;

4, 7;

11;

13;

6, 17;

19;

23;

29;

10, 31;

8, 9, 37;

CROSSREFS

Column k=1 gives A258861.

Last elements of rows give A008578(n+1).

Row lengths give A259154.

Row sums give A259155.

Cf. A000040, A000720, A258851.

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, Jun 19 2015

STATUS

approved

A259155

Sum of all values v whose pi-based arithmetic derivative equals n.

+10
2

1, 2, 3, 5, 11, 11, 13, 23, 19, 23, 29, 41, 54, 41, 43, 61, 53, 59, 61, 82, 83, 95, 79, 83, 89, 123, 122, 103, 107, 109, 138, 161, 167, 155, 139, 187, 151, 190, 163, 167, 173, 260, 181, 191, 260, 197, 199, 211, 223, 285, 229, 233, 263, 333, 278, 308, 312, 269

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

OFFSET

0,2

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..20000

FORMULA

a(n) = Sum_{v>=0, A258851(v)=n} v.

a(n) = Sum_{k=1..A259154(n)} A259153(n,k).

a(n) >= A000040(n) for n>0.

CROSSREFS

Row sums of A259153.

Cf. A000040, A000720, A258851, A259154.