A258197 -id:A258197 - cp's OEIS frontend

A258318 Number of distinct numbers in rows 0 through n of triangle A258197.

1, 1, 2, 2, 4, 5, 7, 9, 12, 16, 20, 24, 29, 34, 41, 44, 50, 57, 66, 74, 83, 91, 102, 112, 124, 135, 148, 161, 174, 187, 201, 214, 230, 246, 263, 279, 296, 313, 331, 349, 369, 388, 408, 428, 450, 471, 494, 516, 540, 563, 588, 612, 637, 662, 689, 715, 741, 769

Offset: 0

Reinhard Zumkeller, May 26 2015

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A253197, A126256.

import Data.Set (singleton, insert, size)
a258318 n = a258318_list !! n
a258318_list = f 2 a258197_tabl $ singleton 0 where
   f k (xs:xss) zs = g (take (div k 2) xs) zs where
     g []     ys = size ys : f (k + 1) xss ys
     g (x:xs) ys = g xs (insert x ys)

A258317 Row sums of triangle A258197.

Original entry on oeis.org

0, 0, 1, 2, 13, 16, 50, 46, 331, 710, 1530, 1968, 6694, 8352, 15550, 24890, 133573, 181552, 486054, 641978, 1840816, 3296140, 6314768, 7452540, 34421218, 59313706, 110347662, 258695534, 627750404, 874913936, 2053073798, 2556088458, 12615058063, 22442790438

Offset: 0

Reinhard Zumkeller, May 26 2015

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A258197.

Haskell
```
a258317 = sum . a258197_row
```

f(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
a(n) = vecsum(vector(n+1, k, f(binomial(n,k-1)))); \\ Michel Marcus, Jan 23 2022

a(n) = Sum_{k=0..n} A258197(n,k).

A068312 Arithmetic derivative of triangular numbers.

Original entry on oeis.org

0, 0, 1, 5, 7, 8, 10, 32, 60, 39, 16, 61, 71, 20, 71, 244, 212, 111, 123, 143, 247, 131, 34, 380, 520, 155, 378, 621, 275, 247, 263, 1008, 1280, 271, 239, 951, 795, 56, 343, 1256, 1004, 431, 451, 581, 1443, 942, 70, 2092, 2492, 840

Offset: 0

Reinhard Zumkeller, Feb 25 2002

			a(7) = d(7*8/2) = d(28) = d(2*14) = d(2)*14 + 2*d(14) = 1*14 + 2*d(2*7) = 14 + 2*(2*d(7) + d(2)*7) = 14 + 2*(2*1 + 1*7) = 14 + 2*9 = 14 + 18 = 32, where d(n) = A003415(n) with d(1) = 0, d(prime) = 1 and d(m*n) = d(m)*n + m*d(n).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A000217, A003415, A007318, A258197.

a068312 = a003415 . a000217  -- Reinhard Zumkeller, May 26 2015

a[0] = a[1] = 0; a[n_] := (n*(n+1)/2) * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n*(n+1)/2]); Array[a, 100, 0] (* Amiram Eldar, May 14 2025 *)

from sympy import factorint
def A068312(n): return 0 if n <= 1 else ((n+1)*sum((n*e//p for p,e in factorint(n).items()))+ sum(((n+1)*e//p for p,e in factorint(n+1).items()))*n - (n*(n+1)//2))//2 # Chai Wah Wu, Jun 24 2022

a(n) = A003415(A000217(n)).

For n > 1: a(n) = A258197(n,2) = A258197(n,n-2). - Reinhard Zumkeller, May 26 2015

a(0)=0 prepended by Reinhard Zumkeller, May 26 2015

A258290 Arithmetic derivative of central binomial coefficients, cf. A000984.

Original entry on oeis.org

0, 1, 5, 24, 59, 456, 1448, 6868, 19749, 69364, 236356, 1526956, 3717440, 22858340, 122553540, 474051984, 954720543, 5726109024, 19329586520, 92051285020, 319059863484, 1271796704788, 4829219746964, 29791326914640, 74372011398840, 340296661452300

Offset: 0

Reinhard Zumkeller, May 26 2015

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A000984, A003415, A258197.

Haskell
```
a258290 = a003415 . a000984
```

ad[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); ad[0] = ad[1] = 0; a[n_] := ad[Binomial[2*n, n]]; Array[a, 26, 0] (* Amiram Eldar, Apr 13 2025 *)

a(n) = A003415(A000984(n)).

Central terms in triangle A258197: a(n) = A258197(2*n,n).

cp's OEIS Frontend

A258318 Number of distinct numbers in rows 0 through n of triangle A258197.

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A258317 Row sums of triangle A258197.

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A068312 Arithmetic derivative of triangular numbers.

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A258290 Arithmetic derivative of central binomial coefficients, cf. A000984.

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