A258851 -id:A258851 - cp's OEIS frontend

A258997 A(n,k) = pi-based antiderivative of n^k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 4, 2, 0, 0, 0, 12, 12, 4, 0, 0, 0, 32, 54, 32, 3, 0, 0, 0, 80, 216, 192, 30, 7, 0, 0, 0, 192, 810, 1024, 225, 84, 4, 0, 0, 0, 448, 2916, 5120, 1500, 756, 56, 12, 0, 0, 0, 1024, 10206, 24576, 9375, 6048, 588, 192, 12, 0

Offset: 0

Alois P. Heinz, Jun 27 2015

			Square array A(n,k) begins:
  0, 0,  0,   0,    0,     0,      0,       0, ...
  0, 0,  0,   0,    0,     0,      0,       0, ...
  0, 1,  4,  12,   32,    80,    192,     448, ...
  0, 2, 12,  54,  216,   810,   2916,   10206, ...
  0, 4, 32, 192, 1024,  5120,  24576,  114688, ...
  0, 3, 30, 225, 1500,  9375,  56250,  328125, ...
  0, 7, 84, 756, 6048, 45360, 326592, 2286144, ...
  0, 4, 56, 588, 5488, 48020, 403368, 3294172, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Rows n=0+1,2,3,4,8 give: A000004, A001787, A212697, A018215, A230539.
Columns k=0,1 give: A000004, A258851.
Main diagonal gives A258846.
Cf. A000720.

with(numtheory):
d:= n-> n*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]):
A:= (n, k)-> `if`(k=0, 0, k*n^(k-1)*d(n)):
seq(seq(A(n, h-n), n=0..h), h=0..14);

A(n,k) = A258851(n^k) = k * n^(k-1) * A258851(n).

A328769 The second primorial based variant of arithmetic derivative: a(p) = A034386(p) for p prime, a(uv) = a(u)v + u*a(v), with a(0) = a(1) = 0.

Original entry on oeis.org

0, 0, 2, 6, 8, 30, 18, 210, 24, 36, 70, 2310, 48, 30030, 434, 120, 64, 510510, 90, 9699690, 160, 672, 4642, 223092870, 120, 300, 60086, 162, 896, 6469693230, 270, 200560490130, 160, 6996, 1021054, 1260, 216, 7420738134810, 19399418, 90168, 360, 304250263527210, 1386, 13082761331670030, 9328, 450, 446185786, 614889782588491410, 288, 2940, 650, 1531632

Offset: 0

Antti Karttunen, Oct 28 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..1001
Index entries for sequences related to primorial numbers

Cf. A002110, A034386, A108951, A328772.

Cf. also A003415, A258851, A328768, A328845, A328846.

A034386(n) = factorback(primes(primepi(n)));
A328769(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*A034386(f[i,1])/f[i, 1]));

A002110(n) = prod(i=1,n,prime(i));
A328769(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*A002110(primepi(f[i,1]))/f[i, 1]));

a(n) = n * Sum e_j * (p_j)#/p_j for n = Product p_j^e_j with (p_j)# = A034386(p_j).

A276150(a(n)) = A328772(n).

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).

Original entry on oeis.org

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

Offset: 0

Alois P. Heinz, Jun 19 2015

			Triangle T(n,k) begins:
   0,  1;
   2;
   3;
   5;
   4,  7;
  11;
  13;
   6, 17;
  19;
  23;
  29;
  10, 31;
   8,  9, 37;

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

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.

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.

A353379 Primepi-based variant of the arithmetic derivative applied to the prime shadow of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 3, 2, 4, 1, 7, 1, 4, 4, 4, 1, 7, 1, 7, 4, 4, 1, 11, 2, 4, 3, 7, 1, 12, 1, 5, 4, 4, 4, 12, 1, 4, 4, 11, 1, 12, 1, 7, 7, 4, 1, 15, 2, 7, 4, 7, 1, 11, 4, 11, 4, 4, 1, 20, 1, 4, 7, 6, 4, 12, 1, 7, 4, 12, 1, 19, 1, 4, 7, 7, 4, 12, 1, 15, 4, 4, 1, 20, 4, 4, 4, 11, 1, 20, 4, 7, 4, 4, 4, 21, 1, 7, 7

Offset: 1

Antti Karttunen, Apr 28 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n

Cf. A181819, A258851.

Cf. also A351942.

a:= n-> (m-> m*add(i[2]*numtheory[pi](i[1])/i[1], i=ifactors(m)[2]))
        (mul(ithprime(i[2]), i=ifactors(n)[2])):
seq(a(n), n=1..120);  # Alois P. Heinz, Apr 28 2022

a[n_] := If[n == 1, 0, #*Sum[i[[2]]*PrimePi[i[[1]]]/i[[1]], {i, FactorInteger[#]}]]&[Product[Prime[i[[2]]], {i, FactorInteger[n]}]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 29 2025, after Alois P. Heinz *)

A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A258851(n) = (n*sum(i=1, #n=factor(n)~, n[2, i]*primepi(n[1, i])/n[1, i])); \\ From A258851
A353379(n) = A258851(A181819(n));

a(n) = A258851(A181819(n)).

A097240 a(n) = (n+1)prime(n) + nprime(n+1).

Original entry on oeis.org

7, 19, 41, 79, 131, 193, 269, 355, 491, 629, 779, 973, 1133, 1303, 1547, 1845, 2099, 2365, 2689, 2951, 3265, 3643, 4039, 4553, 5047, 5405, 5773, 6155, 6547, 7313, 8125, 8707, 9245, 9931, 10649, 11239, 11997, 12703, 13427, 14253, 14939, 15805, 16703

Offset: 1

Reinhard Zumkeller, Aug 02 2004

nonn

Of the first 10000 terms, 1540 are primes. - Antti Karttunen, Jul 30 2022

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from indices in prime factorization

Cf. A000040, A006094, A097241 (primes in this sequence), A258851.

Column 3 of A356155.

Table[(n+1)Prime[n]+n*Prime[n+1],{n,50}] (* Harvey P. Dale, Oct 02 2017 *)

A097240(n) = ((n+1)*prime(n) + n*prime(n+1)); \\ Antti Karttunen, Jul 30 2022

a(n) = A258851(A006094(n)). - Antti Karttunen, Jul 30 2022

A258852 Second pi-based arithmetic derivative of n.

Original entry on oeis.org

0, 0, 0, 1, 4, 2, 4, 4, 20, 20, 5, 3, 32, 7, 19, 8, 80, 4, 37, 12, 80, 25, 26, 12, 76, 53, 30, 135, 64, 11, 16, 5, 208, 12, 11, 13, 188, 20, 41, 64, 188, 6, 21, 15, 192, 88, 13, 19, 448, 116, 86, 58, 108, 32, 351, 49, 156, 53, 56, 7, 260, 33, 16, 332, 704, 73

Offset: 0

Alois P. Heinz, Jun 12 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Column k=2 of A258850.

Cf. A000720, A258851, A258862.

with(numtheory):
d:= n-> n*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]):
A:= proc(n, k) option remember; `if`(k=0, n, d(A(n, k-1))) end:
a:= n-> A(n, 2):
seq(a(n), n=0..100);

a(n) = A258851^2(n).

a(A258862(n)) = n.

A258853 Third pi-based arithmetic derivative of n.

Original entry on oeis.org

0, 0, 0, 0, 4, 1, 4, 4, 32, 32, 3, 2, 80, 4, 8, 12, 208, 4, 12, 20, 208, 30, 25, 20, 108, 16, 53, 351, 192, 5, 32, 3, 512, 20, 5, 6, 248, 32, 13, 192, 248, 7, 26, 19, 704, 172, 6, 8, 1600, 156, 71, 49, 324, 80, 864, 56, 332, 16, 116, 4, 536, 37, 32, 424, 2432

Offset: 0

Alois P. Heinz, Jun 12 2015

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Column k=3 of A258850.

Cf. A000720, A258851, A258995.

with(numtheory):
d:= n-> n*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]):
A:= proc(n, k) option remember; `if`(k=0, n, d(A(n, k-1))) end:
a:= n-> A(n, 3):
seq(a(n), n=0..100);

a(n) = A258851^3(n).

a(A258995(n)) = n.

A259154 Number of values v whose pi-based arithmetic derivative equals n.

Original entry on oeis.org

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

Offset: 0

Alois P. Heinz, Jun 19 2015

nonn

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

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

Row lengths of A259153.

a(n) = |{ v >= 0 : A258851(v) = n }|.

A120272 Numerator of Sum_{k=1..n} k/prime(k).

Original entry on oeis.org

0, 1, 7, 53, 491, 6451, 97723, 1871501, 39642599, 999076987, 31204161323, 1038495626543, 40831064063651, 1770543222362221, 80392862250956443, 3974705945770003271, 220497651647226035923, 13563377141298566879867, 861975691921988407175147, 59980850604601955729416979

Offset: 0

Alexander Adamchuk, Jul 01 2006

Harvey P. Dale, Table of n, a(n) for n = 0..349

Denominators are in A002110.

[0] cat [Numerator((&+[k/NthPrime(k): k in [1..n]])): n in [1..20]]; // G. C. Greubel, Aug 23 2018

Numerator[Table[Sum[i/Prime[i],{i,1,n}],{n,1,20}]]
Numerator[Accumulate[Table[n/Prime[n],{n,20}]]] (* Harvey P. Dale, Jan 21 2013 *)

for(n=0,20, print1(if(n==0, 0, numerator(sum(k=1,n, k/prime(k)))), ", ")) \\ G. C. Greubel, Aug 23 2018

a(n) = A258851(A002110(n)). - Alois P. Heinz, Jun 26 2015

a(0)=0 prepended by Alois P. Heinz, Jun 26 2015

A258845 The pi-based arithmetic derivative of n!.

Original entry on oeis.org

0, 0, 1, 7, 52, 332, 2832, 22704, 242112, 2662848, 30620160, 354965760, 5057925120, 68627036160, 1054183818240, 17469144806400, 321351896678400, 5609441772748800, 112707637036646400, 2192664093342105600, 47745925079924736000, 1065919878891012096000

Offset: 0

Alois P. Heinz, Jun 12 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A000142, A000720, A258851, A259409.

with(numtheory):
d:= n-> n*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]):
a:= proc(n) option remember;
      `if`(n<2, 0, a(n-1)*n+(n-1)!*d(n))
    end:
seq(a(n), n=0..25);

d[n_] := n*Sum[i[[2]]*PrimePi[i[[1]]]/i[[1]], {i, FactorInteger[n]}];
a[n_] := a[n] = If[n < 2, 0, a[n-1]*n + (n-1)!*d[n]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 24 2017, translated from Maple *)

a(n) = A258851(n!) = A258851(A000142(n)).

cp's OEIS Frontend

A258997 A(n,k) = pi-based antiderivative of n^k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

A328769 The second primorial based variant of arithmetic derivative: a(p) = A034386(p) for p prime, a(u*v) = a(u)*v + u*a(v), with a(0) = a(1) = 0.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

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).

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A353379 Primepi-based variant of the arithmetic derivative applied to the prime shadow of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A097240 a(n) = (n+1)*prime(n) + n*prime(n+1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A258852 Second pi-based arithmetic derivative of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula

A258853 Third pi-based arithmetic derivative of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula

A259154 Number of values v whose pi-based arithmetic derivative equals n.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A120272 Numerator of Sum_{k=1..n} k/prime(k).

Views

Author

A328769 The second primorial based variant of arithmetic derivative: a(p) = A034386(p) for p prime, a(uv) = a(u)v + u*a(v), with a(0) = a(1) = 0.

A097240 a(n) = (n+1)prime(n) + nprime(n+1).