cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 15 results. Next

A258851 The pi-based arithmetic derivative of n: a(p) = pi(p) for p prime, a(u*v) = a(u)*v + u*a(v), where pi = A000720.

Original entry on oeis.org

0, 0, 1, 2, 4, 3, 7, 4, 12, 12, 11, 5, 20, 6, 15, 19, 32, 7, 33, 8, 32, 26, 21, 9, 52, 30, 25, 54, 44, 10, 53, 11, 80, 37, 31, 41, 84, 12, 35, 44, 84, 13, 73, 14, 64, 87, 41, 15, 128, 56, 85, 55, 76, 16, 135, 58, 116, 62, 49, 17, 136, 18, 53, 120, 192, 69, 107
Offset: 0

Views

Author

Alois P. Heinz, Jun 12 2015

Keywords

Comments

The pi-based variant of the arithmetic derivative of n (A003415).

Links

Crossrefs

Column k=1 of A258850, A258997.
First differences give A258863.
Partial sums give A258864.
Cf. A000720 (pi), A003415, A020639, A032742, A055396, A056239, A258861, A258862, A258995, A278510.

Programs

  • Maple
    with(numtheory):
a:= n-> n*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]):
seq(a(n), n=0..100);
  • Mathematica
    a[n_] := n*Total[Last[#]*PrimePi[First[#]]/First[#]& /@ FactorInteger[n]]; a[0] = 0; Array[a, 100, 0] (* Jean-François Alcover, Apr 24 2016 *)
  • PARI
    A258851(n)=n*sum(i=1, #n=factor(n)~, n[2, i]*primepi(n[1, i])/n[1, i]) \\ M. F. Hasler, Jul 13 2015
  • Scheme
    (define (A258851 n) (if (<= n 1) 0 (+ (* (A055396 n) (A032742 n)) (* (A020639 n) (A258851 (A032742 n)))))) ;; Antti Karttunen, Mar 07 2017

Formula

a(n) = n * Sum e_j*pi(p_j)/p_j for n = Product p_j^e_j with pi = A000720.
a(A258861(n)) = n; A258861 = pi-based antiderivative of n.
a(a(A258862(n))) = n; A258862 = second pi-based antiderivative of n.
a(a(a(A258995(n)))) = n; A258995 = third pi-based antiderivative of n.
a(0) = a(0*p) = a(0)*p + 0*a(p) = a(0)*p for all p => a(0) = 0.
a(p) = a(1*p) = a(1)*p + 1*a(p) = a(1)*p + a(p) for all p => a(1) = 0.
a(u^v) = v * u^(v-1) * a(u).

A259016 A(n,k) = k-th pi-based antiderivative of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 0, 3, 3, 3, 0, 5, 5, 5, 4, 0, 11, 11, 11, 4, 5, 0, 10, 10, 10, 4, 11, 6, 0, 29, 29, 29, 4, 10, 13, 7, 0, 78, 78, 78, 4, 29, 41, 6, 8, 0, 141, 141, 141, 4, 78, 35, 13, 19, 9, 0, 266, 266, 266, 4, 141, 38, 41, 15, 23, 10, 0, 147, 147, 147, 4, 266, 163, 35, 14, 83, 29, 11
Offset: 0

Views

Author

Alois P. Heinz, Jun 16 2015

Keywords

Examples

			A(5,3) = 29 -> 10 -> 11 -> 5.
A(5,4) = 78 -> 127 -> 31 -> 11 -> 5.
Square array A(n,k) begins:
  0,  0,  0,   0,    0,     0,     0,      0,     0,      0, ...
  1,  2,  3,   5,   11,    10,    29,     78,   141,    266, ...
  2,  3,  5,  11,   10,    29,    78,    141,   266,    147, ...
  3,  5, 11,  10,   29,    78,   141,    266,   147,    194, ...
  4,  4,  4,   4,    4,     4,     4,      4,     4,      4, ...
  5, 11, 10,  29,   78,   141,   266,    147,   194,   1181, ...
  6, 13, 41,  35,   38,   163,   138,    253,   346,   1383, ...
  7,  6, 13,  41,   35,    38,   163,    138,   253,    346, ...
  8, 19, 15,  14,   43,   191,   201,    217,  1113,   1239, ...
  9, 23, 83, 431, 3001, 27457, 10626, 112087, 87306, 172810, ...

Links

Crossrefs

Columns k=0-3 give: A001477, A258861, A258862, A258995.
Rows n=0,1,4,7,8,9 give: A000004, A258975, A010709, A259168, A259169, A259170.
Cf. A000040, A000720, A258850, A258851.

Programs

  • Maple
    with(numtheory):
d:= n-> n*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]):
A:= proc() local t, A; t, A:= proc()-1 end, proc()-1 end;
      proc(n, k) local h;
        while A(n, k) = -1 do
          t(k):= t(k)+1; h:= (d@@k)(t(k));
          if A(h, k) = -1 then A(h, k):= t(k) fi
        od; A(n, k)
      end
    end():
seq(seq(A(n, h-n), n=0..h), h=0..12);
  • Mathematica
    d[n_] := If[n == 0, 0, n*Total[Last[#]*PrimePi[First[#]]/First[#]& /@ FactorInteger[n]]];
A[n_, k_] := For[m = 0, True, m++, If[Nest[d, m, k] == n, Return[m]]];
Table[A[n, k-n], {k, 0, 12}, {n, 0, k}] // Flatten (* Jean-François Alcover, Mar 20 2017 *)

Formula

A(n,k) = min { m >= 0 : A258851^k(m) = n }.
A258850(A(n,k),k) = n.
A(n,k) <= A000040^k(n) for n>0.

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

Views

Author

Alois P. Heinz, Jun 12 2015

Keywords

Links

Crossrefs

Column k=2 of A258850.
Cf. A000720, A258851, A258862.

Programs

  • Maple
    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);

Formula

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

Views

Author

Alois P. Heinz, Jun 12 2015

Keywords

Links

Crossrefs

Column k=3 of A258850.
Cf. A000720, A258851, A258995.

Programs

  • Maple
    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);

Formula

a(n) = A258851^3(n).
a(A258995(n)) = n.

A258848 The n-th pi-based arithmetic derivative of 2^3.

Original entry on oeis.org

8, 12, 20, 32, 80, 208, 512, 2304, 12288, 81920, 622592, 4931584, 38944768, 278380544, 2122727424, 17483677696, 128412352512, 1348723408896, 14768867966976, 188484960780288, 2416519442792448, 30543291749302272, 375877192068366336, 6101345960934506496
Offset: 0

Views

Author

Alois P. Heinz, Jun 12 2015

Keywords

Crossrefs

Row n=8 of A258850.
Cf. A000720, A258851, A259169.

Programs

  • Maple
    with(numtheory):
d:= n-> n*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]):
a:= proc(n) option remember; `if`(n=0, 2^3, d(a(n-1))) end:
seq(a(n), n=0..23);

Formula

a(n) = A258851^n(2^3).

A258854 Fourth pi-based arithmetic derivative of n.

Original entry on oeis.org

0, 0, 0, 0, 4, 0, 4, 4, 80, 80, 2, 1, 208, 4, 12, 20, 512, 4, 20, 32, 512, 53, 30, 32, 324, 32, 16, 864, 704, 3, 80, 2, 2304, 32, 3, 7, 460, 80, 6, 704, 460, 4, 25, 8, 2432, 228, 7, 12, 6720, 332, 20, 56, 1188, 208, 3888, 116, 424, 32, 156, 4, 956, 12, 80, 764
Offset: 0

Views

Author

Alois P. Heinz, Jun 12 2015

Keywords

Links

Crossrefs

Column k=4 of A258850.
Cf. A000720, A258851.

Programs

  • Maple
    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, 4):
seq(a(n), n=0..100);

Formula

a(n) = A258851^4(n).

A258855 Fifth pi-based arithmetic derivative of n.

Original entry on oeis.org

0, 0, 0, 0, 4, 0, 4, 4, 208, 208, 1, 0, 512, 4, 20, 32, 2304, 4, 32, 80, 2304, 16, 53, 80, 1188, 80, 32, 3888, 2432, 2, 208, 1, 12288, 80, 2, 4, 916, 208, 7, 2432, 916, 4, 30, 12, 9536, 476, 4, 20, 32512, 424, 32, 116, 4104, 512, 20736, 156, 764, 80, 332, 4
Offset: 0

Views

Author

Alois P. Heinz, Jun 12 2015

Keywords

Links

Crossrefs

Column k=5 of A258850.
Cf. A000720, A258851.

Programs

  • Maple
    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, 5):
seq(a(n), n=0..100);

Formula

a(n) = A258851^5(n).

A258856 Sixth pi-based arithmetic derivative of n.

Original entry on oeis.org

0, 0, 0, 0, 4, 0, 4, 4, 512, 512, 0, 0, 2304, 4, 32, 80, 12288, 4, 80, 208, 12288, 32, 16, 208, 4104, 208, 80, 20736, 9536, 1, 512, 0, 81920, 208, 1, 4, 1116, 512, 4, 9536, 1116, 4, 53, 20, 30848, 944, 4, 32, 137984, 764, 80, 156, 16092, 2304, 138240, 332, 936
Offset: 0

Views

Author

Alois P. Heinz, Jun 12 2015

Keywords

Links

Crossrefs

Column k=6 of A258850.
Cf. A000720, A258851.

Programs

  • Maple
    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, 6):
seq(a(n), n=0..100);

Formula

a(n) = A258851^6(n).

A258857 Seventh pi-based arithmetic derivative of n.

Original entry on oeis.org

0, 0, 0, 0, 4, 0, 4, 4, 2304, 2304, 0, 0, 12288, 4, 80, 208, 81920, 4, 208, 512, 81920, 80, 32, 512, 16092, 512, 208, 138240, 30848, 0, 2304, 0, 622592, 512, 0, 4, 3000, 2304, 4, 30848, 3000, 4, 16, 32, 114752, 2160, 4, 80, 772352, 936, 208, 332, 52056, 12288
Offset: 0

Views

Author

Alois P. Heinz, Jun 12 2015

Keywords

Links

Crossrefs

Column k=7 of A258850.
Cf. A000720, A258851.

Programs

  • Maple
    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, 7):
seq(a(n), n=0..100);

Formula

a(n) = A258851^7(n).

A258858 Eighth pi-based arithmetic derivative of n.

Original entry on oeis.org

0, 0, 0, 0, 4, 0, 4, 4, 12288, 12288, 0, 0, 81920, 4, 208, 512, 622592, 4, 512, 2304, 622592, 208, 80, 2304, 52056, 2304, 512, 1050624, 114752, 0, 12288, 0, 4931584, 2304, 0, 4, 11900, 12288, 4, 114752, 11900, 4, 32, 80, 423168, 9936, 4, 208, 3679488, 3084
Offset: 0

Views

Author

Alois P. Heinz, Jun 12 2015

Keywords

Links

Crossrefs

Column k=8 of A258850.
Cf. A000720, A258851.

Programs

  • Maple
    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, 8):
seq(a(n), n=0..100);

Formula

a(n) = A258851^8(n).
Showing 1-10 of 15 results. Next

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