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.

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A258846 The pi-based arithmetic derivative of n^n.

Original entry on oeis.org

0, 0, 4, 54, 1024, 9375, 326592, 3294172, 201326592, 4649045868, 110000000000, 1426558353055, 178322008965120, 1817250639553518, 166680102383370240, 8319983917236328125, 590295810358705651712, 5790681833204357349239, 1298431466484785739988992
Offset: 0

Views

Author

Alois P. Heinz, Jun 12 2015

Keywords

Links

Crossrefs

Cf. A000312, A000720, A258851.
Main diagonal of A258997.

Programs

  • Maple
    with(numtheory):
a:= n-> n^(n+1)*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]):
seq(a(n), n=0..20);
  • Mathematica
    a[n_] := n^(n+1)*Sum[i[[2]]*PrimePi[i[[1]]]/i[[1]], {i, FactorInteger[n]}];
a[0] = 0; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 24 2017, translated from Maple *)

Formula

a(n) = A258851(A000312(n)).
a(n) = n^n * A258851(n).
a(n) = A258997(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).

A258859 Ninth pi-based arithmetic derivative of n.

Original entry on oeis.org

0, 0, 0, 0, 4, 0, 4, 4, 81920, 81920, 0, 0, 622592, 4, 512, 2304, 4931584, 4, 2304, 12288, 4931584, 512, 208, 12288, 193644, 12288, 2304, 8322048, 423168, 0, 81920, 0, 38944768, 12288, 0, 4, 37880, 81920, 4, 423168, 37880, 4, 80, 208, 2298880, 43632, 4, 512
Offset: 0

Views

Author

Alois P. Heinz, Jun 12 2015

Keywords

Links

Crossrefs

Column k=9 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, 9):
seq(a(n), n=0..100);

Formula

a(n) = A258851^9(n).

A258860 Tenth pi-based arithmetic derivative of n.

Original entry on oeis.org

0, 0, 0, 0, 4, 0, 4, 4, 622592, 622592, 0, 0, 4931584, 4, 2304, 12288, 38944768, 4, 12288, 81920, 38944768, 2304, 512, 81920, 714096, 81920, 12288, 65719296, 2298880, 0, 622592, 0, 278380544, 81920, 0, 4, 85988, 622592, 4, 2298880, 85988, 4, 208, 512, 13319168
Offset: 0

Views

Author

Alois P. Heinz, Jun 12 2015

Keywords

Links

Crossrefs

Column k=10 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, 10):
seq(a(n), n=0..100);

Formula

a(n) = A258851^10(n).

A258975 a(n) = n-th pi-based antiderivative of 1.

Original entry on oeis.org

1, 2, 3, 5, 11, 10, 29, 78, 141, 266, 147, 194, 1181, 2413, 1834, 6293, 4805, 20290, 28345, 25065, 85334, 87967, 55722, 191559, 385845, 437914, 998758, 396375, 95625, 202043, 341774, 2217782, 1607613, 1333107, 1697893, 1222517, 2277354, 1599111
Offset: 0

Views

Author

Alois P. Heinz, Jun 18 2015

Keywords

Examples

			a(6) = 29 -> 10 -> 11 -> 5 -> 3 -> 2 -> 1.
a(7) = 78 -> 127 -> 31 -> 11 -> 5 -> 3 -> 2 -> 1.

Crossrefs

Row n=1 of A259016.
Cf. A000720, A258850, A258851.

Formula

a(n) = min { m >= 0 : A258851^n(m) = 1 }.
A258850(a(n),n) = 1.
Previous Showing 31-40 of 49 results. Next

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