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 1-8 of 8 results.

A258850 A(n,k) = k-th pi-based arithmetic derivative of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 0, 0, 1, 3, 0, 0, 0, 2, 4, 0, 0, 0, 1, 4, 5, 0, 0, 0, 0, 4, 3, 6, 0, 0, 0, 0, 4, 2, 7, 7, 0, 0, 0, 0, 4, 1, 4, 4, 8, 0, 0, 0, 0, 4, 0, 4, 4, 12, 9, 0, 0, 0, 0, 4, 0, 4, 4, 20, 12, 10, 0, 0, 0, 0, 4, 0, 4, 4, 32, 20, 11, 11, 0, 0, 0, 0, 4, 0, 4, 4, 80, 32, 5, 5, 12
Offset: 0

Views

Author

Alois P. Heinz, Jun 12 2015

Keywords

Examples

			Square array A(n,k) begins:
  0,  0,  0,  0,  0,   0,   0,    0,     0,     0, ...
  1,  0,  0,  0,  0,   0,   0,    0,     0,     0, ...
  2,  1,  0,  0,  0,   0,   0,    0,     0,     0, ...
  3,  2,  1,  0,  0,   0,   0,    0,     0,     0, ...
  4,  4,  4,  4,  4,   4,   4,    4,     4,     4, ...
  5,  3,  2,  1,  0,   0,   0,    0,     0,     0, ...
  6,  7,  4,  4,  4,   4,   4,    4,     4,     4, ...
  7,  4,  4,  4,  4,   4,   4,    4,     4,     4, ...
  8, 12, 20, 32, 80, 208, 512, 2304, 12288, 81920, ...
  9, 12, 20, 32, 80, 208, 512, 2304, 12288, 81920, ...

Links

Crossrefs

Columns k=0-10 give: A001477, A258851, A258852, A258853, A258854, A258855, A258856, A258857, A258858, A258859, A258860.
Rows n=0,1,4,8 give: A000004, A000007, A010709, A258848.
Antidiagonal sums give A258847.
Main diagonal gives A258849.
Cf. A000720, A258975, A259016.

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:
seq(seq(A(n, h-n), n=0..h), h=0..14);
  • Mathematica
    d[n_] := n*Total[Last[#]*PrimePi[First[#]]/First[#]& /@ FactorInteger[n]]; d[0] = 0;
A[n_, k_] := A[n, k] = If[k == 0, n, d[A[n, k-1]]];
Table[Table[A[n, h-n], {n, 0, h}], {h, 0, 14}] // Flatten (* Jean-François Alcover, Apr 24 2016, adapted from Maple *)

Formula

A(n,k) = A258851^k(n).
A(A259016(n,k),k) = n.
A(A258975(n),n) = 1.

A258861 The pi-based antiderivative of n: the least m such that A258851(m) equals n.

Original entry on oeis.org

0, 2, 3, 5, 4, 11, 13, 6, 19, 23, 29, 10, 8, 41, 43, 14, 53, 59, 61, 15, 12, 22, 79, 83, 89, 26, 21, 103, 107, 109, 25, 34, 16, 18, 139, 38, 151, 33, 163, 167, 173, 35, 181, 191, 28, 197, 199, 211, 223, 58, 229, 233, 24, 30, 27, 51, 49, 269, 55, 277, 281, 74
Offset: 0

Views

Author

Alois P. Heinz, Jun 12 2015

Keywords

Links

Crossrefs

Column k=1 of A259016, A259153.
Cf. A000040, A000720, A258851, A258862, A258995.

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:= -1, proc() -1 end;
      proc(n) local h;
        while a(n) = -1 do
          t:= t+1; h:= d(t);
          if a(h) = -1 then a(h):= t fi
        od; a(n)
      end
    end():
seq(a(n), n=0..100);
  • Mathematica
    A258851[n_] := If[n == 0, 0, n*Total[Last[#]*PrimePi[First[#]]/First[#]& /@ FactorInteger[n]]];
a[n_] := For[m = 0, True, m++, If[A258851[m] == n, Return[m]]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Sep 10 2023 *)

Formula

a(n) = min { m >= 0 : A258851(m) = n }.
A258851(a(n)) = n.
a(n) <= A000040(n) for n>0.

A258862 Second pi-based antiderivative of n: the least m such that A258851^2(m) equals n.

Original entry on oeis.org

0, 3, 5, 11, 4, 10, 41, 13, 15, 83, 109, 29, 19, 35, 191, 43, 30, 277, 74, 14, 8, 42, 77, 431, 461, 21, 22, 563, 66, 599, 26, 78, 12, 61, 141, 163, 877, 18, 214, 218, 226, 38, 114, 201, 105, 1201, 215, 1297, 302, 55, 1447, 1471, 89, 25, 103, 170, 58, 291, 51
Offset: 0

Views

Author

Alois P. Heinz, Jun 12 2015

Keywords

Links

Crossrefs

Column k=2 of A259016.
Cf. A000040, A000720, A258851, A258852, A258861, A258995.

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:= -1, proc() -1 end;
      proc(n) local h;
        while a(n) = -1 do
          t:= t+1; h:= d(d(t));
          if a(h) = -1 then a(h):= t fi
        od; a(n)
      end
    end():
seq(a(n), n=0..100);
  • Mathematica
    d[n_] := 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]]];
a[n_] := A[n, 2];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, May 17 2024 *)

Formula

a(n) = min { m >= 0 : A258851^2(m) = n }.
A258851^2(a(n)) = A258852(a(n)) = n.
a(n) <= A000040^2(n) for n>0.
a(n) <= A258861^2(n); a(21) = 42 < A258861^2(21) = A258861(22) = 79; A258851^2(42) = A258851^2(79) = 21.

A258995 Third pi-based antiderivative of n: the least m such that A258851^3(m) equals n.

Original entry on oeis.org

0, 5, 11, 10, 4, 29, 35, 41, 14, 431, 599, 78, 15, 38, 201, 191, 25, 382, 186, 43, 19, 65, 94, 3001, 535, 22, 42, 633, 317, 4397, 21, 141, 8, 74, 574, 214, 1286, 61, 253, 247, 1417, 163, 115, 217, 66, 546, 138, 10631, 1997, 51, 12097, 12301, 362, 26, 563, 1013
Offset: 0

Views

Author

Alois P. Heinz, Jun 16 2015

Keywords

Links

Crossrefs

Column k=3 of A259016.
Cf. A000040, A000720, A258851, A258853, A258861, A258862.

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:= -1, proc() -1 end;
      proc(n) local h;
        while a(n) = -1 do
          t:= t+1; h:= d(d(d(t)));
          if a(h) = -1 then a(h):= t fi
        od; a(n)
      end
    end():
seq(a(n), n=0..100);
  • Mathematica
    d[n_] := 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]]];
a[n_] := A[n, 3];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, May 17 2024 *)

Formula

a(n) = min { m >= 0 : A258851^3(m) = n }.
A258851^3(a(n)) = A258853(a(n)) = n.
a(n) <= A000040^3(n) for n>0.
a(n) <= A258861^3(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.

A259169 a(n) = n-th pi-based antiderivative of 8.

Original entry on oeis.org

8, 19, 15, 14, 43, 191, 201, 217, 1113, 1239, 986, 925, 375, 526, 689, 998, 3642, 3966, 5299, 4090, 7363, 20942, 150161, 117915, 218218, 597199, 472182, 494550, 1075362, 796042, 310086, 444985, 1403783, 1578955, 2702706, 10010173
Offset: 0

Views

Author

Alois P. Heinz, Jun 19 2015

Keywords

Examples

			a(7) = 217 -> 201 -> 191 -> 43 -> 14 -> 15 -> 19 -> 8.
a(8) = 1113 -> 1714 -> 1153 -> 191 -> 43 -> 14 -> 15 -> 19 -> 8.

Crossrefs

Row n=8 of A259016.
Cf. A000720, A258848, A258851.

Formula

a(n) = min { m >= 0 : A258851^n(m) = 8 }.

A259168 a(n) = n-th pi-based antiderivative of 7.

Original entry on oeis.org

7, 6, 13, 41, 35, 38, 163, 138, 253, 346, 1383, 630, 4657, 3210, 5633, 9469, 20838, 22525, 28491, 21035, 16491, 13735, 22114, 54298, 225361, 639070, 479794, 421883, 720634, 461055, 788446, 650762, 688229, 478126, 1465550, 1960533, 2117157
Offset: 0

Views

Author

Alois P. Heinz, Jun 19 2015

Keywords

Crossrefs

Row n=7 of A259016.
Cf. A000720, A258851.

Formula

a(n) = min { m >= 0 : A258851^n(m) = 7 }.

A259170 a(n) = n-th pi-based antiderivative of 9.

Original entry on oeis.org

9, 23, 83, 431, 3001, 27457, 10626, 112087, 87306, 172810, 280217, 390133, 353555, 750243, 1318106, 937023, 367542, 2615090, 3434663, 3281065, 3270774, 4979697, 8021665, 4627825, 5618666, 11169397
Offset: 0

Views

Author

Alois P. Heinz, Jun 19 2015

Keywords

Crossrefs

Row n=9 of A259016.
Cf. A000720, A258851.

Formula

a(n) = min { m >= 0 : A258851^n(m) = 9 }.
Showing 1-8 of 8 results.

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