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 1-4 of 4 results.

A280703 a(n) = A003961(n) / A280702(n) = A003961(n) / gcd(A003961(n),A250469(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 9, 1, 7, 1, 15, 1, 11, 1, 9, 1, 25, 1, 21, 1, 13, 1, 45, 1, 17, 25, 11, 1, 35, 1, 81, 13, 19, 1, 15, 1, 23, 17, 21, 1, 55, 1, 39, 35, 29, 1, 135, 1, 1, 19, 1, 1, 125, 1, 9, 23, 31, 1, 105, 1, 37, 55, 27, 1, 1, 1, 57, 29, 77, 1, 225, 1, 41, 49, 23, 1, 85, 1, 189, 125, 43, 1, 165, 1, 47, 31, 39, 1, 175, 1, 87, 37
Offset: 1

Views

Author

Antti Karttunen, Mar 08 2017

Keywords

Comments

If there are no such n that A250469(n) = k*A003961(n) for some integer k > 1, then A280693 gives the positions of ones in this sequence. Cf. also comment in A280704.

Links

Crossrefs

Cf. A003961, A250469, A280702, A280704, A280693.

Programs

  • Mathematica
    f[n_] := f[n] = Which[n == 1, 1, PrimeQ@ n, NextPrime@ n, True, Times @@ Replace[FactorInteger[n], {p_, e_} :> f[p]^e, 1]]; g[n_] := If[n == 1, 0, PrimePi@ FactorInteger[n][[1, 1]]]; Function[s, MapIndexed[ Function[p, p/GCD[Lookup[s, g@ First@ #2 + 1][[#1]] - Boole[First@ #2 == 1], p]]@ f@ First@ #2 &, #] &@ Map[Position[Lookup[s, g@ #], #][[1, 1]] &, Range@ 120]]@ PositionIndex@ Array[g, 10^4] (* Michael De Vlieger, Mar 08 2017, Version 10 *)
  • Scheme
    (define (A280703 n) (/ (A003961 n) (A280702 n)))

Formula

a(n) = A003961(n) / A280702(n) = A003961(n) / gcd(A003961(n),A250469(n)).

A280704 a(n) = A250469(n) / A280702(n) = A250469(n) / gcd(A003961(n),A250469(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 7, 1, 9, 1, 11, 1, 13, 1, 5, 1, 17, 1, 19, 1, 21, 1, 23, 1, 25, 13, 9, 1, 29, 1, 31, 17, 33, 1, 7, 1, 37, 19, 13, 1, 41, 1, 43, 23, 45, 1, 47, 1, 1, 25, 1, 1, 53, 1, 5, 29, 57, 1, 59, 1, 61, 31, 7, 1, 1, 1, 67, 35, 69, 1, 71, 1, 73, 37, 25, 1, 77, 1, 79, 41, 81, 1, 83, 1, 85, 43, 29, 1, 89, 1, 91, 47, 93, 1, 19, 1, 97, 49, 33, 1
Offset: 1

Views

Author

Antti Karttunen, Mar 08 2017

Keywords

Comments

Note: a(352) = 1 even though A280703(352) = 3 as A003961(352) = 3159 = 3^5 * 13, while A250469(352) = 1053 = 3^4 * 13. (Thus also A266645(352) = 176 = 352/2.) Question: Are there more n for which A003961(n) = k*A250469(n) for some integer k > 1 ? Cf. also comments in A280703.

Links

Crossrefs

Cf. A003961, A250469, A266645, A280701, A280702, A280703.

Programs

  • Mathematica
    f[n_] := f[n] = Which[n == 1, 1, PrimeQ@ n, NextPrime@ n, True, Times @@ Replace[FactorInteger[n], {p_, e_} :> f[p]^e, 1]]; g[n_] := If[n == 1, 0, PrimePi@ FactorInteger[n][[1, 1]]]; Function[s, MapIndexed[Function[t, t/GCD[t, f@ First@ #2]][Lookup[s, g[First@ #2] + 1][[#1]] - Boole[First@ #2 == 1]] &, #] &@ Map[Position[Lookup[s, g@ #], #][[1, 1]] &, Range@ 120]]@ PositionIndex@ Array[g, 10^4] (* Michael De Vlieger, Mar 08 2017, Version 10 *)
  • Scheme
    (define (A280704 n) (/ (A250469 n) (A280702 n)))

Formula

a(n) = A250469(n) / A280702(n) = A250469(n) / gcd(A003961(n),A250469(n)).
A280701(n) = n - a(n).

A280692 a(n) = A003961(n) - A250469(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 6, 0, -6, 0, 12, 0, -6, 0, 36, 0, 24, 0, 6, 0, -24, 0, 66, 0, -24, 60, 18, 0, 18, 0, 150, -20, -42, 0, 120, 0, -42, -10, 72, 0, 42, 0, -12, 60, -48, 0, 264, 0, 0, -30, 0, 0, 216, 0, 132, -30, -78, 0, 138, 0, -72, 120, 540, 0, 0, 0, -30, -30, 24, 0, 462, 0, -96, 60, -18, 0, 24, 0, 330, 420, -114, 0, 246
Offset: 1

Views

Author

Antti Karttunen, Mar 08 2017

Keywords

Links

Crossrefs

Cf. A003961, A250469, A249820, A280492, A280497, A280498, A280702.
Cf. A280693 (gives the positions of zeros).
Cf. also arrays A083221 and A246278.

Programs

  • Mathematica
    f[n_] := f[n] = Which[n == 1, 1, PrimeQ@ n, NextPrime@ n, True, Times @@ Replace[FactorInteger[n], {p_, e_} :> f[p]^e, 1]]; g[n_] := If[n == 1, 0, PrimePi@ FactorInteger[n][[1, 1]]]; Function[s, MapIndexed[ Function[{m, n}, f@ n - Lookup[s, g[n] + 1][[m]] + Boole[n == 1]][#1, First@ #2] &, #] &@ Map[Position[Lookup[s, g@ #], #][[1, 1]] &, Range@ 120]]@ PositionIndex@ Array[g, 10^4] (* Michael De Vlieger, Mar 09 2017, Version 10 *)
  • Scheme
    (define (A280692 n) (- (A003961 n) (A250469 n)))

Formula

a(n) = A003961(n) - A250469(n).

A280701 a(n) = n - A280704(n).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 1, 8, 1, 10, 1, 12, 1, 14, 11, 16, 1, 18, 1, 20, 1, 22, 1, 24, 1, 14, 19, 28, 1, 30, 1, 16, 1, 34, 29, 36, 1, 20, 27, 40, 1, 42, 1, 22, 1, 46, 1, 48, 49, 26, 51, 52, 1, 54, 51, 28, 1, 58, 1, 60, 1, 32, 57, 64, 65, 66, 1, 34, 1, 70, 1, 72, 1, 38, 51, 76, 1, 78, 1, 40, 1, 82, 1, 84, 1, 44, 59, 88, 1, 90, 1, 46
Offset: 1

Views

Author

Antti Karttunen, Mar 08 2017

Keywords

Comments

Questions: Are all terms nonnegative? Where do ones occur?

Links

Crossrefs

Cf. A003961, A250469, A280702, A280704.

Programs

  • Mathematica
    f[n_] := f[n] = Which[n == 1, 1, PrimeQ@ n, NextPrime@ n, True, Times @@ Replace[FactorInteger[n], {p_, e_} :> f[p]^e, 1]]; g[n_] := If[n == 1, 0, PrimePi@ FactorInteger[n][[1, 1]]]; Function[s, MapIndexed[ Function[t, First@ #2 - t/GCD[t, f@ First@ #2]][Lookup[s, g[First@ #2] + 1][[#1]] - Boole[First@ #2 == 1]] &, #] &@ Map[Position[Lookup[s, g@ #], #][[1, 1]] &, Range@ 120]]@ PositionIndex@ Array[g, 10^4] (* Michael De Vlieger, Mar 08 2017, Version 10 *)
  • Scheme
    (define (A280701 n) (- n (A280704 n)))

Formula

a(n) = n - A280704(n) = n - (A250469(n)/gcd(A003961(n),A250469(n))).
Showing 1-4 of 4 results.

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