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 1-3 of 3 results.

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

Original entry on oeis.org

1, 3, 5, 9, 7, 15, 11, 3, 25, 3, 13, 3, 17, 3, 35, 9, 19, 3, 23, 3, 55, 3, 29, 3, 49, 3, 5, 9, 31, 3, 37, 3, 5, 3, 77, 15, 41, 3, 5, 9, 43, 3, 47, 3, 5, 3, 53, 3, 121, 147, 5, 153, 59, 3, 91, 33, 5, 3, 61, 3, 67, 3, 5, 27, 119, 195, 71, 3, 5, 3, 73, 3, 79, 3, 5, 9, 143, 3, 83, 3, 5, 3, 89, 3, 133, 3, 5, 9, 97, 3, 187, 3, 5, 3, 161
Offset: 1

Views

Author

Antti Karttunen, Mar 08 2017

Keywords

Comments

For n > 1, a(n) > 1 because A020639(A003961(n)) = A020639(A250469(n)) = A003961(A020639(n)).

Links

Crossrefs

Cf. A003961, A020639, A250469, A280497, A280498, A280692, A280703, A280704, A283463, A283464.

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[ GCD[ Lookup[s, g[First@ #2] + 1][[#1]] - Boole[First@ #2 == 1], f@ First@ #2] &, #] &@ Map[Position[Lookup[s, g@ #], #][[1, 1]] &, Range@ 120]]@ PositionIndex@ Array[g, 10^4] (* Michael De Vlieger, Mar 08 2017 *)
  • Scheme
    (define (A280702 n) (gcd (A003961 n) (A250469 n)))

Formula

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

A280693 Numbers n such that A003961(n) = A250469(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 50, 52, 53, 55, 59, 61, 65, 66, 67, 71, 73, 77, 79, 83, 85, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 121, 127, 131, 133, 137, 139, 143, 149, 151, 157, 161, 163, 167, 169, 173, 179, 181, 186, 187, 191, 193, 197, 199, 203
Offset: 1

Views

Author

Antti Karttunen, Mar 08 2017

Keywords

Comments

Positions of zeros in A280692. Conjectured to be also the positions of ones in A280703.
For most terms k of this sequence A003961(k) is also included as a term. Exceptions are: 50, 52, 66, 186, 435, 1245, 1445, 2068, 2085, 11605, ... that seems to be a subsequence of all those terms that have more than two prime factors: 50, 52, 66, 186, 435, 1245, 1445, 2068, 2085, 8695, 11605, ...
Note how 8695 = 5*37*47 and A003961(8695) = 7*41*53 = 15211 = A003961(8695) = A250469(8695) (for no apparent reason?).

Links

Crossrefs

Fixed points of permutations A266645 & A266646.
Cf. A001222, A003961, A250469, A280692, A280703.
Cf. A000040, A001248, A006094, A251728 (subsequences).
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]]]; With[{nn = 204}, Function[s, Function[t, Select[Range@ nn, f@ # == t[[#]] &]]@ MapIndexed[Lookup[s, g[First@ #2] + 1][[#1]] - Boole[First@ #2 == 1] &, #] &@ Map[Position[Lookup[s, g@ #], #][[1, 1]] &, Range@ nn]]@ PositionIndex@ Array[g, 10^4]] (* Michael De Vlieger, Mar 08 2017, Version 10 *)

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).
Showing 1-3 of 3 results.

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