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 1-7 of 7 results.

A283464 a(n) = A032742(A266646(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 5, 3, 4, 1, 8, 1, 6, 5, 14, 1, 13, 1, 11, 7, 7, 1, 23, 5, 9, 17, 17, 1, 18, 1, 41, 9, 10, 7, 38, 1, 12, 11, 32, 1, 28, 1, 20, 23, 15, 1, 68, 7, 25, 13, 26, 1, 63, 11, 50, 15, 16, 1, 53, 1, 19, 37, 122, 13, 33, 1, 29, 19, 39, 1, 113, 1, 21, 33, 35, 11, 43, 1, 95, 83, 22, 1, 83, 17, 24, 21, 59, 1, 88, 13, 44, 25, 27, 19, 203
Offset: 1

Views

Author

Antti Karttunen, Mar 08 2017

Keywords

Links

Crossrefs

Cf. A020639, A032742, A266646, A280497, A280498, A283463, A283466.

Programs

Formula

a(n) = A032742(A266646(n)).
a(n) = A266646(n) / A020639(n). [Because A266646 preserves the smallest prime factor of n.]

A283466 a(n) = A046523(A266646(n)).

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 2, 6, 4, 8, 2, 16, 2, 12, 6, 12, 2, 6, 2, 6, 6, 6, 2, 6, 4, 12, 6, 6, 2, 36, 2, 6, 8, 12, 6, 12, 2, 24, 6, 64, 2, 24, 2, 24, 6, 30, 2, 24, 4, 12, 6, 12, 2, 60, 6, 36, 12, 32, 2, 6, 2, 6, 6, 12, 6, 30, 2, 6, 6, 30, 2, 6, 2, 30, 12, 30, 6, 6, 2, 30, 6, 12, 2, 6, 6, 48, 12, 6, 2, 48, 6, 24, 12, 24, 6, 30, 2, 6, 6, 12, 2, 96, 2, 30
Offset: 1

Views

Author

Antti Karttunen, Mar 08 2017

Keywords

Links

Crossrefs

Cf. A046523, A266646, A283464, A283465.

Programs

Formula

a(n) = A046523(A266646(n)).

A266403 Self-inverse permutation of natural numbers: a(n) = A250470(A263273(A250469(n))).

Original entry on oeis.org

1, 2, 5, 4, 3, 8, 17, 6, 13, 10, 11, 20, 9, 14, 71, 22, 7, 26, 19, 12, 23, 16, 21, 24, 41, 18, 53, 28, 31, 56, 29, 38, 107, 58, 67, 74, 61, 32, 197, 40, 25, 68, 59, 50, 137, 64, 73, 62, 49, 44, 227, 76, 27, 80, 55, 30, 89, 34, 43, 66, 37, 48, 91, 46, 69, 60, 35, 42, 65, 70, 15, 78, 47, 36, 119, 52
Offset: 1

Views

Author

Antti Karttunen, Jan 02 2016

Keywords

Links

Crossrefs

Cf. A000035, A250469, A250470, A263273.
Cf. A265369, A265904, A266190, A266401 (other conjugates or similar derivations of A263273).
Cf. also A250471, A250472, A264996, A266404, A266415, A266416, A266645, A266646.

Programs

Formula

a(n) = A250470(A263273(A250469(n))).
As a composition of related permutations:
a(n) = A266415(A266645(n)) = A266646(A266416(n)).
a(n) = A250472(A264996(A250471(n))).
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]

A266645 Permutation of natural numbers: a(n) = A064989(A250469(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 9, 8, 11, 14, 13, 22, 15, 12, 17, 26, 19, 34, 21, 20, 23, 38, 25, 18, 33, 16, 29, 46, 31, 58, 39, 28, 35, 30, 37, 62, 51, 44, 41, 74, 43, 82, 57, 24, 47, 86, 49, 50, 27, 52, 53, 94, 55, 42, 69, 68, 59, 106, 61, 118, 87, 40, 65, 66, 67, 122, 45, 76, 71, 134, 73, 142, 93, 36, 77, 70, 79, 146, 111, 32, 83, 158, 85, 78, 123
Offset: 1

Views

Author

Antti Karttunen, Jan 02 2016

Keywords

Links

Crossrefs

Inverse: A266646.
Cf. A000035, A003961, A020639, A055396, A064989, A250469.
Related permutations: A266403, A266416, A249817, A249818.

Programs

  • Mathematica
    f[n_] := Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n; g[n_] := If[n == 1, 0, PrimePi@ FactorInteger[n][[1, 1]]]; Function[s, MapIndexed[ Function[{m, n}, f[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 *)

Formula

a(n) = A064989(A250469(n)).
a(n) = A064989(A249817(A003961(A249818(n)))).
As a composition of related permutations:
a(n) = A266416(A266403(n)).
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]
A020639(a(n)) = A020639(n). [More generally, it preserves the smallest prime dividing n.]
A055396(a(n)) = A055396(n).

A266415 Permutation of natural numbers: a(n) = A250470(A263273(A003961(n))).

Original entry on oeis.org

1, 2, 5, 4, 3, 8, 17, 10, 13, 6, 11, 22, 9, 20, 71, 28, 7, 18, 19, 16, 23, 14, 21, 64, 41, 26, 227, 58, 31, 74, 29, 82, 53, 12, 67, 52, 61, 24, 107, 46, 25, 30, 59, 40, 65, 56, 73, 190, 49, 44, 197, 76, 27, 230, 55, 172, 137, 38, 43, 220, 37, 32, 571, 244, 69, 60, 35, 34, 89, 72, 15, 154, 47, 68, 479, 70
Offset: 1

Views

Author

Antti Karttunen, Jan 02 2016

Keywords

Links

Crossrefs

Inverse: A266416.
Cf. A000035, A003961, A250470, A263273.
Related permutations: A048673, A250472, A264985, A264996, A266403, A266646.

Programs

Formula

a(n) = A250470(A263273(A003961(n))).
As a composition of related permutations:
a(n) = A266403(A266646(n)).
a(n) = A250472(A264996(A048673(n))) = A250472(1+A264985(-1+A048673(n))).
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]

A269356 Permutation of natural numbers: a(n) = A268674(A269379(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 17, 10, 11, 12, 13, 14, 15, 16, 19, 18, 29, 20, 25, 22, 9, 24, 23, 26, 27, 28, 31, 30, 21, 32, 73, 34, 53, 36, 37, 38, 39, 40, 41, 42, 43, 44, 107, 46, 47, 48, 33, 50, 51, 52, 59, 54, 71, 56, 137, 58, 101, 60, 61, 62, 63, 64, 109, 66, 67, 68, 121, 70, 35, 72, 97, 74, 75, 76, 79, 78, 131, 80, 197
Offset: 1

Views

Author

Antti Karttunen, Mar 03 2016

Keywords

Examples

			For n=9 we first find what number is below 9 in square array A255127, which is 19, then we find what number is above 19 in square array A083221, which is 17, thus a(9) = 17.

Links

Crossrefs

Inverse: A269355.
Cf. A268674, A269379.
Cf. also arrays A083221 & A255127.
More recursed variant: A269358. Cf. also permutations A266646, A255408, A269172.

Programs

Formula

a(n) = A268674(A269379(n)).
Other identities. For all n >= 1:
a(2n) = 2n. [Fixes the even numbers.]

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 *)
Showing 1-7 of 7 results.

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