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 1-9 of 9 results.

A283463 a(n) = A032742(A266645(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 5, 3, 4, 1, 7, 1, 11, 5, 6, 1, 13, 1, 17, 7, 10, 1, 19, 5, 9, 11, 8, 1, 23, 1, 29, 13, 14, 7, 15, 1, 31, 17, 22, 1, 37, 1, 41, 19, 12, 1, 43, 7, 25, 9, 26, 1, 47, 11, 21, 23, 34, 1, 53, 1, 59, 29, 20, 13, 33, 1, 61, 15, 38, 1, 67, 1, 71, 31, 18, 11, 35, 1, 73, 37, 16, 1, 79, 17, 39, 41, 46, 1, 83, 13, 55, 43, 58, 19, 51
Offset: 1

Views

Author

Antti Karttunen, Mar 08 2017

Keywords

Links

Crossrefs

Cf. A020639, A032742, A266645, A280497, A280498, A283464, A283465.

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}, If[# == 1, 1, Divisors[#][[-2]]] &@ 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 *)
  • Scheme
    (define (A283463 n) (A032742 (A266645 n)))

Formula

a(n) = A032742(A266645(n)).
a(n) = A266645(n) / A020639(n). [Because A266645 preserves the smallest prime factor of n.]

A250469 a(1) = 1; and for n > 1, a(n) = A078898(n)-th number k for which A055396(k) = A055396(n)+1, where A055396(n) is the index of smallest prime dividing n.

Original entry on oeis.org

1, 3, 5, 9, 7, 15, 11, 21, 25, 27, 13, 33, 17, 39, 35, 45, 19, 51, 23, 57, 55, 63, 29, 69, 49, 75, 65, 81, 31, 87, 37, 93, 85, 99, 77, 105, 41, 111, 95, 117, 43, 123, 47, 129, 115, 135, 53, 141, 121, 147, 125, 153, 59, 159, 91, 165, 145, 171, 61, 177, 67, 183, 155, 189, 119, 195, 71, 201, 175, 207, 73, 213, 79, 219, 185, 225, 143, 231, 83, 237, 205, 243, 89, 249, 133, 255
Offset: 1

Views

Author

Antti Karttunen, Dec 06 2014

Keywords

Comments

Permutation of odd numbers.
For n >= 2, a(n) = A078898(n)-th number k for which A055396(k) = A055396(n)+1. In other words, a(n) tells which number is located immediately below n in the sieve of Eratosthenes (see A083140, A083221) in the same column of the sieve that contains n.
A250471(n) = (a(n)+1)/2 is a permutation of natural numbers.
Coincides with A003961 in all terms which are primes. - M. F. Hasler, Sep 17 2016. Note: primes are a proper subset of A280693 which gives all n such that a(n) = A003961(n). - Antti Karttunen, Mar 08 2017

Links

Crossrefs

Cf. A000040, A003961, A016945, A046523, A055396, A078898, A083140, A083221, A084967, A249744, A249810, A249820, A249817, A249818, A250471, A266645, A280692, A280693, A283465.
Cf. A250470, A268674 (left inverses, the latter is simpler).

Programs

  • Mathematica
    a[1] = 1; a[n_] := If[PrimeQ[n], NextPrime[n], m1 = p1 = FactorInteger[n][[ 1, 1]]; For[k1 = 1, m1 <= n, m1 += p1; If[m1 == n, Break[]]; If[ FactorInteger[m1][[1, 1]] == p1, k1++]]; m2 = p2 = NextPrime[p1]; For[k2 = 1, True, m2 += p2, If[FactorInteger[m2][[1, 1]] == p2, k2++]; If[k1+2 == k2, Return[m2]]]]; Array[a, 100] (* Jean-François Alcover, Mar 08 2016 *)
g[n_] := If[n == 1, 0, PrimePi@ FactorInteger[n][[1, 1]]]; Function[s, MapIndexed[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 *)

Formula

a(1) = 1, a(n) = A083221(A055396(n)+1, A078898(n)).
a(n) = A249817(A003961(A249818(n))).
Other identities. For all n >= 1:
A250470(a(n)) = A268674(a(n)) = n. [A250470 and A268674 provide left inverses for this function.]
a(2n) = A016945(n-1). [Maps even numbers to the numbers of form 6n+3, in monotone order.]
a(A016945(n-1)) = A084967(n). [Which themselves are mapped to the terms of A084967, etc. Cf. the Example section of A083140.]
a(A000040(n)) = A000040(n+1). [Each prime is mapped to the next prime.]
For all n >= 2, A055396(a(n)) = A055396(n)+1. [A more general rule.]
A046523(a(n)) = A283465(n). - Antti Karttunen, Mar 08 2017

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.]

A266646 Permutation of natural numbers: a(n) = A250470(A003961(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 9, 8, 11, 16, 13, 12, 15, 28, 17, 26, 19, 22, 21, 14, 23, 46, 25, 18, 51, 34, 29, 36, 31, 82, 27, 20, 35, 76, 37, 24, 33, 64, 41, 56, 43, 40, 69, 30, 47, 136, 49, 50, 39, 52, 53, 126, 55, 100, 45, 32, 59, 106, 61, 38, 111, 244, 65, 66, 67, 58, 57, 78, 71, 226, 73, 42, 99, 70, 77, 86, 79, 190, 249, 44, 83, 166, 85
Offset: 1

Views

Author

Antti Karttunen, Jan 02 2016

Keywords

Links

Crossrefs

Inverse: A266645.
Cf. A000035, A020639, A055396, A003961, A250470.
Related permutations: A249817, A249818, A266403, A266415.

Programs

Formula

a(n) = A250470(A003961(n)).
As a composition of related permutations:
a(n) = A266403(A266415(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).

A266416 Permutation of natural numbers: a(n) = A064989(A263273(A250469(n))).

Original entry on oeis.org

1, 2, 5, 4, 3, 10, 17, 6, 13, 8, 11, 34, 9, 22, 71, 20, 7, 18, 19, 14, 23, 12, 21, 38, 41, 26, 53, 16, 31, 42, 29, 62, 107, 68, 67, 142, 61, 58, 197, 44, 25, 122, 59, 50, 137, 40, 73, 118, 49, 82, 227, 36, 33, 146, 55, 46, 89, 28, 43, 66, 37, 86, 91, 24, 45, 106, 35, 74, 65, 76, 15, 70, 47, 30, 119, 52
Offset: 1

Views

Author

Antti Karttunen, Jan 02 2016

Keywords

Links

Crossrefs

Inverse: A266415.
Cf. A000035, A064989, A250469, A263273.
Related permutations: A064216, A250471, A264985, A264996, A266403, A266645.

Programs

Formula

a(n) = A064989(A263273(A250469(n))).
As a composition of related permutations:
a(n) = A266645(A266403(n)).
a(n) = A064216(A264996(A250471(n))) = A064216(1+A264985(-1+A250471(n))).
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]

A269355 Permutation of natural numbers: a(n) = A269380(A250469(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 23, 10, 11, 12, 13, 14, 15, 16, 9, 18, 17, 20, 31, 22, 25, 24, 21, 26, 27, 28, 19, 30, 29, 32, 49, 34, 71, 36, 37, 38, 39, 40, 41, 42, 43, 44, 107, 46, 47, 48, 119, 50, 51, 52, 35, 54, 89, 56, 101, 58, 53, 60, 61, 62, 63, 64, 115, 66, 67, 68, 173, 70, 55, 72, 33, 74, 75, 76, 131, 78, 77, 80, 167
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 A083221, which is 25, then we find what number is above 25 in square array A255127, which is 23, thus a(9) = 23.

Links

Crossrefs

Inverse: A269356.
Cf. A250469, A269380.
Cf. also arrays A083221 & A255127.
More recursed variant: A269357. Cf. also permutations A266645, A255407, A269171.

Programs

Formula

a(n) = A269380(A250469(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 *)

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).

A283465 a(n) = A046523(A250469(n)).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Mar 08 2017

Keywords

Links

Crossrefs

Cf. A046523, A250469, A266645, A283463, A283466.

Programs

Formula

a(n) = A046523(A250469(n)) = A046523(A266645(n)).
Showing 1-9 of 9 results.

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