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.

A305801 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = 0 if n is an odd prime, with f(n) = n for all other n.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 7, 8, 3, 9, 3, 10, 11, 12, 3, 13, 3, 14, 15, 16, 3, 17, 18, 19, 20, 21, 3, 22, 3, 23, 24, 25, 26, 27, 3, 28, 29, 30, 3, 31, 3, 32, 33, 34, 3, 35, 36, 37, 38, 39, 3, 40, 41, 42, 43, 44, 3, 45, 3, 46, 47, 48, 49, 50, 3, 51, 52, 53, 3, 54, 3, 55, 56, 57, 58, 59, 3, 60, 61, 62, 3, 63, 64, 65, 66, 67, 3, 68, 69, 70, 71, 72, 73, 74, 3, 75, 76, 77, 3, 78, 3, 79, 80
Offset: 1

Views

Author

Antti Karttunen, Jun 14 2018

Keywords

Comments

The original name was: "Filter sequence for a(odd prime) = constant sequences", which stemmed from the fact that for all i, j, a(i) = a(j) => b(i) = b(j) for any sequence b that obtains a constant value for all odd primes A065091.
For example, we have for all i, j:
a(i) = a(j) => A305800(i) = A305800(j),
a(i) = a(j) => A007814(i) = A007814(j),
a(i) = a(j) => A305891(i) = A305891(j) => A291761(i) = A291761(j).
There are several filter sequences "above" this one (meaning that they have finer equivalence class partitioning), for example, we have, for all i, j:
[where odd primes are further distinguished by]
A305900(i) = A305900(j) => a(i) = a(j), [whether p = 3 or > 3]
A319350(i) = A319350(j) => a(i) = a(j), [A007733(p)]
A319704(i) = A319704(j) => a(i) = a(j), [p mod 4]
A319705(i) = A319705(j) => a(i) = a(j), [A286622(p)]
A331304(i) = A331304(j) => a(i) = a(j), [parity of A000720(p)]
A336855(i) = A336855(j) => a(i) = a(j). [distance to the next larger prime]

Links

Crossrefs

Cf. A000720, A065091, A305800, A305900.
Cf. A305900, A319350, A319704, A319705, A331304, A336855 (sequences with finer equivalence class partitioning).
Cf. A305800, A305890, A305891, A305896, A318500, A318888, A319346, A319347, A319349, A319701, A322591, A322809, A322810, A323078, A323367, A323082, A323369, A323370, A323371, A323374, A323400, A324401, A326199, A326201, A326203, A326203, A328470, A329608, A331174, A331730, A331301 (sequences with coarser equivalence class partitioning).
Cf. also A003602, A103391, A295300, A305795, A324400, A331300, A336460 (for similar constructions or similarly useful sequences).

Programs

  • Mathematica
    Array[If[# <= 2, #, If[PrimeQ[#], 3, 2 + # - PrimePi[#]]] &, 105] (* Michael De Vlieger, Oct 18 2021 *)
  • PARI
    A305801(n) = if(n<=2,n,if(isprime(n),3,2+n-primepi(n)));

Formula

a(1) = 1, a(2) = 2; for n > 2, a(n) = 3 for odd primes, and a(n) = 2+n-A000720(n) for composite n.
For n > 2, a(n) = 1 + A305800(n).

Extensions

Name changed and Comment section rewritten by Antti Karttunen, Oct 17 2021

A062173 a(n) = 2^(n-1) mod n.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 1, 0, 4, 2, 1, 8, 1, 2, 4, 0, 1, 14, 1, 8, 4, 2, 1, 8, 16, 2, 13, 8, 1, 2, 1, 0, 4, 2, 9, 32, 1, 2, 4, 8, 1, 32, 1, 8, 31, 2, 1, 32, 15, 12, 4, 8, 1, 14, 49, 16, 4, 2, 1, 8, 1, 2, 4, 0, 16, 32, 1, 8, 4, 22, 1, 32, 1, 2, 34, 8, 9, 32, 1, 48, 40, 2, 1, 32, 16, 2, 4, 40, 1, 32, 64, 8, 4, 2, 54, 32, 1, 58, 58, 88, 1, 32, 1, 24, 46
Offset: 1

Views

Author

Henry Bottomley, Jun 12 2001

Keywords

Comments

If p is an odd prime then a(p)=1. However, a(n) is also 1 for pseudoprimes to base 2 such as 341.

Examples

			a(5) = 2^(5-1) mod 5 = 16 mod 5 = 1.

Links

Crossrefs

Cf. A000079, A001567, A015910, A015919, A062172.
Cf. A082495, A106262, A257531, A305890.
Cf. A176997 (after the initial term, gives the positions of ones).

Programs

  • Haskell
    import Math.NumberTheory.Moduli (powerMod)
a062173 n = powerMod 2 (n - 1) n  -- Reinhard Zumkeller, Oct 17 2015
  • Magma
    [Modexp(2,n-1,n): n in [1..110]]; // G. C. Greubel, Jan 11 2023
  • Mathematica
    Array[Mod[2^(# - 1), #] &, 105] (* Michael De Vlieger, Jul 01 2018 *)
Array[PowerMod[2,#-1,#]&,120] (* Harvey P. Dale, May 17 2023 *)
  • PARI
    A062173(n) = if(1==n, 0, lift(Mod(2, n)^(n-1))); \\ Antti Karttunen, Jul 01 2018
  • SageMath
    [power_mod(2,n-1,n) for n in range(1,110)] # G. C. Greubel, Jan 11 2023

Formula

a(n) = A106262(2*n-3, n-2). - G. C. Greubel, Jan 11 2023

Extensions

More terms from Antti Karttunen, Jul 01 2018

A319701 Filter sequence for sequences that are constant for all odd terms >= 3.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 3, 10, 3, 11, 3, 12, 3, 13, 3, 14, 3, 15, 3, 16, 3, 17, 3, 18, 3, 19, 3, 20, 3, 21, 3, 22, 3, 23, 3, 24, 3, 25, 3, 26, 3, 27, 3, 28, 3, 29, 3, 30, 3, 31, 3, 32, 3, 33, 3, 34, 3, 35, 3, 36, 3, 37, 3, 38, 3, 39, 3, 40, 3, 41, 3, 42, 3, 43, 3, 44, 3, 45, 3, 46, 3, 47, 3, 48, 3, 49, 3, 50, 3, 51, 3, 52, 3, 53, 3, 54, 3, 55, 3, 56, 3, 57, 3, 58, 3, 59, 3, 60, 3, 61, 3, 62
Offset: 1

Views

Author

Antti Karttunen, Oct 02 2018

Keywords

Comments

For all i, j:
A305801(i) = A305801(j) => A305890(i) = A305890(j) => a(i) = a(j).
a(i) = a(j) => A007814(i) = A007814(j) => A000035(i) = A000035(j).

Links

Crossrefs

Cf. A305801, A305890, A319702.

Programs

  • PARI
    A319701(n) = if(n<=2, n, if(n%2, 3, 2+(n/2)));

Formula

a(1) = 1, and for n > 1, if n is odd, a(n) = 3, otherwise [when n is even], a(n) = 2+(n/2).
Showing 1-3 of 3 results.

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