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

A318885 If n = p^a * q^b * ... * r^c, with p < q < r primes, with nonzero exponents a, b, c, then a(n) = prime(1+p-p)^a * prime(1+q-p)^b * ... * prime(1+r-p)^c; a(1) = 1.

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 2, 8, 4, 14, 2, 12, 2, 26, 10, 16, 2, 18, 2, 28, 22, 58, 2, 24, 4, 74, 8, 52, 2, 42, 2, 32, 46, 106, 10, 36, 2, 122, 62, 56, 2, 78, 2, 116, 20, 158, 2, 48, 4, 98, 94, 148, 2, 54, 34, 104, 118, 214, 2, 84, 2, 226, 44, 64, 46, 174, 2, 212, 146, 182, 2, 72, 2, 302, 50, 244, 22, 222, 2, 112, 16, 346, 2, 156, 82, 362, 206, 232, 2
Offset: 1

Views

Author

Antti Karttunen, Sep 24 2018

Keywords

Examples

			For n = 10 = 2^1 * 5^1, a(n) = prime(1)^1 * prime(1+5-2)^1 = prime(1) * prime(4) = 2*7 = 14.
For n = 55 = 5^1 * 11^1, a(n) = prime(1)^1 * prime(1+11-5)^1 = prime(1) * prime(7) = 2*17 = 34.
For n = 90 = 2^1 * 3^2 * 5^1, a(n) = prime(1)^1 * prime(1+3-2)^2 * prime(1+5-2)^1 = 2^1 * 3^2 * 7^1 = 126.

Links

Crossrefs

Cf. A318887 (rgs-transform), A318888.

Programs

  • PARI
    A318885(n) = if(1==n,n,my(f=factor(n),m=2^f[1,2],i=1); for(k=2,#f~,i += (f[k,1]-f[k-1,1]); m *= prime(i)^f[k,2]); (m));

A318887 Filter sequence related to the differences of prime factors of n.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Sep 24 2018

Keywords

Comments

Restricted growth sequence transform of A318885.

Links

Crossrefs

Cf. A318885, A318888.

Programs

  • PARI
    up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A318885(n) = if(1==n,n,my(f=factor(n),m=2^f[1,2],i=1); for(k=2,#f~,i += (f[k,1]-f[k-1,1]); m *= prime(i)^f[k,2]); (m));
v318887 = rgs_transform(vector(up_to,n,A318885(n)));
A318887(n) = v318887[n];
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.