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.

A009191 a(n) = gcd(n, d(n)), where d(n) is the number of divisors of n (A000005).

Original entry on oeis.org

1, 2, 1, 1, 1, 2, 1, 4, 3, 2, 1, 6, 1, 2, 1, 1, 1, 6, 1, 2, 1, 2, 1, 8, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 9, 1, 2, 1, 8, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 8, 1, 2, 1, 12, 1, 2, 3, 1, 1, 2, 1, 2, 1, 2, 1, 12, 1, 2, 3, 2, 1, 2, 1, 10, 1, 2, 1, 12, 1, 2, 1, 8, 1, 6, 1, 2, 1, 2, 1, 12, 1, 2, 3, 1, 1, 2, 1, 8, 1
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

a(A046642(n)) = 1.
First occurrence of k: 1, 2, 9, 8, 400, 12, 3136, 24, 36, 80, 123904, 60, 692224, 448, 2025, 384, .... Conjecture: each k is present. - Robert G. Wilson v, Mar 27 2013
Conjecture is true. See David A. Corneth's comment in A324553. - Antti Karttunen, Mar 06 2019

Links

Crossrefs

Cf. A000005, A009194, A009195, A009205, A009213, A009230, A049820, A125168, A138010, A286540, A303781, A318459, A319337, A322979, A322980, A323073.
Cf. A046642 (positions of ones), A324553 (position of the first occurrence of each n).

Programs

Formula

a(n) = gcd(n, A000005(n)) = gcd(n, A049820(n)). - Antti Karttunen, Sep 25 2018

A259935 Infinite sequence of positive integers such that a(n) = A000005(a(1) + a(2) + ... + a(n)) for all n >= 1.

Original entry on oeis.org

2, 4, 6, 6, 4, 8, 4, 8, 4, 8, 4, 4, 8, 8, 12, 4, 8, 4, 8, 4, 3, 4, 4, 15, 8, 10, 4, 8, 8, 8, 4, 16, 4, 8, 8, 6, 6, 8, 4, 16, 4, 8, 12, 4, 4, 8, 4, 16, 12, 4, 8, 4, 8, 8, 16, 4, 8, 8, 8, 8, 8, 8, 4, 16, 4, 8, 12, 8, 16, 12, 8, 16, 12, 4, 4, 8, 8, 8, 8, 8, 24, 8, 12, 8, 4, 8, 8, 8, 16, 8, 6, 6, 8, 4, 8, 4, 8, 8, 12, 8, 18, 8, 32, 24, 18, 4, 8, 16, 4, 16, 4, 8, 12, 8, 8, 8, 8, 8, 8, 12
Offset: 1

Views

Author

Max Alekseyev, Jul 09 2015

Keywords

Comments

V. S. Guba (2015) proved that such an infinite sequence exists. Numerical evidence suggests that it may also be unique (cf. A259934).
If there are infinitely many n with a(n) = a(n+1), then A175304 is infinite (see comment in A259934). - Vladimir Shevelev, Jul 21 2015

Links

Crossrefs

First differences of A259934.
Cf. A000005, A260257, A260085, A260123, A286540.

Formula

a(n) = A000005(A259934(n)) = A259934(n) - A259934(n-1).
gcd(a(n), A259934(n)) = A286540(n) = A009191(A259934(n)). - Antti Karttunen, Nov 26 2017

A323073 Number of iterations of A049820(x) = x - A000005(x) needed to reach either zero or such x that x and A049820(x) are coprime, when starting from x = n.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 2, 0, 1, 3, 3, 0, 3, 0, 4, 0, 0, 0, 4, 0, 5, 0, 5, 0, 1, 0, 6, 0, 6, 0, 6, 0, 7, 0, 7, 0, 1, 0, 8, 0, 8, 0, 8, 0, 9, 1, 9, 0, 9, 0, 10, 0, 10, 0, 10, 0, 10, 0, 11, 0, 10, 0, 12, 1, 0, 0, 12, 0, 13, 0, 13, 0, 11, 0, 14, 1, 14, 0, 14, 0, 14, 0, 15, 0, 12, 0, 16, 0, 15, 0, 15, 0, 17, 0, 16, 0, 13, 0
Offset: 0

Views

Author

Antti Karttunen, Jan 05 2019

Keywords

Links

Crossrefs

Cf. A000005, A009191, A049820, A155043, A259934, A286540, A320009, A322974, A322987, A322996.
Cf. A046642 (positions of zeros after the initial a(0)=0).

Programs

  • PARI
    A323073(n) = if(!n,0,my(nn=(n-numdiv(n))); if(1==gcd(n,nn),0,1+A323073(nn)));
  • PARI
    A323073(n) = if(!n,0,for(j=0,oo,my(nn=(n-numdiv(n))); if((0==nn)||(1==gcd(n,nn)),return(j+(2==n)),n = nn)));

Formula

a(0) = 0; for n > 0, if A009191(n) == 1, a(n) = 0, otherwise a(n) = 1 + a(n-A000005(n)).
a(n) <= A155043(n).
Showing 1-3 of 3 results.

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