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 1-4 of 4 results.

A160752 a(n) is the number of sets of (distinct, not necessarily consecutive) positive divisors of n where each set has all of its elements in arithmetic progression, and where each set contains exactly A067131(n) elements.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 6, 3, 6, 1, 1, 1, 6, 1, 10, 1, 2, 1, 15, 6, 6, 1, 2, 3, 6, 6, 1, 1, 4, 1, 15, 6, 6, 6, 2, 1, 6, 6, 1, 1, 2, 1, 15, 3, 6, 1, 3, 3, 15, 6, 15, 1, 3, 6, 2, 6, 6, 1, 1, 1, 6, 15, 21, 6, 3, 1, 15, 6, 28, 1, 4, 1, 6, 2, 15, 6, 2, 1, 2, 10, 6, 1, 2, 6, 6, 6, 28, 1, 10, 1, 15, 6, 6, 6, 4, 1, 15
Offset: 1

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Author

Leroy Quet, May 25 2009

Keywords

Comments

If A067131(n) = 2, then a(n) = d(n)*(d(n)-1)/2, where d(n) is the number of divisors of n.
a(p) = 1 for all primes p.

Examples

			The divisors of 18 are 1,2,3,6,9,18. There are 2 sets of these divisors, (1,2,3) and (3,6,9), that have their terms in arithmetic progression and that each have the maximal number (3) of such divisors of 18. So a(18) = 2.

Links

Crossrefs

Cf. A067131, A319354.

Programs

  • PARI
    A160752(n) = if(1==n,n,my(d=divisors(n),m=1,counts=vector(#d)); for(i=1,(#d-1), for(j=(i+1),#d,my(c=1,k=d[j],s=(d[j]-d[i])); while(!(n%k), k+=s; c++); counts[c]++; m = max(m,c))); (counts[m])); \\ Antti Karttunen, Sep 21 2018

Formula

a(n) = A071178(A319354(n)). - Antti Karttunen, Sep 21 2018

Extensions

Extended by Ray Chandler, Jun 15 2009

A319354 a(n) = Product prime(k), where k ranges over the lengths of all arithmetic progressions formed from the divisors of n (with at least two distinct terms each); a(1) = 2 by convention.

Original entry on oeis.org

2, 3, 3, 27, 3, 1215, 3, 729, 27, 729, 3, 93002175, 3, 729, 1215, 59049, 3, 39858075, 3, 14348907, 729, 729, 3, 576626970315375, 27, 729, 729, 23914845, 3, 176518460300625, 3, 14348907, 729, 729, 729, 6305415920398625625, 3, 729, 729, 38127987424935, 3, 63546645708225, 3, 14348907, 66430125, 729, 3, 289588836976147679079375, 27, 14348907, 729
Offset: 1

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Author

Antti Karttunen, Sep 21 2018

Keywords

Examples

			For n = 6, the arithmetic progressions found in its divisor set {1, 2, 3, 6} are: {1, 2}, {1, 3}, {2, 3}, {2, 6}, {3, 6} and {1, 2, 3}. Five of these have length 2, and one is of length 3, thus a(6) = prime(2)^5 * prime(3) = 243*5 = 1215.

Links

Crossrefs

Cf. A319355 (rgs-transform).

Programs

  • PARI
    A319354(n) = if(1==n,2,my(d=divisors(n),m=1); for(i=1,(#d-1), for(j=(i+1),#d,my(c=1,k=d[j],s=(d[j]-d[i])); while(!(n%k), k+=s; c++); m *= prime(c))); (m));

Formula

For all n >= 1:
A061395(a(n)) = A067131(n).
A071178(a(n)) = A160752(n).
For all n >= 2, A001222(a(n)) = A066446(n).

A067132 Number of elements in the largest set of divisors of n which are in geometric progression.

Original entry on oeis.org

1, 2, 2, 3, 2, 2, 2, 4, 3, 2, 2, 3, 2, 2, 2, 5, 2, 3, 2, 3, 2, 2, 2, 4, 3, 2, 4, 3, 2, 2, 2, 6, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 2, 3, 3, 2, 2, 5, 3, 3, 2, 3, 2, 4, 2, 4, 2, 2, 2, 3, 2, 2, 3, 7, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 3, 3, 2, 2, 2, 5, 5, 2, 2, 3, 2, 2, 2, 4, 2, 3, 2, 3, 2, 2, 2, 6, 2, 3, 3, 3, 2, 2, 2
Offset: 1

Views

Author

Amarnath Murthy, Jan 09 2002

Keywords

Comments

Also a(n) = minimal 'freeness' of n with regard to squares, cubes, etc: All entries where a(n) = 2 are squarefree (or prime); Entries where a(n) = 3 are cubefree (and thus free of higher powers) but not squarefree, and so on. - Carl R. White, Jul 27 2009
For n > 1, a(n) is asymptotic to A000005(n)/A001221(n). - Eric Desbiaux, Dec 10 2012

Examples

			a(12) = 3 as the divisors of 12 are {1,2,3,4,6,12} and the maximal subsets in geometric progression are {1,2,4} and {3,6,12}.
a(16) = 5; the maximal set is {1,2,4,8,16}.

Links

Crossrefs

Cf. A000961, A067131, A051903.

Programs

  • Maple
    seq(max(0,seq(padic[ordp](n, p), p in numtheory[factorset](n))) + 1, n=1..100); # Ridouane Oudra, Sep 10 2024
  • Mathematica
    a[n_] := If[n==1, 1, Max@@Last/@FactorInteger[n]+1]

Formula

If the prime factorization of n>1 is p_1^e_1 ... p_k^e_k, then a(n) = 1+max(e_1, ..., e_k).
a(n) = A051903(n) + 1. - Ridouane Oudra, Sep 10 2024

Extensions

Edited by Dean Hickerson, Jan 15 2002

A319355 Filter sequence constructed from the lengths of arithmetic progressions occurring among the divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 5, 2, 6, 2, 5, 4, 7, 2, 8, 2, 9, 5, 5, 2, 10, 3, 5, 5, 11, 2, 12, 2, 9, 5, 5, 5, 13, 2, 5, 5, 14, 2, 15, 2, 9, 16, 5, 2, 17, 3, 9, 5, 9, 2, 18, 5, 15, 5, 5, 2, 19, 2, 5, 9, 20, 5, 18, 2, 9, 5, 21, 2, 22, 2, 5, 8, 9, 5, 15, 2, 23, 7, 5, 2, 24, 5, 5, 5, 21, 2, 25, 4, 9, 5, 5, 5, 26, 2, 9, 9, 27, 2, 15, 2, 21, 28
Offset: 1

Views

Author

Antti Karttunen, Sep 21 2018

Keywords

Comments

Restricted growth sequence transform of A319354.
For all i, j:
a(i) = a(j) => A067131(i) = A067131(j).
a(i) = a(j) => A160752(i) = A160752(j).
a(i) = a(j) => A091009(i) = A091009(j).

Links

Crossrefs

Cf. A319354.

Programs

  • PARI
    up_to = 65537;
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; };
A319354(n) = if(1==n,2,my(d=divisors(n),m=1); for(i=1,(#d-1), for(j=(i+1),#d,my(c=1,k=d[j],s=(d[j]-d[i])); while(!(n%k), k+=s; c++); m *= prime(c))); (m));
v319355 = rgs_transform(vector(up_to,n,A319354(n)));
A319355(n) = v319355[n];
Showing 1-4 of 4 results.

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