A160752 -id:A160752 - cp's OEIS frontend

A067131 Number of elements in the largest set of divisors of n which are in arithmetic progression.

1, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 4, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 4, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 4, 2, 2, 2, 3, 2, 3, 2, 2, 3, 2, 2, 4, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 6, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 4, 2, 2, 3, 2, 2, 3, 2, 3, 2, 2, 2, 4, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 3, 2

Offset: 1

Amarnath Murthy, Jan 09 2002

			a(12) = 4 as the divisors of 12 are {1,2,3,4,6,12} and the maximal subset in arithmetic progression is {1,2,3,4}. a(15) = 3; the maximal set is {1,3,5}.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A067132, A091009, A160752, A319354.

lap[s_] := Module[{}, l=Length[s]; If[l<2, Return[l]]; val=2; For[i=1, ival, val=k]]]; val]; lap/@Divisors/@Range[1, 200]

A067131(n) = { 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 = max(m,c))); (m); }; \\ Antti Karttunen, Sep 21 2018

a(n) = A061395(A319354(n)). - Antti Karttunen, Sep 21 2018

Edited by Dean Hickerson, Jan 15 2002

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

Antti Karttunen, Sep 21 2018

nonn

			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.

Antti Karttunen, Table of n, a(n) for n = 1..4096

Cf. A319355 (rgs-transform).

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));

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

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

Antti Karttunen, Sep 21 2018

nonn

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).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A319354.

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];

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A067131 Number of elements in the largest set of divisors of n which are in arithmetic progression.

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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.

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

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PARI