A091009 -id:A091009 - cp's OEIS frontend

A091011 Numbers having at least one divisor, d, such that for some x, d-x and d+x are also divisors.

6, 12, 15, 18, 24, 28, 30, 36, 40, 42, 45, 48, 54, 56, 60, 66, 72, 75, 78, 80, 84, 90, 91, 96, 102, 105, 108, 112, 114, 120, 126, 132, 135, 138, 140, 144, 150, 153, 156, 160, 162, 165, 168, 174, 180, 182, 186, 190, 192, 195, 196, 198, 200, 204, 210, 216, 220

Offset: 1

Reinhard Zumkeller, Dec 13 2003

nonn

A091009(a(n)) > 0; complement of A091010.

Numbers k with at least one pair of divisors, (d1,d2), with d1 < d2, whose (integer) average divides k. - Wesley Ivan Hurt, Aug 23 2020

Cf. A091009, A091010, A337331.

Table[If[Sum[Sum[(1 - Ceiling[(i + k)/2] + Floor[(i + k)/2]) (1 - Ceiling[2 n/(i + k)] + Floor[2 n/(i + k)]) (1 - Ceiling[n/k] + Floor[n/k]) (1 - Ceiling[n/i] + Floor[n/i]), {i, k - 1}], {k, n}] > 0, n, {}], {n, 200}] // Flatten (* Wesley Ivan Hurt, Aug 23 2020 *)

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

Original entry on oeis.org

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

A094518 Number of pairs (x,y) of divisors of n with x

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 5, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 6, 0, 0, 0, 2, 0, 3, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 3, 0, 1, 0, 0, 0, 12, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 11, 0, 0, 0, 0, 0, 2, 0, 3, 0, 0, 0, 9, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 9, 0, 0, 0, 2, 0, 2

Offset: 1

Reinhard Zumkeller, May 06 2004

nonn

a(A094519(n)) > 0, a(A094520(n)) = 0.

a(A097370(n)) = n and a(m) <> n for m < A097370(n).

			n=30 with divisor set {1,2,3,5,6,10,15,30}: a(30)=4, as 1<2 & 3=1+2, 1<5 & 6=1+5, 2<3 & 5=2+3 and 5<10 & 15=5+10.

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

Cf. A091009.

Table[Count[Subsets[#, {2}], ?(Mod[n, Total@ #] == 0 &)] &@ Divisors@ n, {n, 102}] (* _Michael De Vlieger, Sep 10 2018 *)

A094518(n) = if(1==n,0,my(d=divisors(n),c=0); for(i=1,(#d-1),for(j=(i+1),#d,if(!(n%(d[i]+d[j])),c++))); (c)); \\ Antti Karttunen, Sep 10 2018

A091010 Numbers having no divisor d such that also d-x and d+x are divisors for some x.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 73, 74, 76, 77, 79, 81, 82, 83, 85, 86, 87, 88, 89, 92, 93, 94

Offset: 1

Reinhard Zumkeller, Dec 13 2003

nonn

A091009(a(n)) = 0; complement of A091011.

A091012 Numbers having exactly one divisor d such that for some x also d-x and d+x are divisors.

Original entry on oeis.org

6, 15, 28, 40, 91, 153, 190, 220, 325, 496, 544, 561, 572, 627, 703, 861, 897, 935, 946, 1012, 1225, 1287, 1292, 1431, 1581, 1610, 1653, 1768, 1891, 2278, 2300, 2465, 2618, 2701, 2967, 3321, 3344, 3496, 3596, 3655, 3952, 4123, 4301, 4324, 4371

Offset: 1

Reinhard Zumkeller, Dec 13 2003

nonn

A091009(a(n)) = 1.

Cf. A091013, A091011.

A091013 Numbers having more than one divisor d such that for some x also d-x and d+x are divisors.

Original entry on oeis.org

12, 18, 24, 30, 36, 42, 45, 48, 54, 56, 60, 66, 72, 75, 78, 80, 84, 90, 96, 102, 105, 108, 112, 114, 120, 126, 132, 135, 138, 140, 144, 150, 156, 160, 162, 165, 168, 174, 180, 182, 186, 192, 195, 196, 198, 200, 204, 210, 216, 222, 224, 225, 228, 231, 234, 240

Offset: 1

Reinhard Zumkeller, Dec 13 2003

nonn

A091009(a(n)) > 1.

Cf. A091012, A091011, A091010.

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];

A091014 Smallest number having exactly n divisors d such that for some x also d-x and d+x are divisors.

Original entry on oeis.org

1, 6, 18, 12, 30, 24, 36, 48, 132, 84, 72, 60, 312, 264, 816, 144, 270, 168, 252, 300, 288, 120, 810, 432, 1224, 180, 1188, 1140, 792, 630, 1152, 240, 2112, 672, 4104, 504, 420, 600, 3672, 540, 1872, 480, 2592, 1584, 900, 5100, 360, 2760, 2040, 1890

Offset: 0

Reinhard Zumkeller, Dec 13 2003

nonn

A091009(a(n))=n and A091009(m)<>n for m<=a(n).

			a(9)=84: the exactly 9 triples of divisors of 84 with the defining property: (1,2,3), (2,3,4), (2,4,6), (1,4,7), (2,7,12), (3,12,21), (7,14,21), (14,21,28) and (14,28,42).

A360012 a(n) is the number of triples (u,v,w) of divisors of n with u/v = v/w, and u < v < w.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 4, 1, 0, 2, 2, 0, 0, 0, 6, 0, 0, 0, 8, 0, 0, 0, 4, 0, 0, 0, 2, 2, 0, 0, 8, 1, 2, 0, 2, 0, 4, 0, 4, 0, 0, 0, 4, 0, 0, 2, 9, 0, 0, 0, 2, 0, 0, 0, 14, 0, 0, 2, 2, 0, 0, 0, 8, 4, 0, 0, 4, 0, 0, 0

Offset: 1

Rémy Sigrist, Jan 21 2023

nonn

In other words, a(n) is the number of triples of distinct divisors of n in geometric progression.

This sequence is unbounded.

			The first terms, alongside the corresponding triples, are:
  n   a(n)  (u,v,w)'s
  --  ----  ------------------------------------
   1     0  None
   2     0  None
   3     0  None
   4     1  (1,2,4)
   5     0  None
   6     0  None
   7     0  None
   8     2  (1,2,4), (2,4,8)
   9     1  (1,3,9)
  10     0  None
  11     0  None
  12     2  (1,2,4), (3,6,12)
  13     0  None
  14     0  None
  15     0  None
  16     4  (1,2,4), (1,4,16), (2,4,8), (4,8,16)

Index entries for sequences related to divisors

Cf. A002620, A005059, A091009, A132345.

Array[Count[Subsets[#, {3}], _?(#2 / #1 == #3 / #2 & @@ # &)] &@ Divisors@ # &, 87]

a(n) = { my (d=divisors(n), v=0); for (i=1, #d-2, for (j=i+1, #d-1, for (k=j+1, #d, if (d[i]*d[k]==d[j]^2, v++)))); return (v) }

a(n) <= a(n*k) for any n, k > 0.
a(p^k) = A002620(k) for any k >= 0 and any prime number p.
a(s^2) = A005059(k) for any squarefree number s with k prime factors.

cp's OEIS Frontend

A091011 Numbers having at least one divisor, d, such that for some x, d-x and d+x are also divisors.

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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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A094518 Number of pairs (x,y) of divisors of n with x

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A091010 Numbers having no divisor d such that also d-x and d+x are divisors for some x.

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A091012 Numbers having exactly one divisor d such that for some x also d-x and d+x are divisors.

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A091013 Numbers having more than one divisor d such that for some x also d-x and d+x are divisors.

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

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A091014 Smallest number having exactly n divisors d such that for some x also d-x and d+x are divisors.

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A360012 a(n) is the number of triples (u,v,w) of divisors of n with u/v = v/w, and u < v < w.

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