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Search: a091011 -id:a091011
Displaying 1-7 of 7 results found. page 1
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A091009 Number of triples (u,v,w) of divisors of n with v-u = w-v, and u < v < w. +10
10
0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 5, 0, 0, 0, 1, 0, 4, 0, 0, 0, 0, 0, 6, 0, 0, 0, 1, 0, 2, 0, 0, 3, 0, 0, 7, 0, 0, 0, 0, 0, 3, 0, 2, 0, 0, 0, 11, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 10, 0, 0, 2, 0, 0, 2, 0, 2, 0, 0, 0, 9, 0, 0, 0, 0, 0, 10, 1, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,12
COMMENTS
a(A091014(n))=n and a(m)<>n for m<=A091014(n);
a(A091010(n))=0; a(A091011(n))>0; a(A091012(n))=1; a(A091013(n))>1.
Number of pairs (x,y) of divisors of n with x<y such that also 2y-x is a divisor of n. - Antti Karttunen, Sep 10 2018
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..65537
EXAMPLE
a(30)=4, as there are exactly 4 triples of divisors with the defining property: (1,2,3), (1,3,5), (2,6,10) and (5,10,15).
MATHEMATICA
Array[Count[Subsets[#, {3}], _?(#2 - #1 == #3 - #2 & @@ # &)] &@ Divisors@ # &, 105] (* Michael De Vlieger, Sep 10 2018 *)
PROG
(PARI) A091009(n) = if(1==n, 0, my(d=divisors(n), c=0); for(i=1, (#d-1), for(j=(i+1), #d, if(!(n%(d[j]+(d[j]-d[i]))), c++))); (c)); \\ Antti Karttunen, Sep 10 2018
CROSSREFS
Cf. also A094518.
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Dec 13 2003
EXTENSIONS
Definition clarified by Antti Karttunen, Sep 10 2018
STATUS
approved
A091010 Numbers having no divisor d such that also d-x and d+x are divisors for some x. +10
4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
A091009(a(n)) = 0; complement of A091011.
LINKS
Table of n, a(n) for n=1..71.
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Dec 13 2003
STATUS
approved
A091012 Numbers having exactly one divisor d such that for some x also d-x and d+x are divisors. +10
3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
A091009(a(n)) = 1.
LINKS
Table of n, a(n) for n=1..45.
CROSSREFS
Cf. A091013, A091011.
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Dec 13 2003
STATUS
approved
A094529 Numbers with at least one arithmetic progression of four divisors (not necessarily consecutive). +10
3
12, 24, 36, 48, 60, 72, 84, 96, 105, 108, 120, 132, 140, 144, 156, 168, 180, 192, 204, 210, 216, 228, 240, 252, 264, 276, 280, 288, 300, 312, 315, 324, 336, 348, 360, 372, 384, 396, 408, 420, 432, 440, 444, 456, 468, 480, 492, 504, 516, 525, 528, 540, 552, 560 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
All multiples of terms are also terms.
a(n) = m*A094530(k) for appropriate m and k.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..10000
MATHEMATICA
sd4Q[n_]:=Count[Subsets[Divisors[n], {4}], _?(Length[Union[ Differences[ #]]] == 1&)]>0; Select[Range[600], sd4Q] (* Harvey P. Dale, Mar 19 2016 *)
CROSSREFS
Cf. A091011, A094530.
See A270571 for another version.
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, May 07 2004
EXTENSIONS
Edited by Harvey P. Dale and Alois P. Heinz, Mar 18 2016
STATUS
approved
A091013 Numbers having more than one divisor d such that for some x also d-x and d+x are divisors. +10
2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
A091009(a(n)) > 1.
LINKS
Table of n, a(n) for n=1..56.
CROSSREFS
Cf. A091012, A091011, A091010.
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Dec 13 2003
STATUS
approved
A270571 Numbers with at least one arithmetic progression of four consecutive divisors. +10
2
12, 24, 36, 48, 60, 72, 84, 96, 105, 108, 120, 132, 144, 156, 168, 180, 192, 204, 216, 228, 240, 252, 264, 276, 288, 300, 312, 315, 324, 336, 348, 360, 372, 384, 396, 408, 420, 432, 444, 456, 468, 480, 492, 504, 516, 525, 528, 540, 552, 564, 576, 588, 600 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Contrast A094529 where the divisors in arithmetic progression do not have to be consecutive.
LINKS
Harvey P. Dale, Table of n, a(n) for n = 1..1000
EXAMPLE
348 is included because its divisors are 1, 2, 3, 4, 6, 12, 29, 58, 87, 116, 174, and 348, and the first four are in arithmetic progression.
MATHEMATICA
ap4dQ[n_]:=Count[Partition[Divisors[n], 4, 1], _?(Length[ Union[ Differences[ #]]] == 1&)]>0; Select[ Range[700], ap4dQ]
CROSSREFS
Cf. A091011, A094529, A094530.
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Mar 19 2016
EXTENSIONS
Edited by Harvey P. Dale and Alois P. Heinz, Mar 19 2016
STATUS
approved
A212308 Numbers with no proper divisor that is not in an arithmetic progression of at least three proper divisors. +10
1
1, 6, 12, 15, 18, 24, 30, 36, 45, 48, 54, 60, 66, 72, 75, 84, 90, 91, 96, 108, 120, 132, 135, 144, 150, 162, 168, 180, 192, 198, 216, 225, 240, 252, 264, 270, 276, 288, 300, 306, 312, 324, 330, 336, 360, 375, 384, 396, 405, 420, 432, 435, 450, 480, 486, 504 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Equivalently, the numbers with exactly one divisor that is not in an arithmetic progression of at least three divisors.
Contains p^j*(2*p-1)^k for j,k>=1 if p and 2*p-1 are primes. - Robert Israel, Apr 13 2020
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
EXAMPLE
36 appears in this sequence because its proper divisors are 1, 2, 3, 4, 6, 9, 12 and 18, each of which appears in at least one of the following arithmetic progressions of at least three proper divisors of 36: {1, 2, 3, 4}, {3, 6, 9, 12}, {6, 12, 18}.
MAPLE
filter:= proc(n) local S, D, tau, a, b;
S:= numtheory:-divisors(n) minus {n};
D:= sort(convert(S, list));
tau:= nops(D);
for a from 1 to tau-2 do for b from a+1 to tau-1 do
if member(2*D[b]-D[a], D) then
S:= S minus {D[a], D[b], 2*D[b]-D[a]};
if S = {} then return true fi;
fi
od od;
false;
end proc:
filter(1):= true:
select(filter, [$1..1000]); # Robert Israel, Apr 13 2020
MATHEMATICA
filterQ[n_] := Module[{S, D, tau, a, b}, S = Most @ Divisors[n]; D = S; tau = Length[D]; For[a = 1, a <= tau - 2, a++, For[b = a + 1, b <= tau - 1, b++, If [MemberQ[D, 2 D[[b]] - D[[a]]], S = S ~Complement~ {D[[a]], D[[b]], 2 D[[b]] - D[[a]]}; If[S == {}, Return[True]]]]]; False];
filterQ[1] = True;
Select[Range[1000], filterQ] (* Jean-François Alcover, Sep 26 2020, after Robert Israel *)
CROSSREFS
Cf. A091010, A091011.
Contains A033845, A129521.
KEYWORD
nonn
AUTHOR
William Rex Marshall, Oct 24 2013
STATUS
approved
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Last modified August 29 06:09 EDT 2024. Contains 375510 sequences. (Running on oeis4.)