A243865 -id:A243865 - cp's OEIS frontend

A243917 Number of non-twin divisors of n.

1, 2, 0, 1, 2, 2, 2, 2, 1, 4, 2, 1, 2, 4, 1, 3, 2, 4, 2, 4, 2, 4, 2, 2, 3, 4, 2, 4, 2, 5, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 6, 2, 4, 3, 4, 2, 4, 3, 6, 2, 4, 2, 6, 4, 6, 2, 4, 2, 4, 2, 4, 2, 5, 4, 6, 2, 4, 2, 6, 2, 6, 2, 4, 3, 4, 4, 6, 2, 6, 3, 4, 2, 5, 4, 4, 2, 6, 2, 9, 4, 4, 2, 4, 4, 6

Offset: 1

Juri-Stepan Gerasimov, Jun 15 2014

nonn

A divisor k of n is non-twin if neither the positive values of k - 2 nor k + 2 divide n.

			The positive divisors of 12 are: 1, 2, 3, 4, 6, 12. Of these, 1 and 3 are twin divisors, 2, 4 and 6 are also twin divisors. The unique non-twin divisor is therefore 12. So a(12) = the number of these divisors, which is 1.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A000005, A132881, A243865.

a243917[n_Integer] := Length[Select[Divisors[n], If[And[# <= 2 || Divisible[n, # - 2] == False, Divisible[n, # + 2] == False], True, False] &]]; a243917 /@ Range[120] (* Michael De Vlieger, Aug 17 2014 *)
nntd[n_]:=Module[{d=Select[Divisors[n],#>2&],t},t=Count[d,?(!Divisible[ n, #-2] && !Divisible[ n,#+2]&)]; If[!Divisible[ n,3],t++]; If[ Divisible[ n,2] && !Divisible[n,4],t++];t]; Array[nntd,100] (* _Harvey P. Dale, May 27 2016 *)

a(n) = sumdiv(n, d, (((d<=2) || (n % (d-2))) && (n % (d+2)))); \\ Michel Marcus, Jun 25 2014

a(n) = A000005(n) - A243865(n).

Corrected by Michel Marcus, Jun 27 2014

A243932 Positive integers with the same number of twin divisors as non-twin divisors.

Original entry on oeis.org

6, 8, 21, 27, 33, 35, 39, 40, 45, 51, 57, 69, 72, 75, 87, 93, 96, 105, 111, 123, 129, 141, 143, 159, 168, 177, 183, 189, 201, 213, 219, 237, 249, 252, 264, 267, 291, 297, 303, 309, 312, 321, 323, 327, 339, 381, 393, 399, 411, 417, 420, 429, 447, 453, 471, 483, 489, 501

Offset: 1

Juri-Stepan Gerasimov, Jun 15 2014

nonn

A divisor m of n is twin if the positive values of m - 2 and/or m + 2 also divides n.

A divisor k of n is non-twin if the positive values of neither k - 2 nor k + 2 divide n.

			The divisors of 40 are 1, 2, 4, 5, 8, 10, 20, 40. Of these, 2, 4, 8, 10, are twin divisors and 1, 5, 20, 40 are non-twin divisors. These are the same number of twin divisors (4) as non-twin divisors (4), so 40 is in this sequence.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A134321, A243865, A243917.

fQ[n_] := Block[{d = Divisors@ n}, Length@ d == 2Length@ Select[d, MemberQ[d, # + 2] || MemberQ[d, # - 2] &]]; Select[ Range@ 520, fQ] (* Robert G. Wilson v, Jun 22 2014 *)

isOK(n) = t=sumdiv(n, d, (d>2 && n%(d-2)==0) || (d<=n-2 && n%(d+2)==0)); if(t==numdiv(n)-t, 1, 0)
s=[]; for(n=1, 600, if(isOK(n), s=concat(s, n))); s \\ Colin Barker, Jun 30 2014

A243865(a(n)) = A243917(a(n)).

Missing term (168) inserted by Colin Barker, Jun 30 2014

A243983 Sum of twin divisors of n.

Original entry on oeis.org

0, 0, 4, 6, 0, 4, 0, 6, 4, 0, 0, 16, 0, 0, 9, 6, 0, 4, 0, 6, 4, 0, 0, 24, 0, 0, 4, 6, 0, 9, 0, 6, 4, 0, 12, 16, 0, 0, 4, 24, 0, 4, 0, 6, 9, 0, 0, 24, 0, 0, 4, 6, 0, 4, 0, 6, 4, 0, 0, 43, 0, 0, 20, 6, 0, 4, 0, 6, 4, 12, 0, 24, 0, 0, 9, 6, 0, 4, 0, 24, 4, 0, 0, 42, 0, 0, 4, 6, 0, 9

Offset: 1

Juri-Stepan Gerasimov, Jun 16 2014

nonn

See A243865 for definition of twin divisor.

			The positive divisors of 40 are 1, 2, 4, 5, 8, 10, 20, 40. Of these, 2, 4, 8, 10, are twin divisors. So a(40) = the sum of these divisors, which is 24.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf, A000203, A132748, A243865.

a(n) = A000203(n) - A243984(n).

A277994 Number of unordered integer pairs of the form {k | n, (k + 2^m) | n}, where k >= 1, m >= 0.

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 0, 3, 2, 3, 0, 8, 0, 1, 3, 4, 1, 6, 0, 6, 2, 1, 0, 12, 1, 1, 2, 2, 0, 9, 0, 5, 3, 3, 2, 11, 0, 1, 1, 9, 0, 7, 0, 2, 5, 1, 0, 16, 0, 3, 2, 2, 0, 6, 1, 4, 2, 1, 0, 17, 0, 1, 4, 6, 3, 8, 0, 5, 1, 5, 0, 17, 0, 1, 3, 2, 1, 4, 0, 12, 2, 1, 0, 13, 2

Offset: 1

Juri-Stepan Gerasimov, Nov 07 2016

nonn

Number of power-two-difference-divisor pairs of n.

			The positive divisors of 10 are 1, 2, 5, 10. Of these, {1 | 10, (1 + 2^0) | 10} = {1, 2}, {1 | 10, (1 + 2^2) | 10} = {1, 5}, {2 | 10, (2 + 2^3) | 10} = {2, 10}. So a(10) = 3.

Cf. A027750, A092506, A243865.

f:=proc(n) local D,k;
  D:= numtheory:-divisors(n);
  add(nops(D intersect map(`+`,D,2^k)), k=0..ilog2(n-1));
end proc:
map(f, [$1..100]); # Robert Israel, Nov 08 2016

f[n_] := Module[{dd = Divisors[n], k}, Sum[Length[dd ~Intersection~ (dd + 2^k)], {k, 0, Log[2, n - 1]}]];
Array[f, 100] (* Jean-François Alcover, Jul 29 2020, after Robert Israel *)

Dirichlet g.f.: zeta(s) Sum_{k>=0} Sum_{m>=1} 1/lcm(m, m+2^k)^s. - Robert Israel, Nov 08 2016

a(2^n) = n, a(A092506(n)) = 1.

Corrected by Robert Israel, Nov 08 2016

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A243917 Number of non-twin divisors of n.

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A243932 Positive integers with the same number of twin divisors as non-twin divisors.

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A243983 Sum of twin divisors of n.

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A277994 Number of unordered integer pairs of the form {k | n, (k + 2^m) | n}, where k >= 1, m >= 0.

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