A243917 -id:A243917 - cp's OEIS frontend

A243865 Number of twin divisors of n.

0, 0, 2, 2, 0, 2, 0, 2, 2, 0, 0, 5, 0, 0, 3, 2, 0, 2, 0, 2, 2, 0, 0, 6, 0, 0, 2, 2, 0, 3, 0, 2, 2, 0, 2, 5, 0, 0, 2, 4, 0, 2, 0, 2, 3, 0, 0, 6, 0, 0, 2, 2, 0, 2, 0, 2, 2, 0, 0, 8, 0, 0, 4, 2, 0, 2, 0, 2, 2, 2, 0, 6, 0, 0, 3, 2, 0, 2, 0, 4, 2, 0, 0, 7, 0, 0, 2, 2, 0, 3, 0, 2, 2, 0, 0, 6

Offset: 1

Juri-Stepan Gerasimov, Jun 13 2014

nonn

A divisor m of n is a twin divisor if m-2 (for m >= 3) and m+2 (for m <= n-2) also divide n.

			The positive divisors of 20 are 1, 2, 4, 5, 10, 20. Of these, 2 and 4 are twin divisors: (2)+2 = 4, which divides n, and (4)-2 = 2 also divides n. So a(20) = the number of these divisors, which is 2.

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

Cf. A000005, A002162, A099475, A132747, A243917.

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

a(n) = A000005(n) - A243917(n).
a(3n) > 1 for all n >= 1.
a(A099477(n)) = 0, a(A059267(n)) > 0.
A099475(n) <= a(n) <= A000005(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = log(2)/2 + 17/12 = 1.7632402569... . - Amiram Eldar, Mar 22 2024

A243984 Sum of non-twin divisors of n.

Original entry on oeis.org

1, 3, 0, 1, 6, 8, 8, 9, 9, 18, 12, 12, 14, 24, 15, 25, 18, 35, 20, 36, 28, 36, 24, 36, 31, 42, 36, 50, 30, 63, 32, 57, 44, 54, 36, 75, 38, 60, 52, 66, 42, 92, 44, 78, 69, 72, 48, 100, 57, 93, 68, 92, 54, 116, 72, 114, 76, 90, 60, 125, 62, 96, 84, 121, 84, 140, 68, 120

Offset: 1

Juri-Stepan Gerasimov, Jun 16 2014

See A243917 for definition of non-twin divisor.

			The divisors of 40 are 1, 2, 4, 5, 8, 10, 20, 40. Of these, 1, 5, 20, 40 are non-twin divisors. So a(40) = the sum of these divisors, which is 66.

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

Cf. A000203, A132882, A243917, A243983.

f:= proc(n) local d; d:= numtheory[divisors](n); convert(d minus map(`+`,d,2) minus map(`+`,d,-2),`+`) end proc:
map(f, [$1..100]); # Robert Israel, Aug 17 2014

a243984[n_Integer] := Total[Select[Divisors[n], If[And[# <= 2 || Divisible[n, # - 2] == False, Divisible[n, # + 2] == False], True, False] &]]; a243984 /@ Range[68] (* Michael De Vlieger, Aug 17 2014 *)

a(n) = s=0; fordiv(n, d, if(!((d>2 && n%(d-2)==0) || (d<=n-2 && n%(d+2)==0)), s+=d)); s
for(n=1, 200, print1(a(n), ", ")) \\ Colin Barker, Jun 29 2014

a(n) = A000203(n) - A243983(n).

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

A243985 Numbers n such that A243984(n), the sum of non-twin divisors of n, is a square.

Original entry on oeis.org

1, 3, 4, 8, 9, 16, 20, 22, 24, 27, 35, 48, 64, 90, 94, 115, 119, 143, 170, 171, 192, 200, 214, 216, 217, 265, 310, 322, 323, 343, 382, 497, 517, 527, 656, 679, 710, 729, 742, 745, 782, 862, 889, 899, 935, 970, 1066, 1174, 1177, 1207, 1219, 1270, 1393, 1426

Offset: 1

Juri-Stepan Gerasimov, Jun 16 2014

nonn

See A243917 for definition of non-twin divisor.

Squares included in the sequence are : 1, 4, 9, 16, 64, 729, ...

			The positive divisors of 8 are 1, 2, 4, 8. Of these, 1 and 8 are non-twin divisors. So 8 is in this sequence, which is 1 + 8 = 3^2.

Michael De Vlieger, Table of n, a(n) for n = 1..8692 (a(n) < 5,000,000)

Cf. A006532, A243917 , A243984.

a243984[n_Integer] := Total[Select[Divisors[n], If[And[# <= 2 || Divisible[n, # - 2] == False, Divisible[n, # + 2] == False], True, False] &]]; a243985[n_Integer] := Flatten@Select[Position[Sqrt[a243984 /@ Range[n]], ?IntegerQ], If[Length[#] == 1, True, False] &]; a243985[1500] (* _Michael De Vlieger, Aug 17 2014 *)

A243984(n) = s=0; fordiv(n, d, if(!((d>2 && n%(d-2)==0) || (d<=n-2 && n%(d+2)==0)), s+=d)); s
for(n=1, 200, if(issquare(A243984(n)), print1(n, ", "))) \\ Colin Barker, Jun 29 2014

Several terms corrected by Colin Barker, Jun 29 2014

A244259 Number of non-twin isolated divisors of n.

Original entry on oeis.org

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

Offset: 1

Juri-Stepan Gerasimov, Jun 24 2014

Number of divisors d of n such that neither the positive values d - 2 nor d - 1 nor d + 1 nor d + 2 divide n.

An isolated divisor d is non-twin if neither the positive values d - 2 nor d + 2 divide n.

			The isolated divisors of 56 are: 4, 14, 28, 56. The non-twin divisors of 56 are: 1, 7, 8, 14, 28, 56. The non-twin isolated divisors of 56 are therefore 14, 28, 56. There are 3 of these, so a(56) = 3.

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

Cf. A132881 (number of isolated divisors of n), A243917 (number of non-twin divisors of n).

a(n) = sumdiv(n, d, (((d<=2) || (n % (d-2))) && ((d<=1) || (n % (d-1))) && (n % (d+1)) && (n % (d+2)))); \\ Michel Marcus, Jun 25 2014

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A243865 Number of twin divisors of n.

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A243984 Sum 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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A243985 Numbers n such that A243984(n), the sum of non-twin divisors of n, is a square.

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A244259 Number of non-twin isolated divisors of n.

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