A132748 -id:A132748 - cp's OEIS frontend

A132747 a(n) = number of non-isolated divisors of n.

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

Offset: 1

Leroy Quet, Aug 27 2007

nonn

A divisor d of n is non-isolated if either d-1 or d+1 divides n. a(2n-1) = 0 for all n >= 1.

			The positive divisors of 20 are 1,2,4,5,10,20. Of these, 1 and 2 are next to each other and 4 and 5 are next to each other. So a(20) = the number of these divisors, which is 4.

Ray Chandler, Table of n, a(n) for n=1..10000

Cf. A000005, A002162, A129308, A132748, A132881.

Table[Length[Select[Divisors[n], If[ # > 1, IntegerQ[n/(#*(# - 1))]] || IntegerQ[n/(#*(# + 1))] &]], {n, 1, 90}] (* Stefan Steinerberger, Oct 26 2007 *)

a(n) = my(div = divisors(n)); sumdiv(n, d, vecsearch(div, d-1) || vecsearch(div, d+1)); \\ Michel Marcus, Aug 22 2014

a(n) = A000005(n) - A132881(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = log(2) + 1 = A002162 + 1 = 1.693147.... . - Amiram Eldar, Mar 22 2024

More terms from Stefan Steinerberger, Oct 26 2007

Extended by Ray Chandler, Jun 24 2008

A132882 a(n) = the sum of the positive isolated divisors of n.

Original entry on oeis.org

1, 0, 4, 4, 6, 6, 8, 12, 13, 15, 12, 18, 14, 21, 24, 28, 18, 33, 20, 30, 32, 33, 24, 50, 31, 39, 40, 53, 30, 55, 32, 60, 48, 51, 48, 81, 38, 57, 56, 78, 42, 77, 44, 81, 78, 69, 48, 114, 57, 90, 72, 95, 54, 114, 72, 102, 80, 87, 60, 147, 62, 93, 104, 124, 84, 138, 68, 123, 96

Offset: 1

Leroy Quet, Sep 03 2007

nonn

A divisor, d, of n is isolated if neither (d-1) nor (d+1) divides n.

The convention for 1 is that it is an isolated divisor iff n is not even. - Olivier Gérard, Sep 22 2007

			The positive divisors of 56 are: 1,2,4,7,8,14,28,56. Of these, 1 and 2 are adjacent and 7 and 8 are adjacent. The isolated divisors are therefore 4,14, 28,56. So a(56) = 4 +14 +28 +56 = 102.

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

Cf. A132881, A132748.

Table[Plus@@Select[Divisors[n],(#==1||Mod[n,#-1]>0)&&Mod[n,#+1]>0&],{n,1,200}] (* Olivier Gérard, Sep 22 2007 *)

a(n) = A000203(n) - A132748(n), where A000203 is sigma(n), sum of divisors of n.

More terms from Olivier Gérard, Sep 22 2007

A133565 a(1)=1. a(n+1) = sum{k=non-isolated divisors of n} a(k). A non-isolated divisor, k, of n is a positive divisor of n where (k-1) or (k+1) divides n.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 4, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 3, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 4, 0, 1, 0, 1, 0, 4, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 2, 0, 1

Offset: 1

Leroy Quet, Sep 16 2007

nonn

a(2n) = 0 since 2n-1 has no non-isolated divisors. - Ray Chandler

			The positive divisors of 20 are 1,2,4,5,10,20. Of these, 1 and 2 are adjacent and 4 and 5 are adjacent. So the non-isolated divisors of 20 are 1,2, 4,5. Therefore a(21) = a(1) + a(2) + a(4) + a(5) = 1 + 0 + 0 + 1 = 2.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A132748, A133564.

A133565(n) = if(1==n,n,sumdiv(n-1,d,if((!((n-1)%(1+d))) || ((d>1)&&(!((n-1)%(d-1)))), A133565(d), 0))); \\ Antti Karttunen, Dec 19 2018

Extended by Ray Chandler, Jun 25 2008

A133780 Irregular array: n-th row lists the "non-isolated divisors" of (2n). A positive divisor, k, of n is non-isolated if (k-1) or (k+1) also divides n.

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 3, 1, 2, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 1, 2, 3, 1, 2, 4, 5, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 1, 2, 3, 5, 6, 1, 2, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 4, 5, 1, 2, 3, 6, 7, 1, 2, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 1, 2, 3, 1, 2, 7, 8, 1, 2, 1, 2, 3, 4, 5, 6, 1, 2, 1, 2, 1, 2, 3, 1, 2, 1, 2, 1, 2, 3, 4, 8, 9, 1

Offset: 1

Leroy Quet, Sep 23 2007

No odd integer has any non-isolated divisors. The number of terms in the n-th row of the array is A132747(2n).

			The positive divisors of 20 are 1,2,4,5,10,20. Of these, 1 and 2 are adjacent and 4 and 5 are adjacent. So the non-isolated divisors of 20 are 1,2,4,5.

Cf. A133779, A132747, A132748.

Extended by Ray Chandler, Jun 24 2008

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).

cp's OEIS Frontend

A132747 a(n) = number of non-isolated divisors of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A132882 a(n) = the sum of the positive isolated divisors of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A133565 a(1)=1. a(n+1) = sum{k=non-isolated divisors of n} a(k). A non-isolated divisor, k, of n is a positive divisor of n where (k-1) or (k+1) divides n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

A133780 Irregular array: n-th row lists the "non-isolated divisors" of (2n). A positive divisor, k, of n is non-isolated if (k-1) or (k+1) also divides n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A243983 Sum of twin divisors of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula