A109323 -id:A109323 - cp's OEIS frontend

A109324 Gap lengths associated with A109323.

Original entry on oeis.org

1, 3, 5, 9, 11, 15, 17, 19, 23, 29, 31, 35, 47, 53, 55, 57, 59, 67, 71, 73, 81, 83, 85

Offset: 1

Max Alekseyev and Jud McCranie, Aug 08 2005

Cf. A109322, A109323, A110875.

a(19)-a(23) from Donovan Johnson, Jan 25 2012

A083531 First difference sequence of A002191. Differences between possible values for sum of divisors of n.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, May 20 2003

nonn

			8 and 12 are the 6th and 7th possible values for sigma(x), since they are sum of divisors of x = 7 and x = 11 respectively, while 9, 10, 11 are impossible ones so 12 - 8 = 4 = a(6) = A002191(7) - A002191(6).
From _Michael De Vlieger_, Jul 22 2017: (Start)
First position of values:
Value   First position
    1         2
    2         1
    3        10
    4         6
    5        30
    6        24
    7       277
    8       165
    9       509
   10       150
   11       824
   12       400
   13     10970
   14      1400
   15     10448
   16      1182
   17     18731
   18      2218
   19    209237
   20      3420
   21    127385
   22      6910
   23     28899
   24      5377
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A002191, A007609, A007369, A083532, A083533, A083534, A083535, A083536, A109323 (start of record gaps in A002191).

t=Table[DivisorSigma[1, w], {w, 1, 25000}]; u=Union[%]; Delete[u-RotateRight[u], 1]
(* Second program: *)
With[{nn = 300}, Differences@ TakeWhile[Union@ DivisorSigma[1, Range@ nn], # < nn &]] (* Michael De Vlieger, Jul 22 2017 *)

A110875 Minimum positive integer such that length of the gap described at A109322 is exactly n (in contrast to A109322 where the gap length is >= n).

Original entry on oeis.org

2, 16, 9, 64, 49, 872, 481, 1768, 423, 2980, 1333, 49180, 5335, 46666, 4425, 86815, 8763, 1109259, 14089, 658513, 29883, 137539, 22825, 10927365, 259843, 1667974, 46773, 7698572, 40291, 16048081, 178705, 16039804, 1135023, 132082042, 661285, 525395164

Offset: 1

Bojan Basic (bbasic(AT)ptt.yu), Sep 18 2005

nonn

Conjectures and open problems: 1) It is not known whether the sequence is infinite; 2) It is conjectured that for every n there is corresponding a(n). If Conjecture 2) were proved, Conjecture 1) would follow as a direct consequence.
a(50) > 10^10. - Donovan Johnson, Jan 25 2012
Note that the sequence appears to undulate with terms that satisfy a(2n-1) < a(2n) < a(2n+1). Is there an explanation? - Michel Marcus, Nov 21 2013

			a(2)=16 because 16,17 are not contained in values of sigma(k) and 15,18 are; namely: sigma(8)=15 and sigma(10)=18, where sigma(k)=sum of all positive divisors of k.

Donovan Johnson, Table of n, a(n) for n = 1..49
Art of Problem Solving, Gaps in {sigma(n)}...

Cf. A109322, A109323, A109324.

Cf. A231965 (analog for sigma(n) - n).

oksuccs(v, vi, n) = {for (i = 1, n-1, if (! vecsearch(v, vi+i, ) , return (0));); return(! vecsearch(v, vi-1) && !vecsearch(v, vi+n));}
a(n) = {v = readvec("suntouch2.log"); for (i=1, #v, vi = v[i]; if (oksuccs(v, vi, n), return(vi)););} \\ where file read by readvec is the second column of b-file.  Michel Marcus, Nov 21 2013

A109322 a(n) is the minimum positive integer j such that [j, j+n-1] does not contain any values of sigma(k) (i.e., sum of all positive divisors of k).

Original entry on oeis.org

2, 9, 9, 49, 49, 423, 423, 423, 423, 1333, 1333, 4425, 4425, 4425, 4425, 8763, 8763, 14089, 14089, 22825, 22825, 22825, 22825, 40291, 40291, 40291, 40291, 40291, 40291, 178705, 178705, 661285, 661285, 661285, 661285, 4543141, 4543141, 4543141, 4543141, 4543141, 4543141, 4543141, 4543141, 4543141, 4543141, 4543141, 4543141

Offset: 1

Max Alekseyev and Jud McCranie, Aug 08 2005

nonn

Donovan Johnson, Table of n, a(n) for n = 1..85 (terms < 10^10)
Art of Problem Solving, Gaps in {sigma(n)}...

Cf. A002191, A083531, A109323, A109324, A110875.

cp's OEIS Frontend

A109324 Gap lengths associated with A109323.

Views

Author

Keywords

Crossrefs

Extensions

A083531 First difference sequence of A002191. Differences between possible values for sum of divisors of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A110875 Minimum positive integer such that length of the gap described at A109322 is exactly n (in contrast to A109322 where the gap length is >= n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A109322 a(n) is the minimum positive integer j such that [j, j+n-1] does not contain any values of sigma(k) (i.e., sum of all positive divisors of k).

Views

Author

Keywords

Links

Crossrefs