A319044 a(n) is the least prime factor of all sufficiently large numbers of the form n + Sum_{j=1..k} j!.

3, 2, 5, 2, 13, 2, 3, 2

Offset

0,1

Comments

Is it possible that there are some values of n for which no such prime factor exists?

a(8), a(20), a(34), and a(76) exceed 10^7; the sequence begins 3, 2, 5, 2, 13, 2, 3, 2, a(8), 2, 4709681, 2, 3, 2, 89, 2, 7, 2, 3, 2, a(20), 2, 5, 2, 3, 2, 23, 2, 43, 2, 3, 2, 5, 2, a(34), 2, 3, 2, 41, 2, 61, 2, 3, 2, 7, 2, 6197, 2, 3, 2, 24329, 2, 5, 2, 3, 2, 13, 2, 7, 2, 3, 2, 5, 2, 281, 2, 3, 2, 19, 2, 37, 2, 3, 2, 3617, 2, a(76), ....

Links

Table of n, a(n) for n=0..7.

Example

For n=0, numbers of the form n + Sum_{j=1..k} j! are sums of factorials 1! + 2! + ... + k! (A007489). Since 1! + 2! = 1 + 2 = 3, and all additional factorials added to the sum will be multiples of 6, every number of the form n + Sum_{j=1..k} j! for n=0 and k >= 2 will be divisible by 3, but not by 2, so a(0)=3.
More can be said about the divisibility of sufficiently large sums of this form for n=0; for k = 1..10, these sums and their prime factorizations are as follows:
.
   k |    sum   | prime factorization
  ---+----------+------------------------------------------
   1 |        1 | (1)
   2 |        3 |  3
   3 |        9 |  3^2
   4 |       33 |  3 * 11
   5 |      153 |  3^2 * 17
   6 |      873 |  3^2 * 97
   7 |     5913 |  3^4 * 73
   8 |    46233 |  3^2 * 11 * 467
   9 |   409113 |  3^2 * 131 * 347
  10 |  4037913 |  3^2 * 11 * 40787
.
The sum at k=5 is divisible by 3^2 = 9, and for k > 5, each additional factorial added to the sum will also be divisible by 9, so all the sums for n=0 and k >= 6 will be divisible by 9.
Similarly, the sum at k=10 is divisible by 3^2 * 11 = 99, and 99 divides k! for every k > 10, so all the sums for n=0 and k >= 11 will likewise be divisible by 99.
.
               |     prime factors < 10^7 of
               | all sufficiently large numbers
   n |   a(n)  | of the form n + Sum_{j=1..k} j!
  ---+---------+--------------------------------
   0 |       3 | 3, 11
   1 |       2 | 2
   2 |       5 | 5, 7, 274453
   3 |       2 | 2, 3, 23, 67, 227, 10331
   4 |      13 | 13, 71, 77687
   5 |       2 | 2, 17, 113
   6 |       3 | 3, 139, 2437, 4337
   7 |       2 | 2, 5, 349
   8 |       ? | (none)
   9 |       2 | 2, 3, 7, 126323
  10 | 4709681 | 4709681
  11 |       2 | 2, 11, 19, 661
  12 |       3 | 3, 5, 181, 523, 15391
  13 |       2 | 2, 29, 2347, 41011
  14 |      89 | 89, 6271, 362093, 3338117
  15 |       2 | 2, 3, 313, 52289
  16 |       7 | 7
  17 |       2 | 2, 5, 13
  18 |       3 | 3, 97
  19 |       2 | 2, 73, 647, 16229, 3936827
  20 |       ? | (none)

Crossrefs

Cf. A000142, A007489.

Sequence in context: A281006 A046227 A120842 * A230820 A324549 A152178

Adjacent sequences: A319041 A319042 A319043 * A319045 A319046 A319047

Keyword

nonn,more

Author

Jon E. Schoenfield, Dec 15 2018

Status

approved