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%I #17 Dec 20 2018 23:59:07
%S 3,2,5,2,13,2,3,2
%N a(n) is the least prime factor of all sufficiently large numbers of the form n + Sum_{j=1..k} j!.
%C Is it possible that there are some values of n for which no such prime factor exists?
%C 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), ....
%e 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.
%e 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:
%e .
%e k | sum | prime factorization
%e ---+----------+------------------------------------------
%e 1 | 1 | (1)
%e 2 | 3 | 3
%e 3 | 9 | 3^2
%e 4 | 33 | 3 * 11
%e 5 | 153 | 3^2 * 17
%e 6 | 873 | 3^2 * 97
%e 7 | 5913 | 3^4 * 73
%e 8 | 46233 | 3^2 * 11 * 467
%e 9 | 409113 | 3^2 * 131 * 347
%e 10 | 4037913 | 3^2 * 11 * 40787
%e .
%e 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.
%e 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.
%e .
%e | prime factors < 10^7 of
%e | all sufficiently large numbers
%e n | a(n) | of the form n + Sum_{j=1..k} j!
%e ---+---------+--------------------------------
%e 0 | 3 | 3, 11
%e 1 | 2 | 2
%e 2 | 5 | 5, 7, 274453
%e 3 | 2 | 2, 3, 23, 67, 227, 10331
%e 4 | 13 | 13, 71, 77687
%e 5 | 2 | 2, 17, 113
%e 6 | 3 | 3, 139, 2437, 4337
%e 7 | 2 | 2, 5, 349
%e 8 | ? | (none)
%e 9 | 2 | 2, 3, 7, 126323
%e 10 | 4709681 | 4709681
%e 11 | 2 | 2, 11, 19, 661
%e 12 | 3 | 3, 5, 181, 523, 15391
%e 13 | 2 | 2, 29, 2347, 41011
%e 14 | 89 | 89, 6271, 362093, 3338117
%e 15 | 2 | 2, 3, 313, 52289
%e 16 | 7 | 7
%e 17 | 2 | 2, 5, 13
%e 18 | 3 | 3, 97
%e 19 | 2 | 2, 73, 647, 16229, 3936827
%e 20 | ? | (none)
%Y Cf. A000142, A007489.
%K nonn,more
%O 0,1
%A _Jon E. Schoenfield_, Dec 15 2018
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