A350916 Positive integers k such that (k+1)^4 has a divisor congruent to -1 modulo k.

1, 2, 3, 5, 9, 11, 14, 17, 29, 35, 41, 43, 59, 65, 69, 125, 134, 139, 174, 194, 339, 386, 449, 461, 681, 901, 937, 1169, 1322, 1325, 1715, 1971, 2211, 3054, 6395, 7989, 8857, 9077, 10849, 11483, 12545, 13082, 20909, 21506, 23861, 35233, 54734, 62210, 66923, 89045, 129494, 143289, 172899, 174725, 203321, 332315, 375129, 390051, 426389, 493697, 561513, 982094

Offset

1,2

Comments

For (k+1)^3 similar sequence is finite {1, 2, 3, 5, 9, 11, 14}, while for (k+1)^2 it is just {1, 2, 3, 5}. Starting with power 4 (this sequence), the number of values of k is infinite. One series of values for power 6 is given by A001570.

Formed by the union of 10 linear recurrent sequences satisfying b(n) = q*b(n-1) - b(n-2) - 4: A350919 (q=3), A350920 (q=4), A350921 (q=6), A350922 (q=7), A350923 (q=10), A103974 (q=14), A350924 (q=16), A350925 (q=16), A350926 (q=23), A350917 (q=23). Each of them give identities (b(n)+1)^4 = (b(n)*b(n-1)-1) * (b(n)*b(n+1)-1).

Only terms 1, 2, 5, 9, 11, 14, 29 are shared between two or more sequences, all others come from exactly one sequence.

Links

Max Alekseyev, Table of n, a(n) for n = 1..5000

Prog

(PARI) { for(k=1, 10^6, fordiv((k+1)^4, d, if(Mod(d, k)==-1, print1(k, ", "); break)) ); }

Crossrefs

Union of A350917, A350919, A350920, A350921, A350922, A350923, A103974, A350924, A350925, A350926.

Cf. A001570.

Sequence in context: A163292 A137518 A137509 * A014109 A328642 A102940

Adjacent sequences: A350913 A350914 A350915 * A350917 A350918 A350919

Keyword

nonn

Author

Max Alekseyev, Jan 21 2022

Status

approved