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URL: https://oeis.org/A215653

⇱ A215653 - OEIS

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A215653
a(n) = smallest positive m such that m^2 = 1+k*n with positive k.
4
2, 3, 2, 3, 4, 5, 6, 3, 8, 9, 10, 5, 12, 13, 4, 7, 16, 17, 18, 9, 8, 21, 22, 5, 24, 25, 26, 13, 28, 11, 30, 15, 10, 33, 6, 17, 36, 37, 14, 9, 40, 13, 42, 21, 19, 45, 46, 7, 48, 49, 16, 25, 52, 53, 21, 13, 20, 57, 58, 11, 60, 61, 8, 31, 14, 23, 66, 33, 22, 29
OFFSET
1,1
COMMENTS
Apparently a(n) = A070667(n) for n > 2. Note the linear patterns in the graph.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Zak Seidov)
Dorin Andrica and Vlad Crişan, The Smallest Nontrivial Solution to x^k == 1 (mod n) and Related Sequences, Amer. Math. Monthly, Vol. 126, No. 2 (2019), pp. 173-178.
FORMULA
a(n) = sqrt(1+n*A076942(n)).
a(n) = sqrt(A061369(n)). - Amiram Eldar, Mar 16 2025
EXAMPLE
a(1) = 2, k = 3; a(2) = 3, k = 4; a(3) = 2, k = 1; a(1000) = 249, k = 62.
MATHEMATICA
Flatten[{2, Table[Select[Range[2, 1000], PowerMod[#, 2, k]==1&, 1], {k, 2, 1000}]}] (* first 1000 terms *)
PROG
(PARI) a(n) = {my(m = n + 1); while(!issquare(m), m += n); sqrtint(m); } \\ Amiram Eldar, Mar 16 2025
CROSSREFS
KEYWORD
nonn
AUTHOR
Zak Seidov, Aug 19 2012
STATUS
approved