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A298360
Numbers congruent to {3, 7, 13, 27} mod 30.
1
3, 7, 13, 27, 33, 37, 43, 57, 63, 67, 73, 87, 93, 97, 103, 117, 123, 127, 133, 147, 153, 157, 163, 177, 183, 187, 193, 207, 213, 217, 223, 237, 243, 247, 253, 267, 273, 277, 283, 297, 303, 307, 313, 327, 333, 337, 343, 357, 363, 367, 373, 387, 393, 397, 403
OFFSET
1,1
COMMENTS
For any m >= 0, if F(m) = 2^(2^m) + 1 has a factor of the form b = a(n)*2^k + 1 with even k >= m + 2 and n >= 1, then the cofactor of F(m) is equal to F(m)/b = j*2^k + 1, where j is congruent to 7 mod 10 if n is odd, or j is congruent to 3 mod 10 if n is even. That is, the integer a(n) + j must be divisible by 10.
FORMULA
a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5.
a(n) = a(n-4) + 30.
G.f.: x*(3 + 4*x + 6*x^2 + 14*x^3 + 3*x^4)/((1 + x)*(1 + x^2)*(1 - x)^2).
EXAMPLE
37 belongs to this sequence and d = 37*2^16 + 1 is a divisor of F(9) = 2^(2^9) + 1, so 10 | (37 + (F(9)/d - 1)/2^16).
MATHEMATICA
LinearRecurrence[{1, 0, 0, 1, -1}, {3, 7, 13, 27, 33}, 60]
CoefficientList[ Series[(3 + 4x + 6x^2 + 14x^3 + 3x^4)/((-1 + x)^2 (1 + x + x^2 + x^3)), {x, 0, 54}], x] (* Robert G. Wilson v, Feb 08 2018 *)
PROG
(Magma) [n: n in [0..403] | n mod 30 in {3, 7, 13, 27}];
(PARI) Vec(x*(3 + 4*x + 6*x^2 + 14*x^3 + 3*x^4)/((1 + x)*(1 + x^2)*(1 - x)^2 + O(x^55)))
CROSSREFS
Subsequence of A063226.
Sequence in context: A125898 A146928 A068673 * A140465 A333653 A301594
KEYWORD
nonn,easy
AUTHOR
STATUS
approved