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A330770
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a(n) = 19 * 8^n + 17 for n >= 0.
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2
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36, 169, 1233, 9745, 77841, 622609, 4980753, 39845905, 318767121, 2550136849, 20401094673, 163208757265, 1305670058001, 10445360463889, 83562883710993, 668503069687825, 5348024557502481, 42784196460019729, 342273571680157713, 2738188573441261585
(list;
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OFFSET
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0,1
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COMMENTS
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This sequence is the subject of the 4th problem of the 12th British Mathematical Olympiad in 1976 (see the link BMO).
Proposition: a(n) is never a prime number.
Proof:
If n is even, 3 divides a(n),
if n is odd with n = 4*k+1, 13 divides a(n), and
if n is odd with n = 4*k+3, 5 divides a(n).
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REFERENCES
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A. Gardiner, The Mathematical Olympiad Handbook: An Introduction to Problem Solving, Oxford University Press, 1997, reprinted 2011, Problem 4 pp. 70 and 216-217 (1991).
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LINKS
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FORMULA
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G.f.: (36 - 155*x) / ((1 - x)*(1 - 8*x)).
a(n) = 9*a(n-1) - 8*a(n-2) for n>1.
(End)
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EXAMPLE
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a(4) = 19 * 8^4 + 17 = 77841 = 3 * 25947.
a(5) = 19 * 8^5 + 17 = 622609 = 13 * 47893.
a(7) = 19 * 8^7 + 17 = 39845905 = 5 * 7969181.
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MAPLE
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B:=seq(19*8^n+17, n=0..40);
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MATHEMATICA
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Table[19 * 8^n + 17, {n, 0, 19}] (* Amiram Eldar, Feb 23 2020 *)
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PROG
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(PARI) Vec((36 - 155*x) / ((1 - x)*(1 - 8*x)) + O(x^20)) \\ Colin Barker, Feb 25 2020
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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