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A080075
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Proth numbers: of the form k*2^m + 1 for k odd, m >= 1 and 2^m > k.
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13
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3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65, 81, 97, 113, 129, 145, 161, 177, 193, 209, 225, 241, 257, 289, 321, 353, 385, 417, 449, 481, 513, 545, 577, 609, 641, 673, 705, 737, 769, 801, 833, 865, 897, 929, 961, 993, 1025, 1089, 1153, 1217, 1281, 1345, 1409
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OFFSET
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1,1
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COMMENTS
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A Proth number is a square iff it is of the form (2^(m-1)+-1)*2^(m+1)+1 = 4^m+-2^(m+1)+1 = (2^m+-1)^2 for m > 1. See A086341. - Thomas Ordowski, Apr 22 2019
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LINKS
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FORMULA
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Sum_{n>=1} 1/a(n) = 1.09332245643583252894473574405304699874426408312553... (Borsos et al., 2022). - Amiram Eldar, Jan 29 2022
a(n+1) = a(n) + 2^round(L(n)/2), where L(n) is the number of binary digits of a(n); equivalently, floor(log_2(a(n))/2 + 1) in the exponent. [Lemma 2.2 in Borsos et al.] - M. F. Hasler, Jul 07 2022
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MATHEMATICA
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Select[Range[3, 1500, 2], And[OddQ[#[[1]] ], #[[-1]] >= 1, 2^#[[-1]] > #[[1]] ] &@ Append[QuotientRemainder[#1, 2^#2], #2] & @@ {#, IntegerExponent[#, 2]} &[# - 1] &] (* Michael De Vlieger, Nov 04 2019 *)
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PROG
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(PARI) next_A080075(N)=N+2^(exponent(N)\2+1)
(PARI) a(n) = n++; my(L = logint(n, 2), A = bittest(n, L-1)); (1 << (L + A))*(n - 1 << (L - !A)) + 1 \\ Mikhail Kurkov, Mar 17 2024
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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