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A369901 Proth numbers h*2^k+1, with odd h < 2^k, ordered first by k then by h.

%I A369901 #77 Aug 15 2025 21:05:48
%S A369901 3,5,13,9,25,41,57,17,49,81,113,145,177,209,241,33,97,161,225,289,353,
%T A369901 417,481,545,609,673,737,801,865,929,993,65,193,321,449,577,705,833,
%U A369901 961,1089,1217,1345,1473,1601,1729,1857,1985,2113,2241,2369,2497,2625,2753,2881
%N A369901 Proth numbers h*2^k+1, with odd h < 2^k, ordered first by k then by h.
%H A369901 Daniel Sturm, <a href="/A369901/b369901.txt">Table of n, a(n) for n = 1..1023</a>
%H A369901 David A. Corneth, <a href="/A369901/a369901.gp.txt">PARI program</a>.
%H A369901 E. Proth, <a href="https://www.biodiversitylibrary.org/item/24818#page/932/mode/1up">Théorèmes sur les nombres premiers</a>, Comptes rendus hebdomadaires des séances de l'Académie des sciences, 1878, p. 926.
%H A369901 Wikipedia, <a href="https://en.wikipedia.org/wiki/Proth_prime">Proth Primes</a>.
%F A369901 a(2^b+c) = (2c+1)*2^(b+1)+1 for 0 <= c < 2^b. [Corrected by _Thomas Ordowski_, Aug 11 2025]
%F A369901 a(n) = (2n+1-2^m)*2^m+1 = (2n+1)*2^m-4^m+1, where m = floor(log_2(2n+1)). - _Thomas Ordowski_, Aug 11 2025
%e A369901 Displayed as an irregular triangle:
%e A369901   3;
%e A369901   5, 13;
%e A369901   9, 25, 41, 57;
%e A369901   17, 49, 81, 113, 145, 177, 209, 241;
%e A369901   ...
%o A369901 (Python)
%o A369901 def A369901(n):
%o A369901     b = n.bit_length() - 1
%o A369901     c = n - 2**b
%o A369901     return (2*c+1)*2**(b+1)+1
%o A369901 (PARI) \\ See PARI link
%Y A369901 Cf. A080075 (Proth numbers).
%Y A369901 Cf. A000051 (1st column), A020515 (right diagonal).
%K A369901 nonn,easy,tabf
%O A369901 1,1
%A A369901 _Daniel Sturm_, Feb 05 2024
%E A369901 More terms from _David A. Corneth_, Feb 05 2024