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%I #84 Feb 18 2024 02:05:11
%S 3,5,6,7,10,11,12,13,14,17,19,20,22,23,24,26,28,29,31,34,37,38,40,41,
%T 43,44,46,47,48,52,53,56,58,59,61,62,67,68,71,73,74,76,79,80,82,83,86,
%U 88,89,92,94,96,97,101,103,104,106,107,109,112,113,116,118,122,124,127
%N Products of an odd prime and a power of two (sorted).
%C 2007. For example, 37 = 18 + 19; 48 = 15 + 16 + 17; 56 = 5 + 6 + 7 + 8 + 9 + 10 + 11. (Edited by _M. F. Hasler_, Aug 29 2020: "positive" was missing here. If nonnegative integers are allowed, none of the triangular numbers 3, 6, 10, ... would be in the corresponding sequence. If negative integers are also allowed, it would only have powers of 2 (A000079) which are the only positive integers not the sum of more than one consecutive positive integers, since any x > 0 is the sum of 1-x through x.)
%C Numbers that are the difference of two triangular numbers in exactly two ways.
%C Numbers with largest odd divisor a prime number. - _Juri-Stepan Gerasimov_, Aug 16 2016
%C Numbers k for which A001222(A000265(k)) = 1. - _Antti Karttunen_, Jul 09 2020
%H T. D. Noe, <a href="/A038550/b038550.txt">Table of n, a(n) for n = 1..1000</a>
%H T. Verhoeff, <a href="https://cs.uwaterloo.ca/journals/JIS/trapzoid.html">Rectangular and Trapezoidal Arrangements</a>, J. Integer Sequences, Vol. 2, 1999, #99.1.6.
%F A001227(a(n)) = 2. - _Reinhard Zumkeller_, May 01 2012
%F a(n) ~ 0.5 n log n. - _Charles R Greathouse IV_, Apr 30 2013
%F A000265(a(n)) is a prime. - _Juri-Stepan Gerasimov_, Aug 16 2016
%F Sum_{n>=1} 1/a(n)^s = (2^s*P(s) - 1)/(2^s - 1), for s > 1, where P is the prime zeta function. - _Amiram Eldar_, Dec 19 2020
%t Select[Range[127],DivisorSigma[0,Max[Select[Divisors[#],OddQ]]]-1==1&] (* _Jayanta Basu_, Apr 30 2013 *)
%t fQ[n_] := Module[{p, e}, {p, e} = Transpose[FactorInteger[n]]; (Length[p] == 2 && p[[1]] == 2 && e[[2]] == 1) || (Length[p] == 1 && p[[1]] > 2 && e[[1]] == 1)]; Select[Range[2, 127], fQ] (* _T. D. Noe_, Apr 30 2013 *)
%t upto=150;Module[{pmax=PrimePi[upto],tmax=Ceiling[Log[2,upto]]}, Select[ Sort[ Flatten[ Outer[ Times, Prime[ Range[ 2,pmax]], 2^Range[0,tmax]]]],#<=upto&]] (* _Harvey P. Dale_, Oct 18 2013 *)
%t Flatten@Position[PrimeQ[BitShiftRight[#, IntegerExponent[#, 2]]&/@Range[#]], True]&@127 (* _Federico Provvedi_, Dec 14 2021 *)
%o (Haskell)
%o a038550 n = a038550_list !! (n-1)
%o a038550_list = filter ((== 2) . a001227) [1..]
%o -- _Reinhard Zumkeller_, May 01 2012
%o (PARI) is(n)=isprime(n>>valuation(n,2)) \\ _Charles R Greathouse IV_, Apr 30 2013
%Y Cf. A001227, A000265, A237593, A279387.
%Y Subsequences: A334101, A335431, A335911.
%Y Subsequence of A093641 and of A336101.
%K nonn,easy,nice
%O 1,1
%A _Tom Verhoeff_
%E Edited by _N. J. A. Sloane_ at the suggestion of _Zak Seidov_, Sep 15 2007
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