A038550 Products of an odd prime and a power of two (sorted).

3, 5, 6, 7, 10, 11, 12, 13, 14, 17, 19, 20, 22, 23, 24, 26, 28, 29, 31, 34, 37, 38, 40, 41, 43, 44, 46, 47, 48, 52, 53, 56, 58, 59, 61, 62, 67, 68, 71, 73, 74, 76, 79, 80, 82, 83, 86, 88, 89, 92, 94, 96, 97, 101, 103, 104, 106, 107, 109, 112, 113, 116, 118, 122, 124, 127

Offset: 1

Author

Tom Verhoeff

Keywords

• nonn
• easy
• nice

Comments

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.)

Numbers that are the difference of two triangular numbers in exactly two ways.

Numbers with largest odd divisor a prime number. - Juri-Stepan Gerasimov, Aug 16 2016

Numbers k for which A001222(A000265(k)) = 1. - Antti Karttunen, Jul 09 2020

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2, 1999, #99.1.6.

Crossrefs

Cf. A001227, A000265, A237593, A279387.

Subsequences: A334101, A335431, A335911.

Subsequence of A093641 and of A336101.

Programs

Haskell

a038550 n = a038550_list !! (n-1)
a038550_list = filter ((== 2) . a001227) [1..]
-- Reinhard Zumkeller, May 01 2012

Mathematica

Select[Range[127],DivisorSigma[0,Max[Select[Divisors[#],OddQ]]]-1==1&] (* Jayanta Basu, Apr 30 2013 *)
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 *)
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 *)
Flatten@Position[PrimeQ[BitShiftRight[#, IntegerExponent[#, 2]]&/@Range[#]], True]&@127 (* Federico Provvedi, Dec 14 2021 *)

PARI

is(n)=isprime(n>>valuation(n,2)) \\ Charles R Greathouse IV, Apr 30 2013

Formula

A001227(a(n)) = 2. - Reinhard Zumkeller, May 01 2012

a(n) ~ 0.5 n log n. - Charles R Greathouse IV, Apr 30 2013

A000265(a(n)) is a prime. - Juri-Stepan Gerasimov, Aug 16 2016

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

Extensions

Edited by N. J. A. Sloane at the suggestion of Zak Seidov, Sep 15 2007