A174090 Powers of 2 and odd primes; alternatively, numbers that cannot be written as a sum of at least three consecutive positive integers.

1, 2, 3, 4, 5, 7, 8, 11, 13, 16, 17, 19, 23, 29, 31, 32, 37, 41, 43, 47, 53, 59, 61, 64, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 256

Offset

1,2

Comments

From Omar E. Pol, Feb 24 2014: (Start)

Also the odd noncomposite numbers (A006005) and the powers of 2 with positive exponent, in increasing order.

If a(n) is composite and a(n) - a(n-1) = 1 then a(n-1) is a Mersenne prime (A000668), hence a(n-1)*a(n)/2 is a perfect number (A000396) and a(n-1)*a(n) equals the sum of divisors of a(n-1)*a(n)/2.

If a(n) is even and a(n+1) - a(n) = 1 then a(n+1) is a Fermat prime (A019434). (End)

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Jaap Spies, A Bit of Math, The Art of Problem Solving, Jaap Spies Publishers (2019).

Nieuw Archief voor Wiskunde, Problems/UWC, Problem C, Vol. 5/6, No. 2.

Maple

N:= 300: # to get all terms <= N
S:= {seq(2^i, i=0..ilog2(N))} union select(isprime, { 2*i+1 $ i=1..floor((N-1)/2) }):
sort(convert(S, list)); # Robert Israel, Jun 18 2015

Mathematica

a[n_] := Product[GCD[2 i - 1, n], {i, 1, (n - 1)/2}] - 1;
Select[Range[242], a[#] == 0 &] (* Gerry Martens, Jun 15 2015 *)

Crossrefs

Numbers not in A111774.

Equals A000079 UNION A065091.

Equals A067133 \ {6}.

Cf. A000040, A000203, A000396, A000668, A006005, A019434, A092506.

Cf. also A138591, A174069, A174070, A174071.

Sequence in context: A348284 A162721 A176176 * A280083 A020902 A008751

Adjacent sequences: A174087 A174088 A174089 * A174091 A174092 A174093

Keyword

nonn

Author

Vladimir Joseph Stephan Orlovsky, Mar 07 2010, and Omar E. Pol, Feb 24 2014

Extensions

This entry is the result of merging an old incorrect entry and a more recent correct version. N. J. A. Sloane, Dec 07 2015

Status

approved