A080340 First known infinite sequence containing no odd integer of the form 2^m+p (p prime).

7629217, 18814027, 29998837, 41183647, 52368457, 63553267, 74738077, 85922887, 97107697, 108292507, 119477317, 130662127, 141846937, 153031747, 164216557, 175401367, 186586177, 197770987, 208955797, 220140607, 231325417, 242510227

Offset: 0

Author

Benoit Cloitre, Mar 19 2003

Keywords

• nonn
• easy

Comments

To a question of Romanoff: Are there infinitely many odd integers not of the form 2^m+p where p is prime? Erdős answered Yes in 1950 by constructing the present sequence, an infinite arithmetic sequence, using a system of congruences.

Links

• P. Erdős, On integers of form 2^n+p and some related problems, Summa Brasil Math.11 (1950), pp. 1-11.
• Tanya Khovanova, Recursive Sequences.
• T. Zamojski, Survey on covering congruences.
• Index entries for linear recurrences with constant coefficients, signature (2,-1).

Formula

a(n) = n*11184810 + 7629217.

G.f.: (7629217 + 3555593*x)/(1 - x)^2. - Stefano Spezia, Apr 25 2025