A226111 Composite squarefree numbers n such that the ratio (n - 1/2)/(p(i) + 1/2) is an integer, where p(i) are the prime factors of n.

260813, 960323, 4572113, 5991098, 18912713, 37481945, 68688458, 214337813, 1418459963, 1488523838, 1905782603, 1906387718, 2416383938, 3866147051, 6153859058, 6927221438, 10696723538, 12000312419, 24529142138, 43004079563, 43648495313, 54750300413

Offset: 1

Author

Paolo P. Lava, May 27 2013

Keywords

• nonn
• hard

Comments

Also composite squarefree numbers n such that (2*p(i)+1) | (2*n-1).

Examples

			The prime factors of 5991098 are 2, 103, 127 and 229. We see that (5991098 - 1/2)/(2 + 1/2) = 2396439, (5991098 - 1/2)/(103 + 1/2) = 57885, (5991098 - 1/2)/(127 + 1/2) = 46989 and (5991098 - 1/2)/(229 + 1/2) = 26105. Hence 5991098 is in the sequence.
The prime factors of 1123342 are 2, 11 and 51061. We see that(1123342 - 1/2)/(2 + 1/2) = 748895, (1123342 - 1/2)/(11 + 1/2) = 106985 but (1123342 - 1/2)/(51061 + 1/2) = 2246685/102121. Hence 1123342 is not in the sequence.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..62 (terms < 2*10^12)

Crossrefs

Cf. A208728, A225702-A225720, A226020, A226112-A226114.

Programs

Maple

with(numtheory); A226111:=proc(i, j) local c, d, n, ok, p;
for n from 2 to i do if not isprime(n) then p:=ifactors(n)[2]; ok:=1;
for d from 1 to nops(p) do if p[d][2]>1 or not type((n-j)/(p[d][1]+j), integer) then ok:=0; break; fi; od;
if ok=1 then print(n); fi; fi; od; end: A226111(10^9,1/2);

Extensions

a(8)-a(22) from Giovanni Resta, Jun 02 2013