A260872 Squarefree numbers k such that k+1 has no primes of the form 4*m-1 and at most one 2 in its prime factorization.

1, 33, 57, 73, 105, 129, 145, 177, 193, 201, 217, 249, 273, 313, 337, 345, 385, 393, 409, 457, 465, 481, 537, 553, 561, 577, 609, 633, 649, 673, 697, 705, 745, 753, 777, 793, 817, 849, 865, 889, 897, 913, 921, 969, 985, 1009, 1041, 1065, 1081, 1113, 1129

Offset

1,2

Comments

An even number k is congruent to either 0 or 2 mod 4. If congruent to 0, it is divisible by 4 and thus not squarefree. If k is congruent to 2, k+1 will be one less than a multiple of 4, and thus at least one prime factor of k+1 will be one less than a multiple of 4. Thus, there are no even numbers in this sequence.

From the author's comment above, all sequence terms must be odd, so k+1 must always be even and k+1 will always be singly even. - Ray Chandler, Aug 03 2015

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Example

41 + 1 = 42 = 2*3*7 and both 3 and 7 are prime numbers of the form 4*n-1, so 41 is not a term of this sequence.

Mathematica

Select[Range[1100], SquareFreeQ[#]&&IntegerExponent[#+1, 2]<2&&Select[First/@FactorInteger[#+1], Mod[#, 4]==3&]=={}&] (* Ray Chandler, Aug 02 2015 *)

Crossrefs

Cf. A002144, A002145, A004613, A004614, A005117, A016825.

Sequence in context: A242605 A351396 A350598 * A039380 A043203 A043983

Adjacent sequences: A260869 A260870 A260871 * A260873 A260874 A260875

Keyword

nonn

Author

J. Lowell, Aug 01 2015

Status

approved