A004613 -id:A004613 - cp's OEIS frontend

A356794 Odd numbers that have at least one prime factor congruent to 1 (mod 4) and at least one prime factor congruent to 3 (mod 4).

15, 35, 39, 45, 51, 55, 75, 87, 91, 95, 105, 111, 115, 117, 119, 123, 135, 143, 153, 155, 159, 165, 175, 183, 187, 195, 203, 215, 219, 225, 235, 245, 247, 255, 259, 261, 267, 273, 275, 285, 287, 291, 295, 299, 303, 315, 319, 323, 327, 333, 335, 339, 345, 351

Offset: 1

Jon E. Schoenfield, Aug 27 2022

nonn

This sequence consists of the odd positive integers that are in neither A004613 nor A004614.

			15 is a term: its prime factors are 3 and 5; 3 == 3 (mod 4), 5 == 1 (mod 4).

Cf. A004613, A004614.

Select[Range[1, 351, 2], Length[Union[Mod[FactorInteger[#][[;; , 1]], 4]]] == 2 &] (* Amiram Eldar, Aug 28 2022 *)

isok(m) = if (m % 2, my(vf=factor(m)[,1]); (#select(x->((x%4)==1), vf) >= 1) && (#select(x->((x%4)==3), vf) >= 1)); \\ Michel Marcus, Aug 29 2022

A370082 The number of square lattice nodes inside the n-th largest octagon with angles 3*Pi/4, along the perimeter of which there are only 8 lattice nodes - at the vertices of the octagon.

Original entry on oeis.org

4, 32, 88, 116, 172, 200, 256, 284, 368, 424, 452, 508, 592, 620, 676, 704, 760, 788, 872, 956, 1012, 1040, 1096, 1180, 1208, 1264, 1292, 1348, 1376, 1432, 1544, 1600, 1628, 1684, 1796, 1852, 1880, 1936, 1964, 2020, 2048, 2132, 2188, 2216, 2272, 2356, 2440, 2468, 2552, 2608, 2636

Offset: 1

Alexander M. Domashenko, Feb 10 2024

The adjacent sides of an octagon are not equal, the ratio of the larger side to the smaller one is sqrt(2), its area is 7 times the square of the shorter side. Using Pick's formula N = S - V/2 + 1 we obtain N = S - 3 = 7*A004613(n) - 3.

Cf. A004613.

a(n) = 7*A004613(n) - 3.

A387595 Numbers k that divide both x^2 + 1 and 2^x + 1 for some number x.

Original entry on oeis.org

1, 5, 13, 17, 29, 37, 41, 53, 61, 65, 97, 101, 109, 113, 137, 145, 149, 157, 173, 181, 185, 193, 197, 229, 241, 257, 265, 269, 277, 281, 293, 313, 317, 349, 353, 373, 377, 389, 397, 401, 409, 421, 433, 449, 457, 461, 481, 509, 521, 533, 541, 545, 557, 565, 569, 577, 593, 613, 617, 641, 653, 661

Offset: 1

Robert Israel, Sep 02 2025

Numbers that divide some term of A066787.
Every divisor of a term is a term.
The first term that is a square > 1 is 1093^2, with 49351666^2 + 1 and 2^49351666 + 1 both divisible by 1093^2.
All prime divisors of a term that is a square must be Wieferich primes == 1 (mod 4). The only known Wieferich prime == 1 (mod 4) is 1093.

			a(3) = 13 is a term because 13 divides both 18^2 + 1 = 325 = 13 * 25 and 2^18 + 1 = 262145 = 13 * 20165.

Robert Israel, Table of n, a(n) for n = 1..10000
Mathematics StackExchange, Primes where x^2+1 == 2^x+1 == 0 (mod p^2) has a solution

Subset of A004613. Cf. A001220, A066787.

# Note: due to a bug in Maple, this program fails for 1093^2 (at least up to Maple 2025)
filter:= proc(n) local a,b,b0,b1,x,t,tt;
    uses NumberTheory;
    if n::even then return false fi;
    a:=[msolve(x^2 + 1 = 0, n)];
    if a = [] then return false fi;
    a:= map(t -> rhs(op(t)), a);
    b:=msolve(2^x + 1 = 0,t, n);
    if b = {} or b = NULL then return false fi;
    b:= rhs(op(b));
    tt:= indets(b)[1];
  b0:= subs(tt=0,b); b1:= coeff(b,tt);
    for x in a do
    if ChineseRemainder([x,b0],[n,b1]) <> FAIL then return true fi
  od;
    false
end proc:
filter(1):= true:
select(filter, [seq(seq(i+6*j,i=[1,5]),j=0..1000)]);

cp's OEIS Frontend

A356794 Odd numbers that have at least one prime factor congruent to 1 (mod 4) and at least one prime factor congruent to 3 (mod 4).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

A370082 The number of square lattice nodes inside the n-th largest octagon with angles 3*Pi/4, along the perimeter of which there are only 8 lattice nodes - at the vertices of the octagon.

Views

Author

Keywords

Comments

Crossrefs

Formula

A387595 Numbers k that divide both x^2 + 1 and 2^x + 1 for some number x.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple