A002331 -id:A002331 - cp's OEIS frontend

A080680 Integer part of the square root of the n-th prime of the form 4k+1.

2, 3, 4, 5, 6, 6, 7, 7, 8, 9, 9, 10, 10, 10, 11, 12, 12, 13, 13, 13, 14, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 22, 22, 23, 23, 23, 24, 24, 24, 24, 24, 25, 25, 25, 25, 26, 26, 26, 27, 27, 27, 27, 27, 28, 28, 28, 28, 29, 29, 29

Offset: 1

Reinhard Zumkeller, Mar 02 2003

nonn

a(n)^2 < A002330(n+2)^2 + A002331(n+2)^2 < a(n+1)^2.

Floor[Sqrt[#]]&/@Select[Prime[Range[300]],Mod[#,4]==1&] (* Harvey P. Dale, Jun 04 2023 *)

a(n) = A000196(A002144(n)).

A107961 Pythagorean semiprimes: products of two Pythagorean primes (A002313).

Original entry on oeis.org

4, 10, 25, 26, 34, 58, 65, 74, 82, 85, 106, 122, 145, 146, 169, 178, 185, 194, 202, 205, 218, 221, 226, 265, 274, 289, 298, 305, 314, 346, 362, 365, 377, 386, 394, 445, 458, 466, 481, 482, 485, 493, 505, 514, 533, 538, 545, 554, 562, 565, 586, 626, 629, 634

Offset: 1

Jonathan Vos Post, Jun 12 2005

Fermat's 4n+1 theorem, sometimes called Fermat's two-square theorem or simply "Fermat's theorem," states that a prime number p can be represented in an essentially unique manner (up to the order of addends) in the form x^2 + y^2 for integer x and y iff p = 1 (mod 4) or p = 2 (which is a degenerate case with x = y = 1). The theorem was stated by Fermat, but the first published proof was by Euler.

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 146-147 and 220-223, 1996.
Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 13 and 219, 1979.
Seroul, R. "Prime Number and Sum of Two Squares." Section 2.11 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 18-19, 2000.

Eric Weisstein's World of Mathematics, Fermat's 4n Plus 1 Theorem.
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A001358, A002313, A002330, A002331.

{a(n)} = {p*q: p and q both elements of A002313} = {p*q: p and q both of form m^2 + n^2 for integers m, n}.

A383176 If p = A002313(n) is a prime such that p = x^2 + y^2, then a(n) is the largest integer k that satisfies x^2 + y^2 - kxy > 0.

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 2, 3, 2, 3, 2, 2, 10, 3, 2, 3, 2, 2, 6, 2, 2, 14, 7, 2, 4, 16, 2, 2, 3, 8, 2, 2, 2, 3, 2, 2, 2, 3, 20, 6, 2, 2, 3, 5, 2, 4, 2, 2, 2, 2, 24, 3, 5, 2, 2, 6, 2, 4, 2, 26, 5, 2, 13, 3, 2, 2, 2, 2, 5, 2, 3, 2, 7, 5, 2, 2, 2, 3, 2, 7, 5, 2, 2, 3

Offset: 1

Gonzalo Martínez, Apr 18 2025

nonn

Fermat's Christmas theorem states that if p = 2 or if p is congruent to 1 modulo 4 (A002313), then p is written as a sum of 2 squares uniquely. Thus, if A002313(n) = x^2 + y^2, for certain integers x and y, then a(n) is the largest integer k such that x^2 + y^2 - k*x*y > 0.

a(n) >= 2, for n > 1. If p > 2 and p = x^2 + y^2, since x != y, then it is satisfied that 0 < (x - y)^2 = x^2 + y^2 - 2x*y < x^2 + y^2 - x*y. The equality a(n) = 2 is given when |x - y| < phi*min{x, y}.

			Since A002313(8) = 53 and 53 = 2^2 + 7^2, we have that 53 - 3*2*7 > 0 and 53 - 4*2*7 < 0, then a(8) = 3.

Cf. A002313, A002330, A002331.

from itertools import count, islice
from sympy import isprime
from sympy.solvers.diophantine.diophantine import cornacchia
def A383176_gen(): # generator of terms
    yield 1
    for p in count(5,4):
        if isprime(p):
            for x,y in cornacchia(1,1,p):
                yield p//(x*y)
A383176_list = list(islice(A383176_gen(),30)) # Chai Wah Wu, Apr 26 2025

Definition clarified by Chai Wah Wu, Apr 26 2025

cp's OEIS Frontend

A080680 Integer part of the square root of the n-th prime of the form 4k+1.

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A107961 Pythagorean semiprimes: products of two Pythagorean primes (A002313).

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A383176 If p = A002313(n) is a prime such that p = x^2 + y^2, then a(n) is the largest integer k that satisfies x^2 + y^2 - kxy > 0.

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cp's OEIS Frontend

A080680 Integer part of the square root of the n-th prime of the form 4k+1.

Views

Author

Keywords

Comments

Programs

Mathematica

Formula

A107961 Pythagorean semiprimes: products of two Pythagorean primes (A002313).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Formula

A383176 If p = A002313(n) is a prime such that p = x^2 + y^2, then a(n) is the largest integer k that satisfies x^2 + y^2 - k*x*y > 0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Python

Extensions

A383176 If p = A002313(n) is a prime such that p = x^2 + y^2, then a(n) is the largest integer k that satisfies x^2 + y^2 - kxy > 0.