A375361 Odd numbers with at least two prime divisors of the form 4*k + 1 counted with multiplicity.

25, 65, 75, 85, 125, 145, 169, 175, 185, 195, 205, 221, 225, 255, 265, 275, 289, 305, 325, 365, 375, 377, 425, 435, 445, 455, 475, 481, 485, 493, 505, 507, 525, 533, 545, 555, 565, 575, 585, 595, 615, 625, 629, 663, 675, 685, 689, 697, 715, 725, 745, 765, 775

Offset: 1

Author

Arkadiusz Wesolowski, Aug 13 2024

Keywords

• nonn

Comments

Odd numbers k such that k^2 can be expressed as the arithmetic mean of two distinct perfect squares in more than one way. For example, 25^2 = (5^2 + 35^2)/2 = (17^2 + 31^2)/2.

Let x be a squared integer which is the central element of a 3 X 3 magic square in which seven (or more) of the entries are squared integers. If the greatest common divisor of all nine entries is 1, then the square root of x is a composite number that is divisible only by primes congruent to 1 mod 4. For example, sqrt(A221669(5)) = 425 is both in A004613 and in this sequence.

Examples

			65 is in this sequence because 65 has two prime factors of the form 4*k + 1, namely 5 = 4*1 + 1 and 13 = 4*3 + 1.
		

Crossrefs

Cf. A004613, A008844, A221669, A221670, A268855, A309812, A375459.

Programs

Magma

f:=func; nopf:=func; sum:=func; [n: n in [1..775 by 2] | sum(n) gt 1];

PARI

isok(n) = my(v=Vec(factor(n))); n%2&&sum(t=1, omega(n), if((v[1]%4)[t]==1, v[2][t]))>1;

PARI

isok(n) = my(t); if(n%2, for(k=sqrtint(n^2-1)+2, sqrtint(2*n^2-1), if(issquare(2*n^2-k^2)&&t++>1, return(1)))); 0;