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

1,1

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.

Links

Table of n, a(n) for n=1..53.

Example

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.

Prog

(Magma) f:=func<n | Factorization(n)>; nopf:=func<n | [i: i in [1..#PrimeDivisors(n)] | f(n)[i][1] mod 4 eq 1]>; sum:=func<n | IsEmpty(nopf(n)) select 0 else &+[f(n)[nopf(n)[k]][2]: k in [1..#nopf(n)]]>; [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;

Crossrefs

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

Sequence in context: A061970 A159008 A183373 * A354381 A306358 A120961

Adjacent sequences: A375358 A375359 A375360 * A375362 A375363 A375364

Keyword

nonn

Author

Arkadiusz Wesolowski, Aug 13 2024

Status

approved