A375361 - OEIS

%I #12 Sep 13 2024 07:33:35

%S 25,65,75,85,125,145,169,175,185,195,205,221,225,255,265,275,289,305,

%T 325,365,375,377,425,435,445,455,475,481,485,493,505,507,525,533,545,

%U 555,565,575,585,595,615,625,629,663,675,685,689,697,715,725,745,765,775

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

%C 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.

%C 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.

%e 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.

%o (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];

%o (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;

%o (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;

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

%K nonn

%O 1,1

%A _Arkadiusz Wesolowski_, Aug 13 2024