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Numbers n such that a positive number m < n exists such that n-m, n+m, and n*m are oblong numbers (A002378).
2

%I #16 Jun 14 2022 10:46:01

%S 42,146,2850,9074,13464,17640,20502,32406,53892,55352,59124,229284,

%T 340890,344904,507012,562466,1016076,1958082,3938256,6087890,6924960,

%U 7645106,8070342,11694540,11861892,12721884,12960020,15959802,18999750,34863842,40503060,42000420,55377282

%N Numbers n such that a positive number m < n exists such that n-m, n+m, and n*m are oblong numbers (A002378).

%C Sequence of corresponding m's begins: 30, 36, 1980, 2268, 11658, 16770, 3990, 900, 14490, 54540, ...

%C Conjectures:

%C 1. The sequence is infinite.

%C 2. There is only one m for each n (this is true for n < 2^26).

%H Kevin P. Thompson, <a href="/A224954/b224954.txt">Table of n, a(n) for n = 1..77</a>

%o (Java)

%o public class A224954 {

%o public static long sr0 = 1;

%o public static boolean[] Oblong = new boolean[1 << 27];

%o public static boolean isOblong(long a) {

%o long b, s, sr = sr0;

%o while (a < sr*(sr+1)) sr>>=1;

%o b = sr>>1;

%o while (b!=0) {

%o s = sr+b;

%o if (a >= s*(s+1)) sr = s;

%o b>>=1;

%o }

%o if (a == sr*(sr+1)) return true;

%o return false;

%o }

%o public static void main (String[] args) {

%o for (int i = 0; i*(i+1) < (1 << 27); ++i) Oblong[i*(i+1)] = true;

%o for (long a = 0; a < (1 << 26); a+=2) {

%o long s = 1L << 30, obn = 0, count = 0, lastB = 0;

%o while (a*a < s*(s+1)) s>>=1;

%o sr0 = s;

%o for (long i = 2; obn < a; i+=2) {

%o long b = a - obn;

%o if (Oblong[(int)(a*2-obn)])

%o if (isOblong(a*(a-obn))) { ++count; lastB = b; }

%o obn += i;

%o }

%o if (count!=0) System.out.printf("\n%d %d %d ", a, lastB, count);

%o if ((a & 0x3fff)==0) System.out.printf(".");

%o }

%o }

%o }

%Y Cf. A002378, A224935.

%K nonn

%O 1,1

%A _Alex Ratushnyak_, Apr 20 2013