A183150 Semiprimes s such that s^2 is expressible as the sum of two positive cubes.

4, 671, 1261, 6371, 127499, 377567, 897623, 1984009, 4266107, 4870741, 4974061, 5491823, 24923137, 26784757, 28192247, 33601933, 36295069, 44091347, 44988481, 61717319, 95327051, 97587433, 99712367, 142798573, 149982097, 193405967

Offset: 1

Author

Jonathan Vos Post, Feb 05 2011

Keywords

• nonn

Comments

Contains 4 and a subset of A099426.

If s=p*q for primes p < q, then (4*q^2-p^4)/3 is a square. Furthermore, q/p^2 = (m^4 + 6*m^3*n + 18*m^2*n^2 + 18*m*n^3 + 9*n^4)/(m^2 - 3*n^2)^2 for some integers m,n. The underlying identity (up to a common factor) is ( (m^4 + 6*m^3*n + 18*m^2*n^2 + 18*m*n^3 + 9*n^4)*(m^2 - 3*n^2) )^2 = ( (m+3*n)*(m+n)*(m^2+3*n^2) )^3 + ( -4*m*n*(m^2+3*m*n+3*n^2) )^3. - Max Alekseyev, Jun 16 2011

Examples

			a(1) = 4 = 2*2 because 4^2 = 16 = 2^3 + 2^3 . a(2) = 671 = 11 * 61 and 56^3 + 65^3 = 671^2 = 450241. a(3) = 1261 = 13 * 97 and 1261^2 = 57^3 + 112^3. a(6) = 897623 = 107 * 8389.
		

Programs

Mathematica

Select[Range[194*10^6],PrimeOmega[#]==2&&Length[ PowersRepresentations[ #^2,2,3]]>0&] (* The program takes a long time to run. *) (* Harvey P. Dale, Feb 27 2016 *)

Formula

A001358 INTERSECTION A050801.

Extensions

a(9)-a(26) from Donovan Johnson, Feb 11 2011