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

1,1

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

Links

Table of n, a(n) for n=1..26.

Formula

A001358 INTERSECTION A050801.

Example

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.

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 *)

Crossrefs

Sequence in context: A046348 A332164 A334528 * A202368 A114765 A307920

Adjacent sequences: A183147 A183148 A183149 * A183151 A183152 A183153

Keyword

nonn

Author

Jonathan Vos Post, Feb 05 2011

Extensions

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

Status

approved