1,1

19581212842 was discovered by Martin Fuller, Dec 06 2007, and proved to the smallest term by Jack Brennen (jb(AT)brennen.net), Dec 07 2007 (see link below)

139136064116422 is also a term - Dean Hickerson (dean(AT)math.ucdavis.edu), Dec 06 2007

In contrast, it seems to be easier to find numbers n such that n^3+n+1 is abundant. The smallest solution is n=12210244 and there are 19 further solutions up through n=271870216. - Hans Havermann (pxp(AT)rogers.com), Dec 05 2007. See A133373.

I found the first 2331 odd primitive abundant numbers A with no prime factors -1 mod 6, then looked for k*A-1 such that int(sqrt(k*A-1))*(int(sqrt(k*A-1))+1) = k*A-1, where k = 103 to 18000001. The first k found was 4447 for A(2331) = 3*7^2*13*19*31*37*43*61*79*97*103 which gives a(1) = 19581212842 = int(sqrt(4447*A(2331)-1); the second k found was 17744401 = 379*46819 for A(2221) = 3*7^4+13*19*31*37*43*61*67*73 which gives a(2) = 681577964785. No solution primitive abundant numbers were found with prime factors -1 mod 6 in the range I considered. - Pierre CAMI (pierrecami(AT)tele2.fr), Jan 11 2008

Sequence Fans members, Notes on this sequence

For n=19581212842, n^2+n+1 = 3*7^2*13*19*31*37*43*61*79*97*103*4447, sigma(n^2+n+1)/(n^2+n+1) = 2.0031147...

681577964785^2+681577964785+1 = 3*7^4*13*19*31*37*43*61*67*73*379*46819; 3*7^4*13*19*31*37*43*61*67*73 is a primitive abundant number so a(2) is an abundant number

5625634605028^2+5625634605028+1 = 3*7*7*7*13*19*19*31*37*43*61*67*73*919*484621. Note that 3*7*7*7*13*19*19*31*37*43*61*67*73 = A136607(17).

Cf. A136607.

Sequence in context: A101815 A113643 A115537 this_sequence A003810 A003803 A068243

Adjacent sequences: A118463 A118464 A118465 this_sequence A118467 A118468 A118469

nonn

njas, Dec 21 2007, based on discussions on the Sequence Fans mailing list.

a(2)-a(13) from Pierre CAMI (pierrecami(AT)tele2.fr), Jan 17 2008