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A118466
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Numbers n such that n^2+n+1 is abundant.
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3
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19581212842, 681577964785, 5625634605028, 7781640485518, 11064002589565, 15693462387430, 17893966035208, 21389600196136, 29088445512025, 30216634902892, 30508467609727, 31009592020780, 31923017215378
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OFFSET
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1,1
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COMMENTS
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19581212842 was discovered by Martin Fuller, Dec 06 2007 and proved to the smallest term by Jack Brennen, Dec 07 2007 (see link below)
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, 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, Jan 11 2008
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LINKS
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EXAMPLE
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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).
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane, Dec 21 2007, based on discussions on the Sequence Fans mailing list.
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EXTENSIONS
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STATUS
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approved
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