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 A066435 Conjectured values for a(n) = least natural number k such that sigma(n+k) = sigma(n)+sigma(k) if it exists; otherwise 0. 3
 2, 1, 0, 5, 4, 2, 14, 2, 0, 5, 22, 43, 26, 7, 0, 496, 34, 2, 38, 37, 0, 11, 46, 6, 50, 13, 0, 4, 26, 10, 62, 929, 282, 17, 28, 252, 20, 19, 0, 101, 8, 14, 12, 19, 17, 23, 38, 307, 98, 25, 54, 65, 106, 51, 14, 14, 0, 29, 118, 66, 56, 30, 0, 8128, 22, 22, 44, 85, 66, 35, 135, 18 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS It would be nice to remove the word "Conjectured" from the description - N. J. A. Sloane. The values of a(3), a(9), a(15) and a(21) listed above, namely 0, are conjectural. There is no natural number k < 10^6 satisfying the "homomorphic condition" sigma(n+k)=sigma(n)+sigma(k) for n=3,9,15,21. The terms for which there is no solution k < 10^6 are n = 3, 9, 15, 21, 27, 39, 57, 63, 81, 93, 105, 117, 165, 171, 183, 189, 201, 219, 225, 243,..., which all satisfy n=3 (mod 6). - T. D. Noe, Jan 20 2004 All n<1000 and k<10^10 have been tested. The largest term is a(837)=4631925025. Sequence A110108 gives the n for which there is no solution k<10^10. All n<1000 and k<10^11 have been tested. The largest term is a(711)=21004780114. - Donovan Johnson, Aug 29 2012 REFERENCES R. K. Guy, Unsolved Problems in Number Theory, 3rd Ed., New York, Springer-Verlag, 2004, Section B15. LINKS MATHEMATICA a[ n_ ] := Min[ Select[ Range[ 1, 10^6 ], DivisorSigma[ 1, n + # ] == DivisorSigma[ 1, n ] + DivisorSigma[ 1, # ] & ] ]; Table[ a[ i ], {i, 1, 21} ] CROSSREFS Cf. A091554 (primes p such that k=2p is the smallest solution to sigma(p+k)=sigma(p)+sigma(k)). Cf. A110176 (least k such that sigma(n)=sigma(k)+sigma(n-k)). Sequence in context: A128899 A155887 A113368 * A171960 A182376 A030206 Adjacent sequences:  A066432 A066433 A066434 * A066436 A066437 A066438 KEYWORD hard,nonn AUTHOR Joseph L. Pe (joseph_l_pe(AT)hotmail.com), Dec 27 2001 EXTENSIONS More terms from T. D. Noe, Jan 20 2004 STATUS approved

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