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A066435
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Conjectured values for a(n) = least natural number k such that sigma(n+k) = sigma(n)+sigma(k) if it exists; otherwise 0.
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3
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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
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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R. K. Guy, Unsolved Problems in Number Theory, 3rd Ed., New York, Springer-Verlag, 2004, Section B15.
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LINKS
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Table of n, a(n) for n=1..72.
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MATHEMATICA
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a[ n_ ] := Min[ Select[ Range[ 1, 10^6 ], DivisorSigma[ 1, n + # ] == DivisorSigma[ 1, n ] + DivisorSigma[ 1, # ] & ] ]; Table[ a[ i ], {i, 1, 21} ]
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CROSSREFS
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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
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KEYWORD
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hard,nonn
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AUTHOR
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Joseph L. Pe (joseph_l_pe(AT)hotmail.com), Dec 27 2001
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EXTENSIONS
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More terms from T. D. Noe, Jan 20 2004
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STATUS
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approved
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