A066435 Conjectured values for a(n) = least natural number k such that sigma(n+k) = sigma(n)+sigma(k) if it exists; otherwise 0.

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

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

Table of n, a(n) for n=1..72.

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: A155887 A357583 A113368 * A261301 A171960 A330396

Adjacent sequences: A066432 A066433 A066434 * A066436 A066437 A066438

Keyword

hard,nonn

Author

Joseph L. Pe, Dec 27 2001

Extensions

More terms from T. D. Noe, Jan 20 2004

Status

approved