A091554 -id:A091554 - cp's OEIS frontend

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

Joseph L. Pe, Dec 27 2001

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

R. K. Guy, Unsolved Problems in Number Theory, 3rd Ed., New York, Springer-Verlag, 2004, Section B15.

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)).

a[ n_ ] := Min[ Select[ Range[ 1, 10^6 ], DivisorSigma[ 1, n + # ] == DivisorSigma[ 1, n ] + DivisorSigma[ 1, # ] & ] ]; Table[ a[ i ], {i, 1, 21} ]

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

cp's OEIS Frontend

A066435 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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