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.

%I A066435 #11 Feb 11 2014 19:05:23
%S A066435 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,
%T A066435 26,10,62,929,282,17,28,252,20,19,0,101,8,14,12,19,17,23,38,307,98,25,
%U A066435 54,65,106,51,14,14,0,29,118,66,56,30,0,8128,22,22,44,85,66,35,135,18
%N A066435 Conjectured values for a(n) = least natural number k such that sigma(n+k) = sigma(n)+sigma(k) if it exists; otherwise 0.
%C A066435 It would be nice to remove the word "Conjectured" from the description - _N. J. A. Sloane_.
%C A066435 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.
%C A066435 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
%C A066435 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.
%C A066435 All n<1000 and k<10^11 have been tested. The largest term is a(711)=21004780114. - _Donovan Johnson_, Aug 29 2012
%D A066435 R. K. Guy, Unsolved Problems in Number Theory, 3rd Ed., New York, Springer-Verlag, 2004, Section B15.
%t A066435 a[ n_ ] := Min[ Select[ Range[ 1, 10^6 ], DivisorSigma[ 1, n + # ] == DivisorSigma[ 1, n ] + DivisorSigma[ 1, # ] & ] ]; Table[ a[ i ], {i, 1, 21} ]
%Y A066435 Cf. A091554 (primes p such that k=2p is the smallest solution to sigma(p+k)=sigma(p)+sigma(k)).
%Y A066435 Cf. A110176 (least k such that sigma(n)=sigma(k)+sigma(n-k)).
%K A066435 hard,nonn
%O A066435 1,1
%A A066435 _Joseph L. Pe_, Dec 27 2001
%E A066435 More terms from _T. D. Noe_, Jan 20 2004