A050507 - OEIS

%I #34 Jul 28 2025 06:19:03

%S 434,8575,8825

%N Solutions to sigma(x)+2=sigma(x+2) other than the smaller of twin primes.

%C This sequence together with A001359 gives the solutions of sigma(x)+2 = sigma(x+2).

%C No others < 4.29*10^9.

%C No others < 5*10^10. - _Charles R Greathouse IV_, Oct 19 2010

%C They are also the solutions of A001065(x) = A001065(x+2), where A001065(n) is the sum of proper divisors of n. - _Michel Marcus_, Nov 14 2014

%C Makowski found these 3 solutions and verified that there are none others with x <= 9998. Haukkanen extended the bound to 2*10^8. - _Amiram Eldar_, Dec 28 2018

%C a(4) > 10^13, if it exists. - _Giovanni Resta_, Dec 12 2019

%C No more terms < 2.7*10^15. - _Jud McCranie_, Jul 27 2025

%D Richard K. Guy, Unsolved Problems in Number Theory, 3rd ed. New York: Springer-Verlag, 2004, chapter B13, p. 104.

%D R. Sivaramakrishnan, Classical Theory of Arithmetical Functions, M. Dekker Inc., New York-Basel, 1989, p. 81, Problem 12.

%H Pentti Haukkanen, <a href="https://www.researchgate.net/publication/282603270_SOME_COMPUTATIONAL_RESULTS_CONCERNING_THE_DIVISOR_FUNCTIONS_dn_AND_SIGMAn">Some computational results concerning the divisor functions d(n) and sigma(n)</a>, The Mathematics Student, Vol. 62, Nos. 1-4 (1993), pp. 166-168.

%H Andrzej Makowski <a href="https://doi.org/10.2307/2310107">On Some Equations Involving Functions phi(n) and sigma(n)</a>, The American Mathematical Monthly, Vol. 67, No. 7 (1960), pp. 668-670.

%e sigma(434)+2=770=sigma(434+2), so 434 is in the sequence.

%t Select[Range[10000], CompositeQ[#] && DivisorSigma[1, #] + 2 == DivisorSigma[1, # + 2] &] (* _Amiram Eldar_, Dec 28 2018 *)

%o (PARI) is(n)=sigma(n+2)==sigma(n)+2&&!isprime(n) \\ _Charles R Greathouse IV_, Feb 13 2013

%Y Cf. A000203, A001359, A054799.

%K nonn,bref,more

%O 1,1

%A _Jud McCranie_, Dec 27 1999