A050507 Solutions to sigma(x)+2=sigma(x+2) other than the smaller of twin primes.

434, 8575, 8825

Offset

1,1

Comments

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

No others < 4.29*10^9.

No others < 5*10^10. - Charles R Greathouse IV, Oct 19 2010

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

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

a(4) > 10^13, if it exists. - Giovanni Resta, Dec 12 2019

References

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

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

Links

Table of n, a(n) for n=1..3.

Pentti Haukkanen, Some computational results concerning the divisor functions d(n) and sigma(n), The Mathematics Student, Vol. 62, Nos. 1-4 (1993), pp. 166-168.

Andrzej Makowski On Some Equations Involving Functions phi(n) and sigma(n), The American Mathematical Monthly, Vol. 67, No. 7 (1960), pp. 668-670.

Example

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

Mathematica

Select[Range[10000], CompositeQ[#] && DivisorSigma[1, #] + 2 == DivisorSigma[1, # + 2] &] (* Amiram Eldar, Dec 28 2018 *)

Prog

(PARI) is(n)=sigma(n+2)==sigma(n)+2&&!isprime(n) \\ Charles R Greathouse IV, Feb 13 2013

Crossrefs

Cf. A000203, A001359, A054799.

Sequence in context: A219033 A237386 A259294 * A145318 A054987 A054905

Adjacent sequences: A050504 A050505 A050506 * A050508 A050509 A050510

Keyword

nonn,bref,more

Author

Jud McCranie, Dec 27 1999

Status

approved