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

434, 8575, 8825

Offset: 1

Author

Jud McCranie, Dec 27 1999

Keywords

• nonn
• bref
• more

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

No more terms < 2.7*10^15. - Jud McCranie, Jul 27 2025

Examples

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

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

• 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.

Crossrefs

Cf. A000203, A001359, A054799.

Programs

Mathematica

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

PARI

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