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A050507 Solutions to sigma(x)+2=sigma(x+2) other than the smaller of twin primes. 6
434, 8575, 8825 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A219033 A237386 A259294 * A145318 A054987 A054905
KEYWORD
nonn,bref,more
AUTHOR
Jud McCranie, Dec 27 1999
STATUS
approved

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Last modified May 18 15:50 EDT 2024. Contains 372664 sequences. (Running on oeis4.)