A015886 - OEIS

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A015886

a(n) = smallest number k such that sigma(k + n) = sigma(k) + n, or -1 if no such number exists.

1, 2, 3, 2, 3, 2, 5, 74, 3, 2, 3, 2, 5, 4418, 3, 2, 3, 2, 5, 6, 3, 2, 7 (list; graph; refs; listen; history; internal format)

	OFFSET	`0,2`
	COMMENTS	`There are solutions to sigma(k)+n=sigma(k+n) whenever n is the difference between two primes (A030173), e.g. k and k+n are primes. There are other values of n that have solutions (see example).` `The "other" values of n are the odd n such that n+2 is not prime. For these n, in order for sigma(k) or sigma(n+k) to be odd, either k or n+k must be a square or twice a square. Examples: for n=7, n+k=9^2; for n=13, k=2*47^2 and for n=19, n+k=5^2. Using this idea, it is easy to show that if a(23) exists it is greater than 10^12. - T. D. Noe (noe(AT)sspectra.com), Sep 24 2007`
	FORMULA	`a(2n)=A020483(n)=A054906(n) - T. D. Noe (noe(AT)sspectra.com), Sep 24 2007`
	EXAMPLE	`sigma(74+7)=121=sigma(74)+7, so a(7)=74.`
	MATHEMATICA	`Table[k=1; While[DivisorSigma[1, k+n] != DivisorSigma[1, k]+n, k++ ]; k, {n, 22}] - T. D. Noe (noe(AT)sspectra.com), Sep 24 2007`
	CROSSREFS	`Cf. A000203, A030173.` `Sequence in context: A065559 A087317 A086489 * A108656 A164962 A119880` `Adjacent sequences: A015883 A015884 A015885 * A015887 A015888 A015889`
	KEYWORD	`nonn,hard,nice`
	AUTHOR	`Robert G. Wilson v (rgwv(AT)rgwv.com)`
	EXTENSIONS	`a(23) > 4292000000, if it exists - Jud McCranie (JudMcCranie(AT)ugaalum.uga.edu), Jan 05 2000` `Jud McCranie (JudMcCranie(AT)ugaalum.uga.edu), Jan 08 2000 reports that the sequence begins 1,2,3,2,3,2,5,74,3,2,3,2,5,4418,3,2,3,2,5,6,3,2,7,?,5,?,3,2,3,2,7,?,5, 18,3,2,5,44,3,2,3,2,5,?,3,2,7,?,5,3315,3,2,7,?,5,?,3,2,3,2,7,?,5,?,3, 2,5,?,3,2,3,2,7,18,5,?,3,2,5,?,3,2,7,?,5,?,3,2,13,?,7,?,5,32,3,2,5 where the other missing terms (designated by "?") are > 10^9, if they exist.`

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Last modified February 17 18:01 EST 2012. Contains 206061 sequences.