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 A068418 Composite n such that n-phi(n) divides sigma(n)-n. 9
 12, 56, 260, 992, 1320, 1976, 2156, 2754, 3696, 5520, 13800, 16256, 19872, 22560, 23688, 25232, 41072, 87000, 89964, 133984, 145888, 366720, 785808, 851760, 1100864, 1235052, 1270208, 1439552, 1470720, 2129400, 2237888, 4729664 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS If 2^p-1 is prime(a Mersenne prime) then n=2^p*(2^p-1) is in the sequence because 3n-2phi(n)=sigma(n) (see Comments line of the sequence A068414) so sigma(n)-n=2*(n-phi(n)) hence n-phi(n) divides sigma(n)-n. - Farideh Firoozbakht, Dec 31 2005 Also if 3*2^m-1 is a prime greater than 5 then n = 15*2^(m+1)*(3*2^m-1) is in the sequence because 4n-3*phi(n) = 4*15*2^(m+1)*(3*2^m-1)-3*2^(m+3)*(3*2^m-2) = 24*(3*2^m)*(2^(m+2)-1) = sigma(15)*sigma(3*2^m-1)*sigma(2^(m+1)) = sigma(15*(3*2^m-1)*2^(m+1)) = sigma(n) hence sigma(n)-n = 3*(n-phi(n)) and n-phi(n) divides sigma(n)-n. - Farideh Firoozbakht, Dec 31 2005 LINKS Donovan Johnson, Table of n, a(n) for n = 1..68 (terms < 10^11) MATHEMATICA Do[s=(DivisorSigma[1, n]-n)/(n-EulerPhi[n]); If[ !PrimeQ[n]&&IntegerQ[s], Print[n]], {n, 2, 10000000}] PROG PARI : for(n=1, 300000, if((sigma(n)-n)%(n-eulerphi(n))==isprime(n), print1(n, ", "))) CROSSREFS A068414 is the subsequence telling when the quotient is 2. Cf. A068414, A051953, A000203, A000668. Sequence in context: A009430 A035289 A009827 * A068414 A199316 A081756 Adjacent sequences:  A068415 A068416 A068417 * A068419 A068420 A068421 KEYWORD easy,nonn AUTHOR Benoit Cloitre, Mar 03 2002 EXTENSIONS More terms from Labos E. (labos(AT)ana.sote.hu), Apr 02 2002 STATUS approved

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