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 A020492 Balanced numbers: numbers k such that phi(k) (A000010) divides sigma(k) (A000203). 87
 1, 2, 3, 6, 12, 14, 15, 30, 35, 42, 56, 70, 78, 105, 140, 168, 190, 210, 248, 264, 270, 357, 418, 420, 570, 594, 616, 630, 714, 744, 812, 840, 910, 1045, 1240, 1254, 1485, 1672, 1848, 2090, 2214, 2376, 2436, 2580, 2730, 2970, 3080, 3135, 3339, 3596, 3720, 3828 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The quotient A020492(n)/A002088(n) = SummatorySigma/SummatoryTotient as n increases seems to approach Pi^4/36 or zeta(2)^2 [~2.705808084277845]. - Labos Elemer, Sep 20 2004, corrected by Charles R Greathouse IV, Jun 20 2012 If 2^p-1 is prime (a Mersenne prime) then m = 2^(p-2)*(2^p-1) is in the sequence because when p = 2 we get m = 3 and phi(3) divides sigma(3) and for p > 2, phi(m) = 2^(p-2)*(2^(p-1)-1); sigma(m) = (2^(p-1)-1)*2^p hence sigma(m)/phi(m) = 4 is an integer. So for each n, A133028(n) = 2^(A000043(n)-2)*(2^A000043(n)-1) is in the sequence. - Farideh Firoozbakht, Nov 28 2005 Phi and sigma are both multiplicative functions and for this reason if m and n are coprime and included in this sequence then m*n is also in this sequence. - Enrique Pérez Herrero, Sep 05 2010 The quotients sigma(n)/phi(n) are in A023897. - Bernard Schott, Jun 06 2017 There are 544768 balanced numbers < 10^14. - Jud McCranie, Sep 10 2017 a(975807) = 419998185095132. - Jud McCranie, Nov 28 2017 REFERENCES D. Chiang, "N's for which phi(N) divides sigma(N)", Mathematical Buds, Chap. VI pp. 53-70 Vol. 3 Ed. H. D. Ruderman, Mu Alpha Theta 1984. LINKS T. D. Noe and Donovan Johnson, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe) EXAMPLE sigma(35) = 1+5+7+35 = 48, phi(35) = 24, hence 35 is a term. MAPLE with(numtheory); A020492:=proc(q) local n; for n from 1 to q do if (sigma(n) mod phi(n))=0 then print(n); fi; od; end: A020492(10000000); # Paolo P. Lava, Jan 31 2013 MATHEMATICA Select[ Range[ 4000 ], IntegerQ[ DivisorSigma[ 1, # ]/EulerPhi[ # ] ]& ] (* Second program: *) Select[Range@ 4000, Divisible[DivisorSigma[1, #], EulerPhi@ #] &] (* Michael De Vlieger, Nov 28 2017 *) PROG (MAGMA) [ n: n in [1..3900] | SumOfDivisors(n) mod EulerPhi(n) eq 0 ]; // Klaus Brockhaus, Nov 09 2008 (PARI) select(n->sigma(n)%eulerphi(n)==0, vector(10^4, i, i)) \\ Charles R Greathouse IV, Jun 20 2012 (Python) from sympy import totient, divisor_sigma print([n for n in range(1, 4001) if divisor_sigma(n)%totient(n)==0]) # Indranil Ghosh, Jul 06 2017 CROSSREFS Cf. A000010, A000043, A000203, A000668, A011257, A023897, A133028, A291565, A291566, A292422. Sequence in context: A015769 A015765 A015771 * A110590 A329401 A291174 Adjacent sequences:  A020489 A020490 A020491 * A020493 A020494 A020495 KEYWORD nonn AUTHOR EXTENSIONS More terms from Farideh Firoozbakht, Nov 28 2005 STATUS approved

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Last modified May 22 08:21 EDT 2022. Contains 353933 sequences. (Running on oeis4.)