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A020492 Balanced numbers: numbers k such that phi(k) (A000010) divides sigma(k) (A000203).

%I A020492 #105 Aug 12 2024 12:02:14
%S A020492 1,2,3,6,12,14,15,30,35,42,56,70,78,105,140,168,190,210,248,264,270,
%T A020492 357,418,420,570,594,616,630,714,744,812,840,910,1045,1240,1254,1485,
%U A020492 1672,1848,2090,2214,2376,2436,2580,2730,2970,3080,3135,3339,3596,3720,3828
%N A020492 Balanced numbers: numbers k such that phi(k) (A000010) divides sigma(k) (A000203).
%C A020492 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
%C A020492 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
%C A020492 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
%C A020492 The quotients sigma(n)/phi(n) are in A023897. - _Bernard Schott_, Jun 06 2017
%C A020492 There are 544768 balanced numbers < 10^14. - _Jud McCranie_, Sep 10 2017
%C A020492 a(975807) = 419998185095132. - _Jud McCranie_, Nov 28 2017
%D A020492 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.
%H A020492 Donovan Johnson, <a href="/A020492/b020492.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%H A020492 Jud McCranie, <a href="/A020492/a020492.txt">670314 balanced numbers</a> (first 1000 from T. D. Noe, first 10000 from Donovan Johnson)
%e A020492 sigma(35) = 1+5+7+35 = 48, phi(35) = 24, hence 35 is a term.
%t A020492 Select[ Range[ 4000 ], IntegerQ[ DivisorSigma[ 1, # ]/EulerPhi[ # ] ]& ]
%t A020492 (* Second program: *)
%t A020492 Select[Range@ 4000, Divisible[DivisorSigma[1, #], EulerPhi@ #] &] (* _Michael De Vlieger_, Nov 28 2017 *)
%o A020492 (Magma) [ n: n in [1..3900] | SumOfDivisors(n) mod EulerPhi(n) eq 0 ]; // _Klaus Brockhaus_, Nov 09 2008
%o A020492 (PARI) select(n->sigma(n)%eulerphi(n)==0,vector(10^4,i,i)) \\ _Charles R Greathouse IV_, Jun 20 2012
%o A020492 (Python)
%o A020492 from sympy import totient, divisor_sigma
%o A020492 print([n for n in range(1, 4001) if divisor_sigma(n)%totient(n)==0]) # _Indranil Ghosh_, Jul 06 2017
%o A020492 (Python)
%o A020492 from math import prod
%o A020492 from itertools import count, islice
%o A020492 from sympy import factorint
%o A020492 def A020492_gen(startvalue=1): # generator of terms >= startvalue
%o A020492     for m in count(max(startvalue,1)):
%o A020492         f = factorint(m)
%o A020492         if not prod(p**(e+2)-p for p,e in f.items())%(m*prod((p-1)**2 for p in f)):
%o A020492             yield m
%o A020492 A020492_list = list(islice(A020492_gen(),20)) # _Chai Wah Wu_, Aug 12 2024
%Y A020492 Cf. A000010, A000043, A000203, A000668, A011257, A023897, A133028, A291565, A291566, A292422, A351114 (characteristic function).
%Y A020492 Positions of 0's in A063514.
%K A020492 nonn
%O A020492 1,2
%A A020492 _David W. Wilson_
%E A020492 More terms from _Farideh Firoozbakht_, Nov 28 2005