A020492 - OEIS

%I #105 Aug 12 2024 12:02:14

%S 1,2,3,6,12,14,15,30,35,42,56,70,78,105,140,168,190,210,248,264,270,

%T 357,418,420,570,594,616,630,714,744,812,840,910,1045,1240,1254,1485,

%U 1672,1848,2090,2214,2376,2436,2580,2730,2970,3080,3135,3339,3596,3720,3828

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

%C 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 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 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 The quotients sigma(n)/phi(n) are in A023897. - _Bernard Schott_, Jun 06 2017

%C There are 544768 balanced numbers < 10^14. - _Jud McCranie_, Sep 10 2017

%C a(975807) = 419998185095132. - _Jud McCranie_, Nov 28 2017

%D 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 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 Jud McCranie, <a href="/A020492/a020492.txt">670314 balanced numbers</a> (first 1000 from T. D. Noe, first 10000 from Donovan Johnson)

%e sigma(35) = 1+5+7+35 = 48, phi(35) = 24, hence 35 is a term.

%t Select[ Range[ 4000 ], IntegerQ[ DivisorSigma[ 1, # ]/EulerPhi[ # ] ]& ]

%t (* Second program: *)

%t Select[Range@ 4000, Divisible[DivisorSigma[1, #], EulerPhi@ #] &] (* _Michael De Vlieger_, Nov 28 2017 *)

%o (Magma) [ n: n in [1..3900] | SumOfDivisors(n) mod EulerPhi(n) eq 0 ]; // _Klaus Brockhaus_, Nov 09 2008

%o (PARI) select(n->sigma(n)%eulerphi(n)==0,vector(10^4,i,i)) \\ _Charles R Greathouse IV_, Jun 20 2012

%o (Python)

%o from sympy import totient, divisor_sigma

%o print([n for n in range(1, 4001) if divisor_sigma(n)%totient(n)==0]) # _Indranil Ghosh_, Jul 06 2017

%o (Python)

%o from math import prod

%o from itertools import count, islice

%o from sympy import factorint

%o def A020492_gen(startvalue=1): # generator of terms >= startvalue

%o     for m in count(max(startvalue,1)):

%o         f = factorint(m)

%o         if not prod(p**(e+2)-p for p,e in f.items())%(m*prod((p-1)**2 for p in f)):

%o             yield m

%o A020492_list = list(islice(A020492_gen(),20)) # _Chai Wah Wu_, Aug 12 2024

%Y Cf. A000010, A000043, A000203, A000668, A011257, A023897, A133028, A291565, A291566, A292422, A351114 (characteristic function).

%Y Positions of 0's in A063514.

%K nonn

%O 1,2

%A _David W. Wilson_

%E More terms from _Farideh Firoozbakht_, Nov 28 2005