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A140480 RMS numbers: numbers n such that root mean square of divisors of n is an integer. 41
1, 7, 41, 239, 287, 1673, 3055, 6665, 9545, 9799, 9855, 21385, 26095, 34697, 46655, 66815, 68593, 68985, 125255, 155287, 182665, 242879, 273265, 380511, 391345, 404055, 421655, 627215, 730145, 814463, 823537, 876785, 1069895, 1087009, 1166399, 1204281, 1256489 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

For any numbers, A and B, both appearing in the sequence, if GCD(A,B)=1, then A*B is also in the sequence. - Andrew Weimholt, Jul 01 2008

The primes in this sequence are the NSW primes (A088165). For the terms less than 2^31, the only powers greater than 1 appearing in the prime factorization of numbers are 3^3 and 13^2. It appears that all terms are +-1 (mod 8). See A224988 for even numbers. - T. D. Noe, Jul 06 2008, Apr 25 2013

A basis for this sequence is given by the recurrence u(i)=6*u(i-1)-u(i-2), i>=2, u(0)=1,u(1)=7. This can be considered as the convergents of quasiregular continued fractions or a special 6-ary numeration system (see A. S. Fraenkel) which gives the characterization of positions of some heap or Wythoff game. What is the Sprague-Grundy function of this game ?

Sequence generalized : sigma_r-numbers are numbers n for which sigma_r(n)/sigma_0(n) = c^r . Sigma_r(n) denotes sum of r-th powers of divisors of n; c,r positive integers. This sequence are sigma_2-numbers, A003601 are sigma_1-numbers. In a weaker form we have sigma_r(n)/sigma_0(n) = c^t; t is an integer from <1,r>. - Ctibor O. Zizka, Jul 14 2008

The primes in this sequence are prime numerators with an odd index in A001333. The RMS values (A141812) of prime RMS numbers (this sequence) are prime Pell numbers (A000129) with an odd index. [Ctibor O. Zizka, Aug 13 2008]

The set of RMS numbers n could be splitted into subsets according to the number and form of divisors of n. By definition, RMS(n) = sqrt (sigma_2(n) / sigma_0(n)) should be an integer. Now let me show some examples. For n prime number, n has 2 divisors [1,n] and we have to solve Pell`s equation n^2 = 2*C^2 -1 ; C positive integer. The solution is prime n of the form u(i)=6*u(i-1)-u(i-2), i>=2, u(0)=1,u(1)=7, known as NSW prime (A088165). For n = p_1*p_2, p_1 and p_2 primes, n has 4 divisors [1;p_1;p_2;p_1*p_2]. There are 2 possible cases. Firstly p^2=(2*C)^2 - 1 which does not hold for any prime p; secondly p_1^2 = 2*C_1^2 - 1 and p_2^2 = 2*C_2^2 - 1 ;C_1 and C_2 positive integers.

The solution is p_1 and p_2 are different NSW primes. If n=p^3, divisors of n are [1;p;p^2;p^3] and we have to solve Diophantine equation (p^8-1)/(p-1) = (2*C)^2 . This equation has no solution for any prime p. RMS numbers n with 4 divisors are only of the form n = p_1*p_2 ; p_1,p_2 NSW primes. General case is n = p_1*...*p_t, n has 2^t divisors and for t>=3 NSW primes are not the only solution. If some of prime divisors equals, p_i=p_j=...=p_k, the general case n = p_1*...*p_t "degenerate" because multiplicity of prime factors and therefore n has less than 2^t divisors. [Ctibor O. Zizka, Aug 30 2008]

General recurrence is a(n)=(a(1)-1)*a(n-1)-a(n-2), a(1)>=4, lim n->infinity a(n)= x*(k*x+1)^n, k =(a(1)-3), x=(1+sqrt((a(1)+1)/(a(1)-3)))/2. Examples in OEIS: a(1)=4 gives A002878, primes in it A121534. a(1)=5 gives A001834, primes in it A086386. a(1)=6 gives A030221, primes in it not in OEIS {29,139,3191,...}. a(1)=7 gives A002315, primes in it A088165. a(1)=8 gives A033890, primes in it not in OEIS (does there exist any ?). a(1)=9 gives A057080, primes in it not in OEIS {71,34649,16908641,...}. a(1)=10 gives A057081, primes in it not in OEIS {389806471,192097408520951,...}. [Ctibor O. Zizka, Sep 02 2008]

16 of the first 1660 terms are even (the smallest is 2217231104). The first 16 even terms are all divisible by 30976. - Donovan Johnson, Apr 16 2013

LINKS

T. D. Noe and Donovan Johnson, Table of n, a(n) for n = 1..1660 (terms < 10^11, first 455 terms from T. D. Noe)

A. S. Fraenkel, Heap games, numeration systems and sequences, Annals of Combinatorics, 2 (1998), 197-210.

Aviezri S. Fraenkel, On the recurrence f(m+1)= b(m)*f(m)-f(m-1) and applications, Discrete Mathematics 224 (2000), pp. 273-279.

H. W. Lenstra Jr., Solving the Pell Equation, Notices of the AMS, Vol.49, No.2, Feb. 2002, p.182-192. [Ctibor O. Zizka, Aug 30 2008]

Eric Weisstein's World of Math, Root Mean Square

MAPLE

with(numtheory); List140480:=proc(q) local a, k, n;

for n from 1 to q do a:=divisors(n);

if type(sqrt(add(a[k]^2, k=1..nops(a))/tau(n)), integer) then print(n);

fi; od; end: List140480 (10^6); # Paolo P. Lava, Apr 11 2013

MATHEMATICA

rmsQ[n_] := IntegerQ[Sqrt[DivisorSigma[2, n]/DivisorSigma[0, n]]]; m = 160000; sel1 = Select[8*Range[0, m]+1, rmsQ]; sel7 = Select[8*Range[m]-1, rmsQ]; Union[sel1, sel7] (* Jean-Fran├žois Alcover, Aug 31 2011, after T. D. Noe's comment *)

Select[Range[1300000], IntegerQ[RootMeanSquare[Divisors[#]]]&] (* Harvey P. Dale, Mar 24 2016 *)

PROG

(Haskell)

a140480 n = a140480_list !! (n-1)

a140480_list = filter

    ((== 1) . a010052 . (\x -> a001157 x `div` a000005 x)) a020486_list

-- Reinhard Zumkeller, Jan 15 2013

CROSSREFS

Cf. A002315, A001653, A001834, A001835, A001599, A000005, A000040, A003601, A010052, A001157, A020486, A158294, A224988.

Sequence in context: A026002 A173409 A057009 * A002315 A141813 A088165

Adjacent sequences:  A140477 A140478 A140479 * A140481 A140482 A140483

KEYWORD

nonn,nice

AUTHOR

Ctibor O. Zizka, Jun 29 2008, Jul 11 2008

EXTENSIONS

More terms from T. D. Noe and Andrew Weimholt, Jul 01 2008

STATUS

approved

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Last modified March 22 22:05 EDT 2017. Contains 283901 sequences.