%I A140480
%S A140480 1,7,41,239,287,1673,3055,6665,9545,9799,9855,21385,26095,34697,46655,
               66815,
%T A140480 68593,68985,125255,155287,182665,242879,273265,380511,391345,404055,421655,
%U A140480 627215,730145,814463,823537,876785,1069895,1087009,1166399,1204281,1256489
%N A140480 RMS numbers: numbers n such that root mean square of divisors of n is 
               an integer.
%C A140480 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
%C A140480 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). - T. D. Noe, Jul 06 2008
%C A140480 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 characterisation of positions 
               of some heap or Wythoff game. What is the Sprague-Grundy function 
               of this game ?
%C A140480 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 (ctibor.zizka(AT)seznam.cz), 
               Jul 14 2008
%C A140480 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. [From Ctibor 
               O. Zizka (ctibor.zizka(AT)seznam.cz), Aug 13 2008]
%C A140480 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.
%C A140480 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. [From Ctibor O. Zizka (ctibor.zizka(AT)seznam.cz), 
               Aug 30 2008]
%C A140480 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,...}. [From Ctibor O. Zizka (ctibor.zizka(AT)seznam.cz), 
               Sep 02 2008]
%D A140480 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. http:/
               /www.wisdom.weizmann.ac.il/~fraenkel/
%D A140480 Aviezri S. FRAENKEL : Heap games, numeration systems and sequences. Annals 
               of Combinatorics 2 (1998), pp. 197-210. http://www.wisdom.weizmann.ac.il/
               ~fraenkel/
%D A140480 H. W. Lenstra Jr. : Solving the Pell Equation, Notices of the AMS, Vol.49, 
               No.2,Feb.2002,p.182-192. [From Ctibor O. Zizka (ctibor.zizka(AT)seznam.cz), 
               Aug 30 2008]
%H A140480 T. D. Noe, <a href="b140480.txt">Table of n, a(n) for n=1..455</a> (all 
               terms < 2^31)
%H A140480 Eric Weisstein's World of Math, <a href="http://mathworld.wolfram.com/
               Root-Mean-Square.html">Root Mean Square</a>
%Y A140480 Cf. A002315, A001653, A001834, A001835, A001599, A000005, A000040.
%Y A140480 Cf. A003601.
%Y A140480 Sequence in context: A152268 A026002 A057009 this_sequence A002315 A141813 
               A088165
%Y A140480 Adjacent sequences: A140477 A140478 A140479 this_sequence A140481 A140482 
               A140483
%K A140480 nonn,nice
%O A140480 1,2
%A A140480 Ctibor O. Zizka (ctibor.zizka(AT)seznam.cz), Jun 29 2008, Jul 11 2008
%E A140480 More terms from T. D. Noe and Andrew Weimholt, Jul 01 2008