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A239148 Expansion of triangle T(n,k) of p-adic valuations of A052129(n) (Somos' quadratic recurrence sequence).

%I A239148 #20 Sep 07 2014 08:20:50
%S A239148 0,0,1,2,1,6,2,12,4,1,25,9,2,50,18,4,1,103,36,8,2,206,74,16,4,413,148,
%T A239148 33,8,826,296,66,16,1,1654,593,132,2,3308,1186,264,64,4,1,6617,2372,
%U A239148 528,129,8,2,13234,4745,1057,258,16,4,26472,9490,211,516,32,8,52944,18980,4228,1032,64,16,1,105889
%N A239148 Expansion of triangle T(n,k) of p-adic valuations of A052129(n) (Somos' quadratic recurrence sequence).
%C A239148 Sum of triangle rows => A238496(n).
%C A239148 Only repeated values are powers of 2; all others are non-repeating.
%C A239148 When n=2p (p>2): T(n,k)=2^p+1.
%F A239148 T(n,k) = p-adic valuations of n*A052129(n-1)^2 (n>1; p=>(k+1)-th prime).
%F A239148 When k is constant and P' means "p-adic valuations of": P'a(n) = 2*P'a(n-1) + P'(n).
%e A239148 2    3    5    7    11   13...  (p)
%e A239148 0
%e A239148 0
%e A239148 1
%e A239148 2    1
%e A239148 6    2
%e A239148 12   4    1
%e A239148 25   9    2
%e A239148 50   18   4    1
%e A239148 103  36   8    2
%e A239148 206  74   16   4
%e A239148 413  148  33   8
%e A239148 826  296  66   16   1
%e A239148 1654 593  132  32   2
%e A239148 3308 1186 264  64   4    1
%e A239148 6617 2372 528  129  8    2
%e A239148 T(11,2)=66 because the (k+1)-th (3rd) prime is 5, and the 5-adic valuation of A052129(11)=66,
%e A239148 T(14,3)=129=2^7+1; n=2p because the (k+1)-th (4th) prime is 7.
%o A239148 (PARI) T(n,k)=my(p=prime(k+1),s); forstep(i=n%p, n-1, p, s+=valuation(n-i, p)<<i); s
%o A239148 for(n=1,20,for(k=1,max(primepi(n)-1,1),print1(T(n,k)", "));print) \\ _Charles R Greathouse IV_, Mar 12 2014
%Y A239148 Cf. A052129, A238496, A238462 (2-adic valuation of A052129).
%Y A239148 Cf. A001045 (Jacobsthal numbers - see A052129 for relationship with this sequence).
%K A239148 nonn,easy,tabf
%O A239148 0,4
%A A239148 _Bob Selcoe_, Mar 11 2014