cp's OEIS Frontend

A247283 Positions of subrecords in A048673.

%I A247283 #16 Jul 30 2020 20:21:14
%S A247283 5,7,9,15,18,27,36,54,72,108,144,216,288,432,576,864,1152,1728,2304,
%T A247283 3456,4608,6912,9216,13824,18432,27648,36864,55296,73728,110592,
%U A247283 147456,221184,294912,442368,589824,884736,1179648,1769472,2359296,3538944,4718592,7077888
%N A247283 Positions of subrecords in A048673.
%C A247283 Odd bisection seems to be A116453 (i.e. A005010, 9*2^n from a(3)=9 onward).
%C A247283 After terms 7 and 15, even bisection from a(6)=27 onward seems to be A175806 (27*2^n).
%H A247283 Antti Karttunen, <a href="/A247283/b247283.txt">Table of n, a(n) for n = 1..60</a>
%F A247283 a(n) = A064216(A247284(n)).
%F A247283 Conjectures from _Chai Wah Wu_, Jul 30 2020: (Start)
%F A247283 a(n) = 2*a(n-2) for n > 6.
%F A247283 G.f.: x*(3*x^5 - x^3 + x^2 - 7*x - 5)/(2*x^2 - 1). (End)
%e A247283 The fourth (A246360(4) = 5) and the fifth (A246360(5) = 8) record of A048673 (1, 2, 3, 5, 4, 8, ...) occur at A029744(4) = 4 and A029744(5) = 6 respectively. In range between, the maximum must occur at 5, thus a(4-3) = a(1) = 5. (All the previous records of A048673 are in consecutive positions, 1, 2, 3, 4, thus there are no previous subrecords).
%e A247283 The ninth (A246360(9) = 68) and the tenth (A246360(10) = 122) record of A048673 occur at A029744(9) = 24 and A029744(10) = 32 respectively. For n in range 25 .. 31 the values of A048673 are: 25, 26, 63, 50, 16, 53, 19, of which 63 is the maximum, and because it occurs at n = 27, we have a(9-3) = a(6) = 27.
%o A247283 (PARI)
%o A247283 \\ Compute A245449, A246360, A247283 and A247284 at the same time:
%o A247283 default(primelimit,(2^31)+(2^30));
%o A247283 A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From _Michel Marcus_
%o A247283 A048673(n) = (A003961(n)+1)/2;
%o A247283 n = 0; i2 = 0; i3 = 0; ir = 0; prevmax = 0; submax = 0; while(n < 2^32, n++; a_n = A048673(n); if((A048673(a_n) == n), i2++; write("b245449.txt", i2, " ", n)); if((a_n > prevmax), if(submax > 0, i3++; write("b247283.txt", i3, " ", submaxpt); write("b247284.txt", i3, " ", submax)); prevmax = a_n; submax = 0; ir++; write("b029744_empirical.txt", ir, " ", n); write("b246360_empirical.txt", ir, " ", a_n), if((a_n > submax), submax = a_n; submaxpt = n)));
%o A247283 (Scheme)
%o A247283 (definec (A247283 n) (max_pt_in_range A048673 (+ (A029744 (+ n 3)) 1) (- (A029744 (+ n 4)) 1)))
%o A247283 (define (max_pt_in_range intfun lowlim uplim) (let loop ((i (+ 1 lowlim)) (maxnow (intfun lowlim)) (maxpt lowlim)) (cond ((> i uplim) maxpt) (else (let ((v (intfun i))) (if (> v maxnow) (loop (+ 1 i) v i) (loop (+ 1 i) maxnow maxpt)))))))
%Y A247283 A247284 gives the corresponding values.
%Y A247283 Cf. A005010 (A116453), A175806, A029744, A048673, A064216, A246360, A245449.
%K A247283 nonn
%O A247283 1,1
%A A247283 _Antti Karttunen_, Sep 11 2014