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A375007 Numbers t which satisfy the equation: t mod k = floor((t - k)/k) mod k (1 <= k <= t) only for k = 1 and t.

%I A375007 #9 Aug 18 2024 20:20:44
%S A375007 1,2,3,4,8,24,28,40,60,112,316,508,568,760,796,1212,1228,2616,5296,
%T A375007 6220,8016,12456,14620,16888,21772,23116,23356,25656,30312,30712,
%U A375007 30808,32716,33720,38328,46072,52816,59112,61728,67960,69808,72972
%N A375007 Numbers t which satisfy the equation: t mod k = floor((t - k)/k) mod k (1 <= k <= t) only for k = 1 and t.
%C A375007 Every term greater than 3 is divisible by 4.
%C A375007 Let b(z) be the number of elements of this sequence <= z:
%C A375007 --------------
%C A375007    z  |   b(z)
%C A375007 --------------
%C A375007 10^2  |    9
%C A375007 10^3  |   15
%C A375007 10^4  |   21
%C A375007 10^5  |   45
%C A375007 10^6  |  106
%C A375007 10^7  |  296
%C A375007 10^8  |  869
%C A375007 --------------
%C A375007 Conjecture: a(n) + 1 is prime for n > 6. Verified for all terms < 10^8.
%C A375007 Conjecture: nextprime(u(n)) - u(n), where u(n) = Product_{m=1..n} (a(m+1) - a(m)), is a noncomposite number. Verified for all terms < 10^8.
%e A375007 Let T(i,j) be the triangle read by rows: T(i,j) = 1 if i mod j = floor((i - j)/j) mod j, T(i,j) = 0 otherwise, for 1 <= j <= i. The triangle begins:
%e A375007 i\j | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...
%e A375007 -----------------------------------------
%e A375007   1 | 1
%e A375007   2 | 1 1
%e A375007   3 | 1 0 1
%e A375007   4 | 1 0 0 1
%e A375007   5 | 1 1 0 0 1
%e A375007   6 | 1 1 0 0 0 1
%e A375007   7 | 1 0 1 0 0 0 1
%e A375007   8 | 1 0 0 0 0 0 0 1
%e A375007   9 | 1 1 0 1 0 0 0 0 1
%e A375007  10 | 1 1 0 0 0 0 0 0 0  1
%e A375007  11 | 1 0 1 0 1 0 0 0 0  0  1
%e A375007  12 | 1 0 1 0 0 0 0 0 0  0  0  1
%e A375007  13 | 1 1 0 0 0 1 0 0 0  0  0  0  1
%e A375007  14 | 1 1 0 1 0 0 0 0 0  0  0  0  0  1
%e A375007  15 | 1 0 0 0 0 0 1 0 0  0  0  0  0  0  1
%e A375007 ...
%e A375007 The j-th column has period j^2.
%o A375007 (Maxima)
%o A375007 (f(i,j):=mod((i-floor((i-j)/j)),j),
%o A375007 (n:4, for t:4 thru 100000 step 4 do
%o A375007 (for k:2 while f(t,k)#0 and k<t/2+1 do
%o A375007 (if k=t/2 then (print(n, "", t), n:n+1)))));
%Y A375007 Cf. A000040, A005235, A008578, A051731.
%K A375007 nonn
%O A375007 1,2
%A A375007 _Lechoslaw Ratajczak_, Jul 27 2024