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.

1, 2, 3, 4, 8, 24, 28, 40, 60, 112, 316, 508, 568, 760, 796, 1212, 1228, 2616, 5296, 6220, 8016, 12456, 14620, 16888, 21772, 23116, 23356, 25656, 30312, 30712, 30808, 32716, 33720, 38328, 46072, 52816, 59112, 61728, 67960, 69808, 72972

Offset

1,2

Comments

Every term greater than 3 is divisible by 4.

Let b(z) be the number of elements of this sequence <= z:

--------------

z | b(z)

--------------

10^2 | 9

10^3 | 15

10^4 | 21

10^5 | 45

10^6 | 106

10^7 | 296

10^8 | 869

--------------

Conjecture: a(n) + 1 is prime for n > 6. Verified for all terms < 10^8.

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.

Links

Table of n, a(n) for n=1..41.

Example

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:
i\j | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...
-----------------------------------------
  1 | 1
  2 | 1 1
  3 | 1 0 1
  4 | 1 0 0 1
  5 | 1 1 0 0 1
  6 | 1 1 0 0 0 1
  7 | 1 0 1 0 0 0 1
  8 | 1 0 0 0 0 0 0 1
  9 | 1 1 0 1 0 0 0 0 1
 10 | 1 1 0 0 0 0 0 0 0  1
 11 | 1 0 1 0 1 0 0 0 0  0  1
 12 | 1 0 1 0 0 0 0 0 0  0  0  1
 13 | 1 1 0 0 0 1 0 0 0  0  0  0  1
 14 | 1 1 0 1 0 0 0 0 0  0  0  0  0  1
 15 | 1 0 0 0 0 0 1 0 0  0  0  0  0  0  1
...
The j-th column has period j^2.

Prog

(Maxima)
(f(i, j):=mod((i-floor((i-j)/j)), j),
(n:4, for t:4 thru 100000 step 4 do
(for k:2 while f(t, k)#0 and k<t/2+1 do
(if k=t/2 then (print(n, "", t), n:n+1)))));

Crossrefs

Cf. A000040, A005235, A008578, A051731.

Sequence in context: A214700 A100997 A368989 * A128657 A105421 A273063

Adjacent sequences: A375004 A375005 A375006 * A375008 A375009 A375010

Keyword

nonn

Author

Lechoslaw Ratajczak, Jul 27 2024

Status

approved