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

Author

Lechoslaw Ratajczak, Jul 27 2024

Keywords

• nonn

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

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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.

Examples

			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.
		

Crossrefs

Cf. A000040, A005235, A008578, A051731.

Programs

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