A298482 -id:A298482 - cp's OEIS frontend

A301801 a(n) = smallest integer not yet in the sequence with no digits in common with a(n-1), a(n-2), a(n-3), and a(n-4); a(0)=0, a(1)=1, a(2)=2, a(3)=3.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 22, 33, 44, 55, 11, 20, 36, 47, 58, 19, 200, 63, 74, 85, 91, 202, 66, 34, 57, 18, 29, 60, 43, 75, 81, 92, 600, 333, 45, 17, 28, 69, 30, 54, 71, 82, 96, 300, 444, 15, 27, 68, 39, 40, 51, 72, 86, 93, 400, 111, 25, 67, 38, 49, 100

Offset: 0

Enrique Navarrete, Mar 26 2018

The first differences of this sequence are symmetrically distributed in a distribution that has a larger kurtosis than the Normal distribution.
It seems that appart from the initial terms, 39 and 40 are the only consecutive terms.
Unlike A298482, 3-digit terms appear as early as a(22)=200.

Rémy Sigrist, PARI program for A301801

Cf. A067581, A276633, A298482.

Nest[Append[#, Block[{k = 4, d}, While[Nand[FreeQ[#, k], ! IntersectingQ[Union@ Apply[Join, Take[#[[All, -1]], -4] ], Set[d, IntegerDigits[k]]] ], k++]; {k, d}]] &, Transpose@ {#, IntegerDigits@ #} &@ Range[0, 3], 62][[All, 1]] (* Michael De Vlieger, Apr 12 2018 *)

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A298483 Numbers n, the smallest of three consecutive numbers that share the property mu(n) <> chi(n).

Original entry on oeis.org

13, 25, 37, 61, 73, 109, 113, 117, 121, 153, 157, 169, 173, 181, 245, 257, 273, 277, 285, 289, 297, 313, 317, 325, 333, 353, 361, 369, 373, 385, 389, 401, 405, 409, 421, 425, 457, 509, 513, 525, 529, 541, 601, 609, 621, 637, 653, 673, 677, 693, 705, 709, 729, 733, 761, 765, 769, 777, 797, 801, 805, 829, 833, 841, 853

Offset: 1

Torlach Rush, Jan 19 2018

nonn

mu and chi share the same property in that they both evaluate to {-1, 0, 1}.
This sequence admits 5 possible outcomes as follows:
- a(n) are of the form 4k + 1, and are either divisible by an odd number of primes, or are nonsquarefree.
- a(n) + 1 are squarefree even numbers.
- a(n) + 2 are of the form 4k + 3, and are either divisible by an even number of primes, or are nonsquarefree.
3 is the largest number of consecutive integers that satisfy the condition mu(n) <> chi(n). Since a(n) + 3  = 4k + 4 = 4(k+1), which is both nonsquarefree and even, then mu(4(k+1))= chi(4(k+1)), and the sequence terminates.
If a(n) is prime then k - 2 is not divisible by 3.
Conjecture: Every prime a(n) has a multiple a(j), with j > n, the result of a multiplication by a number of the form 4k + 1, a multiple a(m) + 1, with m > n, the result of multiplication by a squarefree even number, and lastly a multiple a(k) + 2, with k > n, the result of multiplication by a prime. Example; a(1) = 13, a(8) = 117, a(2) + 1 = 26, and a(3) + 2 = 39.
If a(n) + 1 is a totient then k - 2 is not divisible by 3.
Observation: Of the 72762 triples up to 10^6, only 19 of the middle terms, which are always even, are totients.

			13 is in the sequence because mu(13)=-1 and chi(13)=1, mu(14)=1 and chi(14)=0, and mu(15)=1 and chi(15)=-1.

Cf. A298482, A016813, A002144, A101455 (chi), A008683 (mu).

With[{nn = 10^3, w = {1, 0, -1, 0}}, Position[Map[UnsameQ @@ # & /@ # &, Partition[Transpose@ {Array[MoebiusMu, nn], Array[w[[Mod[#, 4, 1] ]] &, nn]}, 3, 1]], {True, True, True}]][[All, 1]] (* Michael De Vlieger, Jan 28 2018, after Michael Somos at A101455 *)

isok(n) = (moebius(n) != kronecker( -4, n)) && (moebius(n+1) != kronecker( -4, n+1)) && (moebius(n+2) != kronecker( -4, n+2)); \\ Michel Marcus, Jan 28 2018

0 < min({|mu(a(n))| + |chi(a(n))|, |mu(a(n) + 1)| + |chi(a(n) + 1)|, |mu(a(n) + 2)| + |chi(a(n) + 2)|}).

cp's OEIS Frontend

A301801 a(n) = smallest integer not yet in the sequence with no digits in common with a(n-1), a(n-2), a(n-3), and a(n-4); a(0)=0, a(1)=1, a(2)=2, a(3)=3.

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