A125717 -id:A125717 - cp's OEIS frontend

A125718 a(1)=1. a(n) = the smallest positive integer not occurring earlier in the sequence such that the n-th prime is congruent to a(n) (mod n).

1, 3, 2, 7, 6, 13, 10, 11, 5, 9, 20, 25, 15, 29, 17, 21, 8, 43, 48, 31, 52, 35, 14, 41, 22, 23, 49, 51, 80, 53, 34, 67, 38, 37, 44, 79, 46, 87, 50, 93, 56, 55, 19, 61, 62, 107, 70, 127, 129, 179, 131, 83, 82, 89, 92, 39, 98, 97, 100, 101, 161, 45, 118, 119, 183, 185, 63, 65

Offset: 1

Leroy Quet, Feb 01 2007

nonn

This sequence seems likely to be a permutation of the positive integers. It will be if every positive number appears in A004648 (cf. A127149, A127150).

If this is a permutation of the positive integers, then A249678 is the inverse permutation. - M. F. Hasler, Nov 03 2014

Ferenc Adorjan, Table of n,a(n) for n=1,10000
Ferenc Adorjan, Some characteristics of _Leroy Quet_'s permutation sequences
Ferenc Adorjan, More about the structure of _Leroy Quet_'s sequences A125715, A125717, A125718 & A125727

Cf. A004648.

f[l_List] := Block[{n = Length[l] + 1, k = Mod[Prime[n], n, 1]},While[MemberQ[l, k], k += n];Append[l, k]];Nest[f, {1}, 70] (* Ray Chandler, Feb 04 2007 *)

{Quet_p3(n)= /* Permutation sequence a'la Leroy Quet, A125718 */local(x=[1],k=0,w=1); for(i=2,n,if((k=prime(i)%i)==0,k=i);while(bittest(w,k-1)>0,k+=i);x=concat(x,k);w+=2^(k-1));return(x)}

A125718(n,show=0,u=1)={for(n=1,n,p=prime(n)%n;while(bittest(u,p),p+=n);u+=1<M. F. Hasler, Nov 03 2014

Extended by Ray Chandler, Feb 04 2007

A370981 Triangle T(n, k), n >= 0, k = 0..n, read and filled in the greedy way by rows with distinct nonnegative integers, such that for any n > 0 and any k in 0..n, T(n, k) is congruent modulo n to T(n-1, k-1) (provided that k > 0) or to T(n-1, k) (provided that k < n).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 9, 7, 10, 12, 5, 11, 14, 8, 16, 15, 20, 19, 13, 18, 21, 27, 26, 25, 31, 24, 30, 33, 34, 40, 32, 17, 38, 23, 37, 47, 42, 48, 56, 41, 22, 39, 29, 45, 55, 51, 57, 65, 50, 49, 58, 66, 36, 28, 46, 61, 67, 35, 60, 59, 68, 76, 86, 78, 88, 96

Offset: 0

Rémy Sigrist, Mar 06 2024

This sequence can be seen as a two-dimensional variant of A125717.

Will every integer appear in the sequence?

			Triangle T(n, k) begins:
                         0
                       1   2
                     3   4   6
                   9   7  10  12
                 5  11  14   8  16
              15  20  19  13  18  21
            27  26  25  31  24  30  33
          34  40  32  17  38  23  37  47
        42  48  56  41  22  39  29  45  55
      51  57  65  50  49  58  66  36  28  46
    61  67  35  60  59  68  76  86  78  88  96
  72  83  79  71  70  81  43  53  64  44  52  63

Rémy Sigrist, Table of n, a(n) for n = 0..10010 (rows for n = 0..140 flattened)
Rémy Sigrist, Colored representation of the first 500 rows
Rémy Sigrist, PARI program

Cf. A125717.

PARI
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A125718 a(1)=1. a(n) = the smallest positive integer not occurring earlier in the sequence such that the n-th prime is congruent to a(n) (mod n).

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A370981 Triangle T(n, k), n >= 0, k = 0..n, read and filled in the greedy way by rows with distinct nonnegative integers, such that for any n > 0 and any k in 0..n, T(n, k) is congruent modulo n to T(n-1, k-1) (provided that k > 0) or to T(n-1, k) (provided that k < n).

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