A134204 -id:A134204 - cp's OEIS frontend

A134206 a(n) = A134205(n)/n.

5, 4, 4, 5, 6, 6, 6, 8, 8, 6, 6, 4, 6, 9, 8, 9, 8, 7, 8, 9, 10, 9, 14, 14, 6, 11, 14, 9, 14, 13, 8, 9, 8, 5, 6, 9, 18, 20, 16, 12, 14, 15, 8, 12, 12, 11, 16, 15, 12, 15, 14, 9, 8, 11, 20, 19, 12, 13, 8, 5, 12, 11, 6, 12, 12, 8, 12, 13, 12, 10, 18, 19, 16, 17, 14, 12, 12, 14, 22, 21, 10, 9, 6, 8

Offset: 1

Leroy Quet, Oct 14 2007

nonn

A134205(n) is divisible by n for every n, by definition of A134204. But it is unknown whether A134204(n), and therefore also A134205 and A134206, are defined for all n>0.

Conjecture: Apart from a(2)=a(3)=a(12)=4 and a(1)=a(4)=a(34)=a(60)=5, all a(n) exceed 5. - M. F. Hasler, Feb 12 2013. Reply from David Applegate, Dec 11 2013: The conjecture is false: A134206(73397) = A134206(213138) = A134206(790306) = 2.(those are the only 2's for n <= 10^6).

M. F. Hasler and Michael De Vlieger, Table of n, a(n) for n = 1..10000 (First 1000 terms from M. F. Hasler).

Cf. A134204, A134205.

With[{nn = 84}, MapIndexed[Total[#1]/First@ #2 &, Partition[#, 2, 1]] &@ Fold[Append[#1, SelectFirst[Prime@ Range[2, Ceiling@ Log2[nn] nn], Function[p, And[FreeQ[#1, p], Divisible[Last@ #1 + p, #2]]]]] &, {2}, Range@ nn]] (* Michael De Vlieger, Oct 16 2017 *)

More terms from Robert Israel, Oct 14 2007

A224223 a(0)=2; for n>0, a(n) = smallest prime not occurring earlier in the sequence such that a(n-1)+a(n) is a multiple of n^2. If no such prime exists, the sequence terminates.

Original entry on oeis.org

2, 3, 5, 13, 19, 31, 41, 449, 127, 197, 103, 139, 149, 1879, 277, 173, 83, 5119, 389, 1777, 223, 659, 1277, 839, 313, 937, 1091, 367, 1201, 5527, 773, 4993, 1151, 7561, 2843, 4507, 677, 4799, 977, 5107, 4493, 15679, 7253, 26029, 3011, 1039, 5309, 3527

Offset: 0

Daniel Drucker and N. J. A. Sloane, Apr 05 2013

Is this sequence infinite and, if so, is it a permutation of the primes? The answers are probably Yes and No (7 has not appeared after 10000 terms). Compare A134204.

N. J. A. Sloane, Table of n, a(n) for n = 0..9999

Cf. A134204, A224221, A224222, A224229.

A224229 a(0)=2; for n>0, a(n) = smallest prime not occurring earlier in the sequence such that a(n-1)+a(n) is a multiple of floor(sqrt(n)). If no such prime exists, the sequence terminates.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 31, 29, 37, 41, 43, 47, 61, 59, 53, 67, 73, 71, 89, 79, 97, 83, 107, 103, 127, 113, 137, 163, 157, 173, 167, 193, 197, 109, 101, 139, 131, 151, 149, 181, 179, 199, 191, 211, 227, 223, 239, 251, 281, 293, 337, 307, 379, 349, 421, 419, 449, 433, 463, 461, 491, 229, 283, 269, 331, 277, 347, 317, 443, 373, 467, 389, 499, 397, 523

Offset: 0

N. J. A. Sloane, Apr 10 2013

Is this sequence infinite and, if so, is it a permutation of the primes? For this sequence the answers are probably both Yes. A134204 and A224223 are similar sequences whose status is also unknown, while A224221 and A224222 are similar sequences which terminate after about 20 terms.

N. J. A. Sloane, Table of n, a(n) for n = 0..9999

Cf. A134204, A224221, A224222, A224223.

# A224229
Digits:=100;
M1:=100000; hit:=Array(1..M1);
M2:=1000;
a:=[2]; hit[1]:=1;
p:=2;
for n from 1 to M2 do
t1:=floor(sqrt(n));
sw1:=-1;
   for i from 2 to M1  do
      q:=ithprime(i);
      if ( (p+q) mod t1 ) = 0 and hit[i] <> 1 then sw1:=1; break; fi;
   od:
if sw1 < 0 then lprint("ERROR", n, a); break; fi;
a:=[op(a),q];
hit[i]:=1;
p:=q;
                  od:
a;

A131261 a(0)=3; for n>0, a(n) = smallest odd prime not occurring earlier in the sequence such that a(n-1)+a(n) is a multiple of n.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 41, 31, 29, 37, 47, 83, 43, 107, 53, 151, 101, 89, 71, 97, 79, 59, 61, 139, 173, 313, 163, 127, 113, 73, 311, 283, 193, 157, 131, 239, 103, 443, 197, 541, 257, 431, 229, 401, 887, 241, 191, 397, 353, 463, 109, 421, 227, 433, 631, 167

Offset: 0

David Applegate, Oct 26 2007

nonn

Is this sequence infinite and, if so, is it a permutation of the odd primes?

An analog of A134204, but using only the odd primes.

Olivier Gérard, Table of n, a(n) for n = 0..1000

Cf. A134204.

a = {3}; For[n = 1, n < 60, n++, i = 2; While[Length[Intersection[{Prime[i]}, a]] == 1 || Not[Mod[a[[ -1 ]] + Prime[i], n] == 0], i++ ]; AppendTo[a, Prime[i]]]; a (* Stefan Steinerberger, Oct 30 2007 *)

A294639 a(n) = least prime p such that n divides p + prime(n).

Original entry on oeis.org

2, 3, 7, 5, 19, 5, 11, 5, 13, 11, 2, 11, 11, 13, 13, 11, 43, 11, 47, 29, 11, 31, 101, 7, 3, 3, 5, 5, 7, 7, 59, 29, 61, 31, 61, 29, 139, 103, 67, 67, 67, 29, 67, 71, 73, 31, 71, 17, 67, 71, 73, 73, 607, 19, 73, 17, 73, 19, 313, 19, 83, 17, 71, 73, 337, 13, 71

Offset: 1

Rémy Sigrist, Nov 05 2017

This sequence was inspired by A134204.
The logarithmic scatterplot of the sequence has interesting features (see Links section).
We observe runs of consecutive equal terms:
- first pair: a(12) = a(13) = 11,
- first triple: a(39) = a(40) = a(41) = 67,
- first quadruple: a(24980) = a(24981) = a(24982) = a(24983) = 12983.
a(1) = prime(1).
a(2) = prime(2).

			For n=3:
- prime(3) = 5,
- 3 does not divide 2 + 5,
- 3 does not divide 3 + 5,
- 3 does not divide 5 + 5,
- 3 divides 7 + 5,
- hence a(3) = 7.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Logarithmic scatterplot of the sequence for n=1..50000
Rémy Sigrist, Colored logarithmic scatterplot of the sequence for n=1..50000 (where the color is function of (a(n) + prime(n))/n)

Cf. A134204, A254862.

a(n) = my (q=prime(n)); forprime(p=2,, if ((p+q)%n==0, return (p)))

A294734 Lexicographically earliest sequence of distinct positive odd numbers such that, for any n > 1, n divides a(n-1) + a(n).

Original entry on oeis.org

1, 3, 9, 7, 13, 5, 23, 17, 19, 11, 33, 15, 37, 47, 43, 21, 81, 27, 49, 31, 53, 35, 57, 39, 61, 69, 93, 75, 41, 79, 45, 51, 147, 91, 119, 25, 123, 29, 127, 73, 173, 121, 137, 83, 97, 87, 101, 139, 155, 95, 109, 99, 113, 103, 117, 107, 235, 55, 63, 177, 67, 181

Offset: 1

Rémy Sigrist, Nov 07 2017

nonn

This sequence is a variant of A099506 and of A134204.
The variant where "n divides a(n) + a(n+1)" corresponds to A005408 (the odd numbers).
Empirically, the scatterplot of the first terms shows a bundle of curves that correspond to terms sharing the same value of Sum_{k=2.. n} ((-1)^k (a(k+1) + a(k))/k) (see Links section); the sequence A134204 has similar features.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Colored scatterplot of the first 10000 terms
Rémy Sigrist, Colored scatterplot of the first 100000 terms
Rémy Sigrist, PARI program for A294734

Cf. A005408, A099506, A134204.

PARI
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See Links section.
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cp's OEIS Frontend

A134206 a(n) = A134205(n)/n.

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A224223 a(0)=2; for n>0, a(n) = smallest prime not occurring earlier in the sequence such that a(n-1)+a(n) is a multiple of n^2. If no such prime exists, the sequence terminates.

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A224229 a(0)=2; for n>0, a(n) = smallest prime not occurring earlier in the sequence such that a(n-1)+a(n) is a multiple of floor(sqrt(n)). If no such prime exists, the sequence terminates.

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Maple

A131261 a(0)=3; for n>0, a(n) = smallest odd prime not occurring earlier in the sequence such that a(n-1)+a(n) is a multiple of n.

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Mathematica

A294639 a(n) = least prime p such that n divides p + prime(n).

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PARI

A294734 Lexicographically earliest sequence of distinct positive odd numbers such that, for any n > 1, n divides a(n-1) + a(n).

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PARI