A084662 -id:A084662 - cp's OEIS frontend

A258451 a(n) = 1 + a(n-1)/gcd(a(n-1),n) with a(0)=3.

3, 4, 3, 2, 2, 3, 2, 3, 4, 5, 2, 3, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 2, 3, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 2, 3, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 2, 3, 2, 3, 4, 5, 2, 3, 2, 3, 4, 5, 6, 7, 8

Offset: 0

Ctibor O. Zizka, May 30 2015

nonn

The behavior of this sequence depends on a(0).
For a(0) in {1,4,7,9,19,27,34,47,52,59,63,66,69,71,105,133,147,178,183,202,...} we have a(n) = n.
For a(0) in {2,5,10,21,26,35,50,51,53,82,91,96,101,111,122,154,165,170,193,...} we have a(n) = n+2.
For other a(0) the sequence is "saw"-like with small irregular periods between saw teeths. For such a(0), a(n) < n/2.

			a(0)=3. a(1)=1+3/gcd(3,1)=4. a(2)=1+4/gcd(4,2)=3. a(3)=1+3/gcd(3,3)=2. etc

Indranil Ghosh, Table of n, a(n) for n = 0..50000

Cf. A106108, A084662.

FoldList[1+#1/GCD[#1,#2+1]&,3,Range[0,107]] (* Ivan N. Ianakiev, Jun 05 2015 *)
nxt[{n_,a_}]:={n+1,1+a/GCD[a,n+1]}; NestList[nxt,{0,3},110][[All,2]] (* Harvey P. Dale, Jul 27 2019 *)

Typo in data corrected by Ivan N. Ianakiev, Jun 05 2015

A291528 First term s_n(1) of equivalence classes of prime sequences {s_n(k)} for k > 0 derived by records of first differences of Rowland-like recurrences with increasing even starting values e(n) >= 4.

Original entry on oeis.org

2, 7, 17, 19, 31, 43, 53, 71, 67, 79, 97, 103, 109, 113, 127, 137, 151, 163, 181, 173, 191, 197, 199, 211, 229, 239, 241, 251, 269, 257, 271, 283, 293, 317, 331, 337, 349, 367, 373, 419, 409, 431, 433, 439, 443, 463, 491, 487, 499, 523, 557, 547, 577, 593, 607, 599, 601

Offset: 1

Ralf Steiner, Aug 25 2017

nonn

These kinds of equivalence classes {s_n(k)} were defined by Shevelev, see Crossrefs.
Some equivalence classes of prime sequences {s_n(k)} have the same tail for a constant C_n < k, such as {s_2(k)} = {a(2),...} = {7,13,29,59,131,...} and {s_5(k)} = {a(5),...} = {31,61,131,...} with common tail {131,...}. Thus it seems that all terms are leaves of a kind of an inverse prime-tree with branches in A291620 and the root at infinity.
In each equivalence class {s_n(k)} the terms hold: s_n(k+1)-2*s_n(k) >= -1;
(s_n(k+1)+1)/s_n(k) >= 2; lim_{k -> inf} (s_n(k+1)+1)/s_n(k) = 2.

			For n=1 the Rowland recurrence with e(1)=4 is A084662 with first differences A134734 and records {2,3,5,11,...} gives the least new prime a(1)=2 as the first term of a first equivalence class {2,3,5,11,...} of prime sequences.
For n=2 with e(2)=8 and records {2,7,13,29,59,...} gives the least new prime a(2)=7 as the first term of a second equivalence class {7,13,29,59,...} of prime sequences.
For n=3 with e(3)=16, a(3)=17 the third equivalence class is {17,41,83,167,...}.

Cf. A291620 (branches), A167168 (equivalence classes), A134734 (first differences of A084662), A134162.

For[i = 2; pl = {}; fp = {}, i < 350, i++,
ps = Union@FoldList[Max, 1, Rest@# - Most@#] &@
   FoldList[#1 + GCD[#2, #1] &, 2 i, Range[2, 10^5]];
p = Select[ps, (i <= #) && ! MemberQ[pl, #] &, 1];
If[p != {}, fp = Join[fp, {p}];
  pl = Union[pl,
    Drop[ps, -1 + Position[ps, p[[1]]][[1]][[1]]]]]]; Flatten@fp

a(n) >= 2*n; a(n) > 10*n - 50; a(n) < 12*n.

a(n) >= e(n) - 1, for n > 1; a(n) < e(n) + n.

A168144 First differences of A168143 which are different from 1, incremented by 14.

Original entry on oeis.org

19, 23, 31, 47, 79

Offset: 1

Vladimir Shevelev, Nov 19 2009

nonn

All terms of the sequence are primes greater than 17.

Are there more than 5 terms?

E. S. Rowland, A natural prime-generating recurrence, Journal of Integer Sequences, 11 (2008), Article 08.2.8.
V. Shevelev, An infinite set of generators of primes based on the Rowland idea and conjectures concerning twin primes, arXiv:0910.4676 [math.NT], 2009.
V. Shevelev, Generalizations of the Rowland theorem, arXiv:0911.3491 [math.NT], 2009-2010.

Cf. A168143, A167495, A167494, A167493, A167197, A167195, A167170, A167168, A106108, A132199, A167054, A167053, A166944, A166945, A116533, A163961, A163963, A084662, A084663, A134162, A135506, A135508, A118679, A120293.

A168143[17] = 37;
A168143[n_] := A168143[n] = If[GCD[n, A168143[n - 1]] > 1 && FactorInteger[n][[1, 1]] > 17, 3 n - 14, A168143[n - 1] + 1]
DeleteCases[Differences[A168143 /@ Range[17, 100]], 1] + 14 (* Eric Rowland, Jan 27 2019 *)

Corrected and edited by Eric Rowland, Jan 27 2019

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A258451 a(n) = 1 + a(n-1)/gcd(a(n-1),n) with a(0)=3.

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A291528 First term s_n(1) of equivalence classes of prime sequences {s_n(k)} for k > 0 derived by records of first differences of Rowland-like recurrences with increasing even starting values e(n) >= 4.

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A168144 First differences of A168143 which are different from 1, incremented by 14.

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