A132199 -id:A132199 - cp's OEIS frontend

A167494 List of first differences of A167493 that are different from 1.

2, 3, 3, 5, 3, 13, 5, 3, 31, 61, 7, 5, 3, 7, 139, 5, 3, 283, 5, 3, 571, 7, 5, 3, 1153, 5, 3, 2311, 31, 4651, 17, 5, 13, 3, 3, 5, 3, 9343, 5, 3, 11, 3, 59, 3, 29, 3, 19, 7, 5, 3, 7, 19, 5, 3, 17, 3, 113

Offset: 1

Vladimir Shevelev, Nov 05 2009

nonn

Conjecture. All terms of the sequence are primes.

The conjecture is false: a(144)=27, a(146)=25, a(158)=45, etc., which are composite numbers. - Harvey P. Dale, Dec 05 2015

Michel Marcus, Table of n, a(n) for n = 1..211
E. S. Rowland, A natural prime-generating recurrence, Journal of Integer Sequences, Vol. 11 (2008), Article 08.2.8.
V. Shevelev, A new generator of primes based on the Rowland idea, arXiv:0910.4676 [math.NT], 2009.
V. Shevelev, Three theorems on twin primes, arXiv:0911.5478 [math.NT], 2009-2010. [From _Vladimir Shevelev_, Dec 03 2009]

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

nxt[{n_,a_}]:={n+1,If[EvenQ[n],a+GCD[n+1,a],a+GCD[n-1,a]]}; DeleteCases[ Differences[ Transpose[NestList[nxt,{1,2},20000]][[2]]],1] (* Harvey P. Dale, Dec 05 2015 *)

lista(nn) = {my(va = vector(nn)); va[1] = 2; for (n=2, nn, va[n] = if (n%2, va[n-1] + gcd(n, va[n-1]), va[n-1] + gcd(n-2, va[n-1]));); select(x->(x!=1), vector(nn-1, n, va[n+1] - va[n]));} \\ Michel Marcus, Dec 13 2018

A221869 New primes found by Rowland's recurrence in the order of their appearance.

Original entry on oeis.org

5, 3, 11, 23, 47, 101, 7, 13, 233, 467, 941, 1889, 3779, 7559, 15131, 53, 30323, 60647, 121403, 242807, 19, 37, 17, 199, 29, 486041, 421, 972533, 577, 1945649, 163, 3891467, 127, 443, 31, 7783541, 15567089, 5323, 31139561, 41, 62279171, 83, 1103, 124559609

Offset: 1

Bill McEachen, Apr 10 2013

nonn

The terms up to 1103 required examining numbers produced by Rowland's recurrence up to n = 10^8. - T. D. Noe, Apr 11 2013
Exactly 177789368686545736460055960459780707068552048703463291 iterations to find the first 1000 terms of this sequence. - T. D. Noe, Apr 13 2013
The first 10^100 terms of Rowland's sequence generate 18321 primes, 3074 of which are distinct. - Giovanni Resta, Apr 08 2016
Same as A137613 with duplicates deleted; same as A132199 with 1s and duplicates deleted. - Jonathan Sondow, May 03 2013

			b(5)-b(4) = 15-10 = 5, so a(1)=5.
b(6)-b(5) = 18-15 = 3, so a(2)=3.
b(11)-b(10) = 33-22 =11, so a(3)=11.

Giovanni Resta, Table of n, a(n) for n = 1..3074[Terms 1 to 1000 were computed by T. D. Noe; terms 1001 to 3074 by _Giovanni Resta_, Apr 08 2016]
Ivars Peterson, A new formula for generating primes
Eric Rowland, A simple recurrence that produces complex behavior — and primes!
Eric Rowland, A natural prime-generating recurrence, Journal of Integer Sequences, Vol. 11 (2008), Article 08.2.8
Jeffrey Shallit, Recursivity: Rutgers graduate student finds new prime-generating formula
Wikipedia, Formula for primes

Cf. A132199, A137613.

Cf. A106108.

import Data.Set (singleton, member, insert)
a221869 n = a221869_list !! (n-1)
a221869_list = f 2 7 (singleton 1) where
   f u v s | d `member` s = f (u + 1) (v + d) s
           | otherwise    = d : f (u + 1) (v + d) (d `insert` s)
           where d = gcd u v
-- Reinhard Zumkeller, Nov 15 2013

t = {}; b1 = 7; Do[b0 = b1; b1 = b0 + GCD[n, b0]; d = b1 - b0; If[d > 1 && !MemberQ[t, d], AppendTo[t, d]], {n, 2, 10^6}]; t (* T. D. Noe, Apr 10 2013 *)
Rest[ DeleteDuplicates[ f[1] = 7; f[n_] := f[n] = f[n - 1] + GCD[n, f[n - 1]]; Differences[ Table[ f[n], {n, 10^6}]]]] (* Jonathan Sondow, May 03 2013 *)

Entries stem from new adjacent differences b(n) = b(n - 1) + GCD(n, b(n - 1)) where b(1)=7.

More terms from T. D. Noe, Apr 11 2013

Edited by N. J. A. Sloane, Apr 12 2013 at the suggestion of Eric Rowland.

A168143 a(17)=37; for n>=17, a(n)=3n-14 if gcd(n,a(n-1))>1 and all prime divisors of n more than 17; a(n)=a(n-1)+1, otherwise.

Original entry on oeis.org

37, 38, 43, 44, 45, 46, 55, 56, 57, 58, 59, 60, 61, 62, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157

Offset: 17

Vladimir Shevelev, Nov 19 2009

nonn

a(n+1)-a(n)+14 is either 15 or a prime > 17. For a generalization, see the second Shevelev link. - Edited by Robert Israel, Aug 21 2017

Robert Israel, Table of n, a(n) for n = 17..10000
E. S. Rowland, A natural prime-generating recurrence, Journal of Integer Sequences, Vol. 11 (2008), Article 08.2.8.
V. Shevelev, A new generator of primes based on the Rowland idea, arXiv:0910.4676 [math.NT], 2009.
V. Shevelev, Generalizations of the Rowland theorem, arXiv:0911.3491 [math.NT], 2009-2010.

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

A[17]:= 37:
q:= convert(select(isprime,[$2..17]),`*`);
for n from 18 to 100 do
  if igcd(n,A[n-1]) > 1 and igcd(n,q) = 1 then A[n]:= 3*n-14
    else A[n]:= A[n-1]+1 fi
od:
seq(A[i],i=17..100); # Robert Israel, Aug 21 2017

nxt[{n_,a_}]:={n+1,If[GCD[n+1,a]>1&&FactorInteger[n+1][[1,1]]>17,3(n+1)-14,a+1]}; NestList[nxt,{17,37},60][[All,2]] (* Harvey P. Dale, Aug 15 2017 *)

Corrected by Harvey P. Dale, Aug 15 2017

A225487 Duplicate primes found by Rowland's recurrence in the order of their reappearance.

Original entry on oeis.org

3, 5, 11, 7, 13, 101, 47, 53, 23, 19, 29, 37, 31, 41, 83, 73, 17, 43, 67, 157, 179, 167, 79, 443, 139, 113, 137, 97, 233, 61, 823, 71, 103, 151, 199, 499, 181, 229, 353, 313, 1889, 271, 317, 197, 613, 607, 127, 257, 89, 367, 223, 433, 239, 911, 109, 107, 557

Offset: 1

Jonathan Sondow, May 08 2013

nonn

Among the first 10^8 terms of A132199 (Rowland's sequence of 1s and primes), 121 terms are prime. Eleven of them appear more than once, and so are a(1), ..., a(11).
Among the first 10^100 terms of A132199 there are 18321 primes; of these, 3074 are distinct and 351 repeated. - Giovanni Resta, Apr 08 2016
See the crossrefs for references, links, and additional comments.

			The first duplicate in Rowland's sequence of primes A137613 = 5, 3, 11, 3, 23, 3, 47, 3, 5, ... is 3, so a(1) = 3. The second duplicate is 5, so a(2) = 5.

Giovanni Resta, Table of n, a(n) for n = 1..351

Cf. A132199, A137613, A221869.

t = {}; b1 = 7; Do[b0 = b1; b1 = b0 + GCD[n, b0]; d = b1 - b0; If[d > 1, AppendTo[t, d]], {n, 2, 10^8}]; L = {}; Do[ If[MemberQ[Take[t, n - 1], t[[n]]], AppendTo[L, t[[n]]]], {n, 2, Length[t]}]; DeleteDuplicates[L]

a(12)-a(57) from Giovanni Resta, Apr 08 2016

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

A291153 a(n) is the prime index of A191304(n+1).

Original entry on oeis.org

3, 5, 9, 15, 26, 51, 91, 160, 290, 526, 959, 1767, 3279, 6113, 11426, 21456, 40448, 76548, 145205, 276032, 526142, 1004977, 1924032, 3689162, 7086486, 13633821, 26269617, 50680636, 97899691, 189336057, 366569494, 710444878, 1378224063, 2676107406, 5200648226, 10114912373, 19687771058, 38348128843, 74746149884, 145785668141, 284517554507, 555594884599, 1085551499862, 2122142209034, 4150687469435

Offset: 1

Ralf Steiner, Aug 19 2017

nonn

The left point (x,y) of intersection of quadratic fits of log(a(n)) and log(A191304(n+1)) is about (-1, 0).
a(n+1) < 2 a(n) for all n, and lim_{n->inf} a(n+1)/a(n) = 2.
With A167168(1)=3 and s_1 = {3,5,11,23,...}, p_(a(n)) = s_1(n+1) in a two-index notation for every prime p_i for i > 1 based on Shevelev's equivalence classes of Rowland-like prime sequence recurrences. These equivalence classes {s_n(k)} were defined by Shevelev, see Crossrefs.

			p_(a(3)) = A000040(a(3)) = A000040(9) = 23 = s_1(3+1) with
s_1 = {3,5,11,23,...}.

Cf. A191304, A167168 (equivalence classes), A000040 (prime numbers).

Rest@ PrimePi@ Union@ FoldList[Max, 1, Rest@ # - Most@ #] &@ FoldList[#1 + GCD[#2, #1] &, 7, Range[2, 10^7]] (* after Michael De Vlieger, Aug 19 2017, after Robert G. Wilson v at A132199 *)

a(n) = pi(A191304(n+1)).

(4/5)^2 (n - 1) < log(a(n)) < (4/5)^2 (n + 1), for at least n < 46.

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A167494 List of first differences of A167493 that are different from 1.

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A221869 New primes found by Rowland's recurrence in the order of their appearance.

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A168143 a(17)=37; for n>=17, a(n)=3n-14 if gcd(n,a(n-1))>1 and all prime divisors of n more than 17; a(n)=a(n-1)+1, otherwise.

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A225487 Duplicate primes found by Rowland's recurrence in the order of their reappearance.

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

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A291153 a(n) is the prime index of A191304(n+1).

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