A132199 -id:A132199 - cp's OEIS frontend

A167053 a(1)=3; for n > 1, a(n) = 1 + a(n-1) + gcd( a(n-1)*(a(n-1)+2), A073829(a(n-1)) ).

3, 19, 39, 81, 165, 333, 335, 673, 1347, 1349, 1351, 1353, 1355, 1357, 1359, 2721, 2723, 2725, 2727, 5457, 5459, 5461, 5463, 5465, 5467, 5469, 10941, 10943, 10945, 10947, 21897, 21899, 21901, 21903, 21905, 21907, 21909, 43821, 43823, 43825, 43827, 43829, 43831

Offset: 1

Vladimir Shevelev, Oct 27 2009

nonn

The first differences are 16, 20, 42, etc. They are either 2 or in A075369 or in A008864, see A167054.

A proof follows from Clement's criterion of twin primes.

			a(2) = 1 + 3 + gcd(3*5, 4*(2! + 1) + 3) = 19.

E. Trost, Primzahlen, Birkhäuser-Verlag, 1953, pages 30-31.

Amiram Eldar, Table of n, a(n) for n = 1..206
P. A. Clement, Congruences for sets of primes, Amer. Math. Monthly, 56 (1949), 23-25.

Cf. A073829, A008864, A167054.

Cf. A166944, A166945, A116533, A163961, A163963, A084662, A084663, A106108, A132199, A134162, A135506, A135508, A118679, A120293.

A073829 := proc(n) n+4*((n-1)!+1) ; end proc:
A167053 := proc(n) option remember ; local aprev; if n = 1 then 3; else aprev := procname(n-1) ; 1+aprev+gcd(aprev*(aprev+2),A073829(aprev)) ; end if; end proc:
seq(A167053(n),n=1..60) ; # R. J. Mathar, Dec 17 2009

A073829[n_] := 4((n-1)! + 1) + n;
a[1] = 3;
a[n_] := a[n] = 1 + a[n-1] + GCD[a[n-1] (a[n-1] + 2), A073829[a[n-1]]];
Array[a, 60] (* Jean-François Alcover, Mar 25 2020 *)

Definition shortened and values from a(4) on replaced by R. J. Mathar, Dec 17 2009

A134734 First differences of A084662.

Original entry on oeis.org

2, 3, 1, 5, 3, 1, 1, 1, 1, 11, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 23, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 47, 3, 1, 5, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 101, 3, 1, 1

Offset: 1

N. J. A. Sloane, Jan 28 2008

nonn

Eric S. Rowland, A simple prime-generating recurrence, Abstracts Amer. Math. Soc., 29 (No. 1, 2008), p. 50 (Abstract 1035-11-986).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric S. Rowland, A simple prime-generating recurrence, arXiv:0710.3217 [math.NT], 2007-2008.

See A106108 for other cross-references.

Cf. A132199, A134743, A134744.

a134734 n = a134734_list !! (n-1)
a134734_list = zipWith (-) (tail a084662_list) a084662_list
-- Reinhard Zumkeller, Nov 15 2013

b[1] = 4; b[n_] := b[n] = b[n-1] + GCD[n, b[n-1]];
Table[b[n], {n, 104}] // Differences (* Jean-François Alcover, Sep 28 2018 *)

a(n) = A132199(n), n > 2. - R. J. Mathar, Mar 28 2012

A134744 First differences of A084663.

Original entry on oeis.org

2, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 13, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 29, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 59, 3, 1, 1, 7, 1, 5, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

N. J. A. Sloane, Jan 28 2008

nonn

Eric S. Rowland, A simple prime-generating recurrence, Abstracts Amer. Math. Soc., 29 (No. 1, 2008), p. 50 (Abstract 1035-11-986).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric S. Rowland, A simple prime-generating recurrence, arXiv:0710.3217 [math.NT], 2007-2008.

Cf. A084663. See A106108 for other cross-references.

Cf. A132199, A134734, A134743.

a134744 n = a134744_list !! (n-1)
a134744_list = zipWith (-) (tail a084663_list) a084663_list
-- Reinhard Zumkeller, Nov 15 2013

b[1] = 8; b[n_] := b[n] = b[n-1] + GCD[n, b[n-1]];
Table[b[n], {n, 105}] // Differences (* Jean-François Alcover, Sep 28 2018 *)

A167054 Values of A167053(k)-A167053(k-1)-1 not equal to 1.

Original entry on oeis.org

15, 19, 41, 83, 167, 337, 673, 1361, 2729, 5471, 10949, 21911, 43853, 87719, 175447, 350899, 701819, 1403641, 2807303, 5614657, 11229331, 22458671, 44917381, 89834777, 179669557, 359339171, 718678369

Offset: 1

Vladimir Shevelev, Oct 27 2009

All terms of the sequence are primes or products of twin primes (A037074).

Cf. A167053, A166944, A166945, A116533, A163961, A163963, A084662, A084663, A106108, A132199, A134162, A135506, A135508, A118679, A120293

Values from a(3) on replaced by R. J. Mathar, Dec 17 2009

More terms from Amiram Eldar, Sep 13 2019

A134743 First differences of A134736.

Original entry on oeis.org

1, 3, 1, 5, 3, 1, 1, 1, 1, 11, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 23, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 47, 3, 1, 5, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 101, 3, 1, 1

Offset: 1

N. J. A. Sloane, Jan 28 2008

nonn

Eric S. Rowland, A simple prime-generating recurrence, Abstracts Amer. Math. Soc., 29 (No. 1, 2008), p. 50 (Abstract 1035-11-986).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric S. Rowland, A natural prime-generating recurrence, arXiv:0710.3217 [math.NT], 2007-2008.

See A106108 for other cross-references.

Cf. A134734, A134744.

a134743 n = a134743_list !! (n-1)
a134743_list = zipWith (-) (tail a134736_list) a134736_list
-- Reinhard Zumkeller, Nov 15 2013

b[1] = 5; b[n_] := b[n] = b[n-1] + GCD[n, b[n-1]];
Array[b, 104] // Differences (* Jean-François Alcover, Oct 01 2018 *)

a(n) = A132199(n), n>2. [R. J. Mathar, Dec 13 2008]

A167170 a(6) = 14, for n >= 7, a(n) = a(n-1) + gcd(n, a(n-1)).

Original entry on oeis.org

14, 21, 22, 23, 24, 25, 26, 39, 40, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 87, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 177, 180, 181, 182, 189, 190, 195

Offset: 6

Vladimir Shevelev, Oct 29 2009, Nov 06 2009

nonn

For every n >= 7, a(n) - a(n-1) is 1 or prime. This Rowland-like "generator of primes" is different from A106108 (see comment to A167168).

G. C. Greubel, Table of n, a(n) for n = 6..1000
Eric S. Rowland, A natural prime-generating recurrence, J. 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.

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

A167170 := proc(n) option remember; if n = 6 then 14; else procname(n-1)+igcd(n,procname(n-1)) ; end if; end proc: seq(A167170(i),i=6..80) ; # R. J. Mathar, Oct 30 2010

RecurrenceTable[{a[n] == a[n - 1] + GCD[n, a[n - 1]], a[6] == 14}, a, {n, 6, 100}] (* G. C. Greubel, Jun 04 2016 *)
nxt[{n_,a_}]:={n+1,a+GCD[a,n+1]}; NestList[nxt,{6,14},60][[All,2]] (* Harvey P. Dale, Nov 03 2019 *)

first(n)=my(v=vector(n-5)); v[1]=14; for(k=7,n, v[k-5]=v[k-6]+gcd(k,v[k-6])); v \\ Charles R Greathouse IV, Aug 22 2017

Terms > 91 from R. J. Mathar, Oct 30 2010

A167195 a(2)=3, for n>=3, a(n)=a(n-1)+gcd(n, a(n-1)).

Original entry on oeis.org

3, 6, 8, 9, 12, 13, 14, 15, 20, 21, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 44, 45, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 92, 93, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116

Offset: 2

Vladimir Shevelev, Oct 30 2009, Nov 06 2009

For every n>=3, a(n)-a(n-1) is 1 or prime. This Rowland-like "generator of primes" is different from A106108 and from generators A167168. Generalization: Let p be a prime. Let N(p-1)=p and for n>=p, N(n)=N(n-1)+gcd(n, N(n-1)). Then, for every n>=p, N(n)-N(n-1) is 1 or prime.

G. C. Greubel, Table of n, a(n) for n = 2..1000
E. S. Rowland, A natural prime-generating recurrence, Journal of Integer Sequences, 11 (2008), Article 08.2.8.
Vladimir Shevelev, A new generator of primes based on the Rowland idea, arXiv:0910.4676 [math.NT], 2009.

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

RecurrenceTable[{a[n] == a[n - 1] + GCD[n, a[n - 1]], a[2] == 3}, a, {n, 2, 100}] (* G. C. Greubel, Jun 05 2016 *)

a(n) = a(n-1) + 1 if gcd(a(n-1), n) = 1, or a(n) = 2*n otherwise. - Yifan Xie, Aug 20 2025

Edited by Charles R Greathouse IV, Nov 02 2009

A167495 Records in A167494.

Original entry on oeis.org

2, 3, 5, 13, 31, 61, 139, 283, 571, 1153, 2311, 4651, 9343, 19141, 38569, 77419, 154873, 310231, 621631, 1243483, 2486971, 4974721

Offset: 1

Vladimir Shevelev, Nov 05 2009

Conjecture: each term > 3 of the sequence is the greater member of a twin prime pair (A006512).
Indices of the records are 1, 2, 4, 6, 9, 10, 15, 18, 21, 25, 28, 30, 38, 72, 90, ... [R. J. Mathar, Nov 05 2009]
One can formulate a similar conjecture without verification of the primality of the terms (see Conjecture 4 in my paper). [Vladimir Shevelev, Nov 13 2009]

E. S. Rowland, A natural prime-generating recurrence, Journal of Integer Sequences, Vol.11 (2008), Article 08.2.8.
E. S. Rowland, A natural prime-generating recurrence, arXiv:0710.3217 [math.NT], 2007-2008.
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. [_Vladimir Shevelev_, Dec 03 2009]

Cf. A167494, 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]]};
A167494 = DeleteCases[Differences[Transpose[NestList[nxt, {1, 2}, 10^7]][[2]]], 1];
Tally[A167494][[All, 1]] //. {a1___, a2_, a3___, a4_, a5___} /; a4 <= a2 :> {a1, a2, a3, a5} (* Jean-François Alcover, Oct 29 2018, using Harvey P. Dale's code for A167494 *)

Simplified the definition to include all records; one term added by R. J. Mathar, Nov 05 2009
a(16) to a(21) from R. J. Mathar, Nov 19 2009
a(22) from Jean-François Alcover, Oct 29 2018

A167197 a(6) = 7, for n >= 7, a(n) = a(n - 1) + gcd(n, a(n - 1)).

Original entry on oeis.org

7, 14, 16, 17, 18, 19, 20, 21, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 52, 53, 54, 55, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 116, 117, 120, 121, 122, 123, 124, 125, 126, 127, 128

Offset: 6

Vladimir Shevelev, Oct 30 2009, Nov 06 2009

nonn

For every n >= 7, a(n) - a(n - 1) is 1 or prime. This Rowland-like "generator of primes" is different from A106108 (see comment to A167168) and from A167170. Note that, lim sup a(n) / n = 2, while lim sup A106108(n) / n = lim sup A167170(n) / n = 3.

Going up to a million, differences of two consecutive terms of this sequence gives primes about 0.009% of the time. The rest are 1's. [Alonso del Arte, Nov 30 2009]

G. C. Greubel, Table of n, a(n) for n = 6..1000
E. S. Rowland, A natural prime-generating recurrence, Journal of Integer Sequences, 11 (2008), Article 08.2.8.
Vladimir Shevelev, An infinite set of generators of primes based on the Rowland idea and conjectures concerning twin primes, arXiv:0910.4676 [math.NT], 2009.

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

A[6]:= 7:
for n from 7 to 100 do A[n]:= A[n-1] + igcd(n,A[n-1]) od:
seq(A[i],i=6..100); # Robert Israel, Jun 05 2016

a[6] = 7; a[n_ /; n > 6] := a[n] = a[n - 1] + GCD[n, a[n - 1]]; Table[a[n], {n, 6, 58}]

from math import gcd
def aupton(nn):
    alst = [7]
    for n in range(7, nn+1): alst.append(alst[-1] + gcd(n, alst[-1]))
    return alst
print(aupton(68)) # Michael S. Branicky, Jul 14 2021

Verified and edited by Alonso del Arte, Nov 30 2009

A167493 a(1) = 2; thereafter a(n) = a(n-1) + gcd(n, a(n-1)) if n is odd, and a(n) = a(n-1) + gcd(n-2, a(n-1)) if n is even.

Original entry on oeis.org

2, 4, 5, 6, 7, 8, 9, 12, 15, 16, 17, 18, 19, 20, 25, 26, 27, 28, 29, 30, 33, 34, 35, 36, 37, 38, 39, 52, 53, 54, 55, 60, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 124, 125, 126

Offset: 1

Vladimir Shevelev, Nov 05 2009

nonn

Conjectures. 1) For n >= 2, every difference a(n) - a(n-1) is 1 or prime; 2) Every record of differences a(n) - a(n-1) greater than 3 belongs to the sequence of the greater of twin primes (A006512).
Conjecture #1 above fails at n = 620757, with a(n) = 1241487 and a(n-1) = 1241460, difference = 27. Additionally, the terms of related A167495(m) quickly tend to index n/2. So for example, A167495(14) = 19141 is seen at n = 38284. - Bill McEachen, Jan 20 2023
It seems that, for n > 4, (3*n-3)/2 <= a(n) <= 2n - 3. Can anyone find a proof or disproof? - Charles R Greathouse IV, Jan 22 2023

Bill McEachen, Table of n, a(n) for n = 1..100000
E. S. Rowland, A natural prime-generating recurrence, Journal of Integer Sequences, Vol.11 (2008), Article 08.2.8.
Vladimir Shevelev, A new generator of primes based on the Rowland idea, arXiv:0910.4676 [math.NT], 2009.
Vladimir Shevelev, Three theorems on twin primes, arXiv:0911.5478 [math.NT], 2009-2010.

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

Cf also A006512, A167494.

nxt[{n_,a_}]:={n+1,If[EvenQ[n],a+GCD[n+1,a],a+GCD[n-1,a]]}; Transpose[ NestList[nxt,{1,2},70]][[2]] (* 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]));); va; \\ Michel Marcus, Dec 13 2018

from math import gcd
from itertools import count, islice
def agen(): # generator of terms
    an = 2
    for n in count(2):
        yield an
        an = an + gcd(n, an) if n&1 else an + gcd(n-2, an)
print(list(islice(agen(), 66))) # Michael S. Branicky, Jan 22 2023

For n > 3, n < a(n) < n*(n-1)/2. - Charles R Greathouse IV, Jan 22 2023

More terms from Harvey P. Dale, Dec 05 2015

cp's OEIS Frontend

A167053 a(1)=3; for n > 1, a(n) = 1 + a(n-1) + gcd( a(n-1)*(a(n-1)+2), A073829(a(n-1)) ).

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A134734 First differences of A084662.

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A134744 First differences of A084663.

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A167054 Values of A167053(k)-A167053(k-1)-1 not equal to 1.

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A134743 First differences of A134736.

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A167170 a(6) = 14, for n >= 7, a(n) = a(n-1) + gcd(n, a(n-1)).

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A167195 a(2)=3, for n>=3, a(n)=a(n-1)+gcd(n, a(n-1)).

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A167495 Records in A167494.

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