A163961 -id:A163961

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A006992

Bertrand primes: a(n) is largest prime < 2*a(n-1) for n > 1, with a(1) = 2.
(Formerly M0675)

+10
32

2, 3, 5, 7, 13, 23, 43, 83, 163, 317, 631, 1259, 2503, 5003, 9973, 19937, 39869, 79699, 159389, 318751, 637499, 1274989, 2549951, 5099893, 10199767, 20399531, 40799041, 81598067, 163196129, 326392249, 652784471, 1305568919, 2611137817

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

a(n) < a(n+1) by Bertrand's postulate (Chebyshev's theorem). - Jonathan Sondow, May 31 2014

Let b(n) = 2^n - a(n). Then b(n) >= 2^(n-1) - 1 and b(n) is a B_2 sequence: 0, 1, 3, 9, 19, 41, 85, 173, 349, ... - Thomas Ordowski, Sep 23 2014 See the link for B_2 sequence.

These primes can be obtained of exclusive form using a restricted variant of Rowland's prime-generating recurrence (A106108), making gcd(n, a(n-1)) = -1 when GCDs are greater than 1 and less than n (see program). These GCDs are also a divisor of each odd number from a(n) + 2 to 2*a(n-1) - 1 in reverse order, so that this subtraction with -1's invariably leads to the prime. - Manuel Valdivia, Jan 13 2015

First row of array in A229607. - Robert Israel, Mar 31 2015

Named after the French mathematician Joseph Bertrand (1822-1900). - Amiram Eldar, Jun 10 2021

REFERENCES

Martin Aigner and Günter M. Ziegler, Proofs from The Book, Springer-Verlag, Berlin, 1999; see p. 7.

Martin Griffiths, The Backbone of Pascal's Triangle, United Kingdom Mathematics Trust (2008), page 115. [From Martin Griffiths, Mar 28 2009]

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 344.

Ivan Niven and Herbert S. Zuckerman, An Introduction to the Theory of Numbers. 2nd ed., Wiley, NY, 1966, p. 189.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Robert G. Wilson v, Table of n, a(n) for n = 1..1001 (first 100 terms from T. D. Noe)

Paul Erdős, Beweis eines Satzes von Tschebyschef (in German), Acta Litt. Sci. Szeged, Vol. 5 (1932), pp. 194-198.

Paul Erdős, A theorem of Sylvester and Schur, J. London Math. Soc., Vol. 9 (1934), pp. 282-288.

Srinivasa Ramanujan, A proof of Bertrand's postulate, J. Indian Math. Soc., Vol. 11 (1919), pp. 181-182.

Vladimir Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq., Vol. 15 (2012) Article 12.5.4.

Jonathan Sondow, Ramanujan primes and Bertrand's postulate, arXiv:0907.5232 [math.NT], 2009-2010.

Jonathan Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly, Vol. 116, No. 7 (2009), pp. 630-635.

Jonathan Sondow and Eric Weisstein, MathWorld: Bertrand's Postulate.

Eric Weisstein's World of Mathematics, B2 Sequence.

Robert G. Wilson, V, Letter to N. J. A. Sloane, Oct. 1993.

FORMULA

a(n+1) = A007917(2*a(n)). - Reinhard Zumkeller, Sep 17 2014

Limit_{n -> infinity} a(n)/2^n = 0.303976447924... - Thomas Ordowski, Apr 05 2015

MAPLE

A006992 := proc(n) option remember; if n=1 then 2 else prevprime(2*A006992(n-1)); fi; end;

MATHEMATICA

bertrandPrime[1] = 2; bertrandPrime[n_] := NextPrime[ 2*a[n - 1], -1]; Table[bertrandPrime[n], {n, 40}]

(* Second program: *)

NestList[NextPrime[2#, -1] &, 2, 40] (* Harvey P. Dale, May 21 2012 *)

k = 3; a[n_] := If[GCD[n, k] > 1 && GCD[n, k] < n, -1, GCD[n, k]]; Select[Differences@Table[k = a[n] + k, {n, 2611137817}], # > 1 &] (* Manuel Valdivia, Jan 13 2015 *)

PROG

(PARI) print1(t=2); for(i=2, 60, print1(", "t=precprime(2*t))) \\ Charles R Greathouse IV, Apr 01 2013

(Haskell)

a006992 n = a006992_list !! (n-1)

a006992_list = iterate (a007917 . (* 2)) 2

-- Reinhard Zumkeller, Sep 17 2014

(Python)

from sympy import prevprime

l = [2]

for i in range(1, 51):

l.append(prevprime(2 * l[i - 1]))

print(l) # Indranil Ghosh, Apr 26 2017

CROSSREFS

Cf. A055496, A163961.

See A185231 for another version.

Cf. A007917, A229607, A295262.

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane, Robert G. Wilson v

EXTENSIONS

Definition completed by Jonathan Sondow, May 31 2014

B_2 sequence link added by Wolfdieter Lang, Oct 09 2014

STATUS

approved

A163963

First differences of A080735.

+10
13

1, 2, 1, 5, 1, 11, 1, 23, 1, 47, 1, 1, 1, 97, 1, 1, 1, 197, 1, 1, 1, 397, 1, 1, 1, 797, 1, 1, 1, 1597, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3203, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6421, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12853, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 25717, 1, 1, 1, 51437, 1, 1, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Ignoring the 1 terms we obtain A055496. If we consider sequences A_i={a_i(n)}, i=1,2,... with the same constructions as A080735, but with initials a_1(1)=2, a_2(1)=3, a_3(1)=13,..., a_m(1)=A080359(m),..., then the union of A_1,A_2,... contains all primes.

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..5000

MAPLE

A080735 := proc(n) option remember; local p ; if n = 1 then 1; else p := procname(n-1) ; if isprime(p) then 2*p; else p+1 ; end if; end if; end proc: A163963 := proc(n) A080735(n+1)-A080735(n) ; end: seq(A163963(n), n=1..100) ; # R. J. Mathar, Nov 05 2009

MATHEMATICA

Differences@ NestList[If[PrimeQ@ #, 2 #, # + 1] &, 1, 87] (* Michael De Vlieger, Dec 06 2018, after Harvey P. Dale at A080735 *)

PROG

(PARI) lista(nn) = {my(va = vector(nn)); va[1] = 1; for (n=2, nn, va[n] = if (isprime(va[n-1]), 2*va[n-1], va[n-1]+1); ); vector(nn-1, n, va[n+1] - va[n]); } \\ Michel Marcus, Dec 06 2018

CROSSREFS

Cf. A116533, A163961, A080735, A006992, A055496, A080359, A104272, A106108, A132199.

KEYWORD

nonn

AUTHOR

Vladimir Shevelev, Aug 07 2009

EXTENSIONS

More terms from R. J. Mathar, Nov 05 2009

STATUS

approved

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)) ).

+10
11

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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.

REFERENCES

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

LINKS

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.

EXAMPLE

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

MAPLE

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

MATHEMATICA

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 *)

CROSSREFS

Cf. A073829, A008864, A167054.

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

KEYWORD

nonn

AUTHOR

Vladimir Shevelev, Oct 27 2009

EXTENSIONS

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

STATUS

approved

A167054

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

+10
9

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

Table of n, a(n) for n=1..27.

CROSSREFS

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

KEYWORD

nonn,more

AUTHOR

Vladimir Shevelev, Oct 27 2009

EXTENSIONS

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

More terms from Amiram Eldar, Sep 13 2019

STATUS

approved

A167170

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

+10
8

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

6,1

COMMENTS

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).

LINKS

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.

MAPLE

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

MATHEMATICA

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 *)

PROG

(PARI) 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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Vladimir Shevelev, Oct 29 2009, Nov 06 2009

EXTENSIONS

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

STATUS

approved

A167195

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

+10
8

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,1

COMMENTS

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.

LINKS

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.

V. Shevelev, A new generator of primes based on the Rowland idea, arXiv:0910.4676 [math.NT], 2009.

MATHEMATICA

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

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Vladimir Shevelev, Oct 30 2009, Nov 06 2009

EXTENSIONS

Edited by Charles R Greathouse IV, Nov 02 2009

STATUS

approved

A167495

Records in A167494.

+10
8

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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]

LINKS

Table of n, a(n) for n=1..22.

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]

MATHEMATICA

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 *)

CROSSREFS

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

KEYWORD

nonn,more

AUTHOR

Vladimir Shevelev, Nov 05 2009

EXTENSIONS

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

STATUS

approved

A167197

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

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

6,1

COMMENTS

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]

LINKS

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.

MAPLE

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

MATHEMATICA

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

PROG

(Python)

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Vladimir Shevelev, Oct 30 2009, Nov 06 2009

EXTENSIONS

Verified and edited by Alonso del Arte, Nov 30 2009

STATUS

approved

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.

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

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.

FORMULA

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

MATHEMATICA

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 *)

PROG

(PARI) 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

(Python)

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

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

Vladimir Shevelev, Nov 05 2009

EXTENSIONS

More terms from Harvey P. Dale, Dec 05 2015

STATUS

approved

A167494

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

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]

MATHEMATICA

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 *)

PROG

(PARI) 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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Vladimir Shevelev, Nov 05 2009

STATUS

approved

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