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Search: a163963 -id:a163963
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a(1) = 2; a(n) is smallest prime > 2*a(n-1).
+10
34
2, 5, 11, 23, 47, 97, 197, 397, 797, 1597, 3203, 6421, 12853, 25717, 51437, 102877, 205759, 411527, 823117, 1646237, 3292489, 6584983, 13169977, 26339969, 52679969, 105359939, 210719881, 421439783, 842879579, 1685759167, 3371518343
OFFSET
1,1
COMMENTS
It appears that lim_{n->infinity} a(n)/2^n exists and is approximately 1.569985585.... - Franklin T. Adams-Watters, Nov 11 2011
This is a B_2 sequence. - Thomas Ordowski, Sep 23 2014 See the link.
Conjecture: lim_{n->infinity} a(n)/A006992(n) = 5.1648264... - Thomas Ordowski, Apr 05 2015
LINKS
Zak Seidov and Michael De Vlieger, Table of n, a(n) for n = 1..1000 (First 100 terms from Zak Seidov)
Eric Weisstein's World of Mathematics, B_2-Sequence.
FORMULA
a(n+1) = A060264(a(n)). - Peter Munn, Oct 23 2017
MAPLE
A055496 := proc(n) option remember; if n=1 then 2 else nextprime(2*A055496(n-1)); fi; end;
MATHEMATICA
NextPrim[n_Integer] := Block[ {k = n + 1}, While[ !PrimeQ[k], k++ ]; Return[k]]; a[1] = 2; a[n_] := NextPrim[ 2*a[n - 1]]; Table[ a[n], {n, 1, 31} ]
a[1]=2; a[n_]:=a[n]=Prime[PrimePi[2*a[n-1]]+1]; Table[a[n], {n, 40}] (* Zak Seidov, Feb 16 2006 *)
NestList[ NextPrime[2*# ]&, 2, 100] (* Zak Seidov, Jul 28 2009 *)
PROG
(PARI) print1(a=2); for(n=2, 20, print1(", ", a=nextprime(a+a))) \\ Charles R Greathouse IV, Jul 19 2011
CROSSREFS
Values of a(n)-2*a(n-1) in A163469. - Zak Seidov, Jul 28 2009
Cf. A065545 (with a(1)=3). - Zak Seidov, Feb 04 2016
Row 1 of A229608.
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jul 07 2000
EXTENSIONS
Mathematica updated by Jean-François Alcover, Jun 19 2013
STATUS
approved
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
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
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 *)
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
Sequence of prime gaps which characterize Rowland sequences of prime-generating recurrences.
+10
11
3, 7, 17, 19, 31, 43, 53, 67, 71, 79, 97, 103, 109, 113, 127, 137, 151, 163, 173, 181, 191, 197, 199, 211, 229, 239, 241, 251, 257, 269, 271, 283, 293, 317, 331, 337, 349, 367, 373
OFFSET
1,1
COMMENTS
Consider the Rowland sequences with recurrence N(n)= N(n-1)+gcd(n,N(n-1)).
For some of these, like the prototypical A106108, the first differences N(n)-N(n-1) are always 1 or primes.
If for some position p (a prime) N(p-1)=2*p, then the arXiv preprint shows that N is indeed in that class of prime-generating sequences.
Since then N(p)=N(p-1)+p, the prime p characterizes at the same time the gap (first difference) and location in the sequence.
In the same sequence at some larger value of p, we may again have N(p-1)=2*p. In these cases, we put all these p's satisfying that equation into a generator class.
For each of the generator classes, the OEIS sequence shows the smallest member (prime) in that class. So this is a trace of how many essentially different sequences with this N(p-1)=2*p property exist.
LINKS
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.
EXAMPLE
We put a(1)=3 since the N-sequence 4, 6, 9, 10, 15, 18, 19, 20.. = A084662 (essentially the same as A106108) has a first difference of p=3 at position p-1=2, N(2)=2*3.
It has a first difference of p=5 at p-1=4, a first difference of p=11 at p=10, so we put {3,5,11,23,..} into that class. This leaves p=7=a(2) as the lowest prime to be covered by the next class. This is first realized by N = 8, 10, 11, 12, 13, 14, 21, 22, 23, 24, 25, 26, 39.. = A084663. Here N(12)=2*13, so p=13 is in the same class as p=7, namely {7,13,29,59,131,..}. This leaves p=17=a(3) to be the smallest member in a new class, namely {17,41,83,167,..}.
KEYWORD
nonn
AUTHOR
Vladimir Shevelev, Oct 29 2009
EXTENSIONS
Edited, a(1) set to 3, 37 replaced by 31, and extended beyond 53 by R. J. Mathar, Dec 17 2009
STATUS
approved
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
OFFSET
1,1
COMMENTS
All terms of the sequence are primes or products of twin primes (A037074).
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
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
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
Eric S. Rowland, A natural prime-generating recurrence, J. of Integer Sequences 11 (2008), Article 08.2.8.
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
KEYWORD
nonn
AUTHOR
Vladimir Shevelev, Oct 29 2009, Nov 06 2009
EXTENSIONS
Terms > 91 from R. J. Mathar, Oct 30 2010
STATUS
approved
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
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
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 *)
KEYWORD
nonn,easy
AUTHOR
Vladimir Shevelev, Oct 30 2009, Nov 06 2009
EXTENSIONS
Edited by Charles R Greathouse IV, Nov 02 2009
STATUS
approved
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
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
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 *)
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
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
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
E. S. Rowland, A natural prime-generating recurrence, Journal of Integer Sequences, 11 (2008), Article 08.2.8.
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
KEYWORD
nonn
AUTHOR
Vladimir Shevelev, Oct 30 2009, Nov 06 2009
EXTENSIONS
Verified and edited by Alonso del Arte, Nov 30 2009
STATUS
approved
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
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
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
KEYWORD
nonn
AUTHOR
Vladimir Shevelev, Nov 05 2009
EXTENSIONS
More terms from Harvey P. Dale, Dec 05 2015
STATUS
approved
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
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
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
KEYWORD
nonn
AUTHOR
Vladimir Shevelev, Nov 05 2009
STATUS
approved

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