Displaying 1-10 of 12 results found.
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
+0
2
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
COMMENTS
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
MAPLE
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:
MATHEMATICA
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 *)
CROSSREFS
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(6) = 7, for n >= 7, a(n) = a(n - 1) + gcd(n, a(n - 1)).
+0
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
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]
MAPLE
A[6]:= 7:
for n from 7 to 100 do A[n]:= A[n-1] + igcd(n, A[n-1]) od:
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
CROSSREFS
Cf. A167195, A167170, A167168, A106108, A132199, A167054, A167053, A166944, A166945, A116533, A163961, A163963, A084662, A084663, A134162, A135506, A135508, A118679, A120293.
a(1)=3; for n > 1, a(n) = 1 + a(n-1) + gcd( a(n-1)*(a(n-1)+2), A073829(a(n-1)) ).
+0
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
COMMENTS
A proof follows from Clement's criterion of twin primes.
REFERENCES
E. Trost, Primzahlen, Birkhäuser-Verlag, 1953, pages 30-31.
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:
MATHEMATICA
a[1] = 3;
a[n_] := a[n] = 1 + a[n-1] + GCD[a[n-1] (a[n-1] + 2), A073829[a[n-1]]];
CROSSREFS
Cf. A166944, A166945, A116533, A163961, A163963, A084662, A084663, A106108, A132199, A134162, A135506, A135508, A118679, A120293.
EXTENSIONS
Definition shortened and values from a(4) on replaced by R. J. Mathar, Dec 17 2009
Bertrand primes: a(n) is largest prime < 2*a(n-1) for n > 1, with a(1) = 2.
(Formerly M0675)
+0
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
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 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).
FORMULA
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
(Haskell)
a006992 n = a006992_list !! (n-1)
a006992_list = iterate (a007917 . (* 2)) 2
(Python)
from sympy import prevprime
l = [2]
for i in range(1, 51):
l.append(prevprime(2 * l[i - 1]))
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.
+0
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
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
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)
CROSSREFS
Cf. A167197, A167195, A167170, A167168, A106108, A132199, A167054, A167053, A166944, A166945, A116533, A163961, A163963, A084662, A084663, A134162, A135506, A135508, A118679, A120293.
a(6) = 14, for n >= 7, a(n) = a(n-1) + gcd(n, a(n-1)).
+0
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
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).
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.
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
COMMENTS
All terms of the sequence are primes or products of twin primes ( A037074).
CROSSREFS
Cf. A167053, A166944, A166945, A116533, A163961, A163963, A084662, A084663, A106108, A132199, A134162, A135506, A135508, A118679, A120293
First differences of A168143 which are different from 1, incremented by 14.
+0
0
COMMENTS
All terms of the sequence are primes greater than 17.
Are there more than 5 terms?
CROSSREFS
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.
List of first differences of A167493 that are different from 1.
+0
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
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
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.
2, 3, 5, 13, 31, 61, 139, 283, 571, 1153, 2311, 4651, 9343, 19141, 38569, 77419, 154873, 310231, 621631, 1243483, 2486971, 4974721
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]
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.
EXTENSIONS
Simplified the definition to include all records; one term added by R. J. Mathar, Nov 05 2009
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