A167495 -id:A167495 - cp's OEIS frontend

A174216 a(1)=15; for n>1, a(n) = the smallest number k >a(n-1) such that 2A174214(k)= 3(k-1).

15, 27, 63, 123, 279, 567, 1143, 2307, 4623, 9447, 18927, 38283, 77139, 154839, 309747, 620463, 1241823, 2483847, 4967739, 9935607, 19892547, 39785199

Offset: 1

Vladimir Shevelev, Mar 12 2010

Theorem: If the sequence is infinite, then there exist infinitely many twin primes.

Conjecture. a(n+1)/a(n) tends to 2.

V. Shevelev, Theorems on twin primes-dual case, arXiv:0912.4006 [math.GM], 2009-2014.

Cf. A174214, A174215, A166945, A167495.

A174216 := proc(n) option remember ; if n =1 then 15 ; else for k from procname(n-1)+1 do if 2*A173214(k) = 3*(k-1) then return k; end if; end do ; end if; end proc: # R. J. Mathar, Mar 16 2010

(* b = A174214 *) b[n_] := b[n] = Which[n==9, 14, CoprimeQ[b[n-1], n-1- (-1)^n], b[n-1]+1, True, 2n-4]; a[n_] := a[n] = If[n==1, 15, For[k = a[n- 1]+1, True, k++, If[2b[k] == 3(k-1), Return[k]]]]; Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 22}] (* Jean-François Alcover, Feb 02 2016 *)

Terms from a(11) on corrected by R. J. Mathar, Mar 16 2010
I corrected the terms beginning with a(11) and added some new terms. - Vladimir Shevelev, Mar 27 2010
Terms from a(11) onwards were corrected according to independent calculations by R. Mathar, M. Alekseyev, M. Hasler and A. Heinz (SeqFan lists 30 Oct and 1 Nov 2010). - Vladimir Shevelev, Nov 02 2010

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

A174214 a(n) = a(n-1)+1, if the previous term a(n-1) and n-1-(-1)^n are coprime, else a(n)=2*n-4.

Original entry on oeis.org

14, 16, 17, 18, 19, 20, 26, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 52, 53, 54, 55, 60, 62, 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, 127, 128, 129, 130, 131, 132, 133, 134

Offset: 9

Vladimir Shevelev, Mar 12 2010

V. Shevelev, Theorems on twin primes-dual case, arXiv:0912.4006 [math.GM], 2009-2014.

Cf. A166945, A167495.

A174214 := proc(n) option remember ; if n = 9 then 14 ; elif gcd(procname(n-1),n-1-(-1)^n) = 1 then procname(n-1)+1 ; else 2*n-4 ; end if; end proc:
seq(A174214(n),n=9..100) ; # R. J. Mathar, Mar 16 2010

a[n_] := a[n] = Which[n==9, 14, CoprimeQ[a[n-1], n-1-(-1)^n], a[n-1]+1, True, 2n-4]; Table[a[n], {n, 9, 100}] (* Jean-François Alcover, Feb 02 2016 *)

a(15) corrected and sequence extended by R. J. Mathar, Mar 16 2010

a(15) corrected and a(35)-a(74) added by John W. Layman, Mar 16 2010

A174217 a(n) = (A174216(n)-1)/2.

Original entry on oeis.org

7, 13, 31, 61, 139, 283, 571, 1153, 2311, 4723, 9463, 19141, 38569, 77419, 154873, 310231, 620911, 1241923, 2483869, 4967803, 9946273, 19892599

Offset: 1

Vladimir Shevelev, Mar 12 2010

Related to the generation of twin primes according to section 6 of the preprint.

V. Shevelev, Theorems on twin primes-dual case, arXiv:0912.4006 [math.GM], 2009-2014.

Cf. A174214, A174215, A174216, A166945, A167495.

(* b = A174214 *) b[n_] := b[n] = Which[n == 9, 14, CoprimeQ[b[n - 1], n - 1 - (-1)^n], b[n - 1] + 1, True, 2 n - 4];
(* c = A174216 *) c[n_] := c[n] = If[n == 1, 15, For[k = c[n - 1] + 1, True, k++, If[2 b[k] == 3 (k - 1), Return[k]]]];
Table[a[n] = (c[n] - 1)/2; Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 22}] (* Jean-François Alcover, Jan 29 2019 *)

A174214(A174216(n)) = 3*a(n), n>1.

Terms after a(11) corrected by Vladimir Shevelev, Nov 02 2010

A174215 First differences of A174214.

Original entry on oeis.org

2, 1, 1, 1, 1, 6, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 5, 2, 2, 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, 31, 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, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 9

Vladimir Shevelev, Mar 12 2010

If a(n) is odd, then it is 1 or prime; if a(n) is even, then 2+a(n)/2 is prime.

V. Shevelev, Theorems on twin primes-dual case, arXiv:0912.4006

Cf. A167495, A166945, A174214

a(n) = A174214(n+1)-A174214(n).

Terms corrected, using the Mathar-Layman corrections of A174214, by Vladimir Shevelev, Mar 26 2010

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

A174453 a(n) is the smallest k >= 1 for which gcd(m + (-1)^m, m + n - 4) > 1, where m = n + k - 1.

Original entry on oeis.org

1, 2, 1, 1, 1, 5, 1, 2, 1, 1, 1, 12, 1, 2, 1, 1, 1, 18, 1, 2, 1, 1, 1, 4, 1, 2, 1, 1, 1, 30, 1, 2, 1, 1, 1, 5, 1, 2, 1, 1, 1, 42, 1, 2, 1, 1, 1, 6, 1, 2, 1, 1, 1, 4, 1, 2, 1, 1, 1, 60, 1, 2, 1, 1, 1, 5, 1, 2, 1, 1, 1, 72, 1, 2, 1, 1, 1, 9, 1, 2, 1, 1, 1, 4, 1, 2, 1, 1, 1, 6, 1, 2, 1, 1, 1, 5, 1, 2, 1, 1, 1, 102

Offset: 5

Vladimir Shevelev, Mar 20 2010

nonn

If a(n) > sqrt(n), then n-3 is the larger of twin primes. In these cases we have a(10)=5 and, for n > 10, a(n) = n-4. For odd n and for n == 2 (mod 6), a(n)=1; for n == 0 (mod 6), a(n)=2; for {n == 4 (mod 6)} & {n == 8 (mod 10)}, a(n)=4, etc. The problem is to develop this sieve for the excluding n for which a(n) <= sqrt(n) and to obtain nontrivial lower estimates for the counting function of the larger of twin primes.

Paul Tek, Table of n, a(n) for n = 5..10000
V. Shevelev, Theorems on twin primes-dual case, arXiv:0912.4006 [math.GM], 2009-2014.

Cf. A173980, A020639, A173978, A173977, A173979, A174217, A174216, A174214, A174215, A166945, A167495.

A174453 := proc(n) local k,m ; for k from 1 do m := n+k-1 ; if igcd(m+(-1)^m,m+n-4) > 1 then return k; end if; end do: end proc: seq(A174453(n),n=5..120); # R. J. Mathar, Nov 04 2010

a[n_] := For[k=1, True, k++, m=n+k-1; If[GCD[m+(-1)^m, m+n-4]>1, Return[k]] ];
Table[a[n], {n, 5, 106}] (* Jean-François Alcover, Nov 29 2017 *)

Terms beyond a(34) from R. J. Mathar, Nov 04 2010

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

cp's OEIS Frontend

A174216 a(1)=15; for n>1, a(n) = the smallest number k >a(n-1) such that 2*A174214(k)= 3*(k-1).

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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.

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A174214 a(n) = a(n-1)+1, if the previous term a(n-1) and n-1-(-1)^n are coprime, else a(n)=2*n-4.

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A174217 a(n) = (A174216(n)-1)/2.

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A174215 First differences of A174214.

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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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A174453 a(n) is the smallest k >= 1 for which gcd(m + (-1)^m, m + n - 4) > 1, where m = n + k - 1.

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

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A174216 a(1)=15; for n>1, a(n) = the smallest number k >a(n-1) such that 2A174214(k)= 3(k-1).