A118679 -id:A118679 - cp's OEIS frontend

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

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

A120292 Absolute value of numerator of determinant of n X n matrix with elements M[i,j] = prime(i)/(1+prime(i)) if i=j and 1 otherwise.

Original entry on oeis.org

2, 1, 1, 5, 1, 23, 1, 1, 1, 23, 17, 13, 5, 1, 1, 1, 1, 37, 293, 47, 61, 29, 1, 29, 271, 593, 43, 233, 29, 811, 1, 941, 101, 1, 1, 1231, 131, 29, 1, 521, 1, 109, 1, 149, 509, 89, 59, 107, 617, 1, 1, 47, 173, 3067, 47, 1, 3463, 3599, 89, 431, 4021, 521, 2161, 2239, 103, 1, 1

Offset: 1

Alexander Adamchuk, Jul 08 2006, Jul 04 2008

Some a(n) are equal to 1 (n = 2, 3, 5, 7, 8, 9, 14, 15, 16, 17, 23, 31, 34, 35, 39, 41, 43, 50, 51, 56, ...).
a(58) = 3599 = 59*61 is not prime. - T. D. Noe, Nov 15 2006
Most terms are prime or 1.
Numbers n such that a(n)>1 and is not prime are listed in A141779(n) = {58, 282, 367, 743, 808, 1015, 1141, 1299, 1962, 2109, 2179, 2397, 2501, ...}.
Composite terms are listed in A141781 = {3599, 118477, 210589, 971573, 1164103, 1901959, 2446681, 3230069, ...}.
Note that all listed terms of A141781 are semiprime, for example: 3599 = 59*61, 118477 = 257*461, 210589 = 251*839, 971573 = 643*1511.
Conjecture: All composite terms are semiprime.

Alexander Adamchuk, Jul 04 2008, Table of n, a(n) for n = 1..282

Cf. A024528, A118679, A125716, A141780, A141779, A141781.

Abs[Numerator[Table[Det[DiagonalMatrix[Table[Prime[i]/(Prime[i]+1)-1,{i,1,n}]]+1],{n,1,60}]]]
Table[Numerator[Abs[(1 - Sum[Prime[k] + 1,{k, 1, n}])/Product[Prime[k] + 1, {k, 1, n}] ]],{n,1,282}]

a(n)=abs(numerator(matdet(matrix(n,n,i,j,if(i==j,prime(i)/(1+prime(i)),1))))) \\ Charles R Greathouse IV, Feb 07 2013

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

A167494 List of first differences of A167493 that are different from 1.

Original entry on oeis.org

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

Vladimir Shevelev, Nov 05 2009

nonn

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

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]

Cf. 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]]}; DeleteCases[ Differences[ Transpose[NestList[nxt,{1,2},20000]][[2]]],1] (* 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]));); select(x->(x!=1), vector(nn-1, n, va[n+1] - va[n]));} \\ Michel Marcus, Dec 13 2018

A127853 Numbers n such that A118680(n) = 1.

Original entry on oeis.org

10, 17, 26, 36, 37, 45, 50, 59, 61, 65, 67, 78, 82, 90, 91, 94, 101, 102, 105, 108, 110, 122, 136, 138, 145, 147, 149, 153, 155, 165, 170, 173, 181, 183, 188, 189, 193, 197, 210, 213, 220, 224, 226, 231, 232, 239, 249, 250, 257, 262, 263, 266, 268, 276, 279

Offset: 1

Alexander Adamchuk, Feb 03 2007

nonn

Also a(n) are the numbers n such that 1 + Sum[ k, {k,1,n} ] = 1 + n(n+1)/2 divides Product[ k, {k,1,n} ] = n!. A118680[ a(n) ] = 1, where A118680(n) = {2, 2, 7, 11, 2, 11, 29, 37, 23, 1, 67, 79, 23, 53, 11, 137, 1, ...} = Absolute value of numerator of determinant of n X n matrix with M(i,j) = (i+1)/i if i=j otherwise 1. A118680(n) = Numerator[ (1 + n(n+1)/2) / n! ].

Cf. A118680, A118679, A127852.

Select[Range[1000],Numerator[(1 + #(#+1)/2)/#! ]==1&]

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

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

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Python

Extensions

A120292 Absolute value of numerator of determinant of n X n matrix with elements M[i,j] = prime(i)/(1+prime(i)) if i=j and 1 otherwise.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A167494 List of first differences of A167493 that are different from 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A127853 Numbers n such that A118680(n) = 1.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions