A261301 -id:A261301 - cp's OEIS frontend

A186253 Indices of zeros of the sequence u(n)=abs(u(n-1)-gcd(u(n-1),n-1)), u(1)=1.

2, 5, 11, 23, 47, 79, 157, 313, 619, 1237, 2473, 4909, 9817, 19603, 39199, 78193, 156019, 311347, 622669, 1244149, 2487739, 4975111, 9950221, 19900399, 39800797, 79601461, 159202369, 318404629, 636788881, 1273577761, 2547155419, 5094310069, 10188620041

Offset: 1

Benoit Cloitre, Feb 16 2011

nonn

For any fixed integer m>=1 define u(1)=1 and u(n)=abs(u(n-1)-gcd(u(n-1),m*n-1)). Then (b_m(k))_{k>=1} is the sequence of integers such that u(b_m(k))=0 and we conjecture that for k large enough m*b_m(k)+m-1 is a prime number. Here for m=1 it appears a(n) is prime for n>=1.
See A261301 for the sequence u relevant here (m=1). - M. F. Hasler, Aug 14 2015
A261301(a(n)-1) = 1; A261301(a(n)) = 0; A261301(a(n)+1) = a(n). - Reinhard Zumkeller, Sep 07 2015

Moritz Firsching, Table of n, a(n) for n = 1..315
B. Cloitre, 10 conjectures in additive number theory, arXiv:1101.4274 [math.NT], 2011.
M. F. Hasler, Rowland-Cloître type prime generating sequences, OEIS Wiki, August 2015.

Cf. A106108.

Cf. A261301 - A261310; A186254 - A186263.

a186253 n = a186253_list !! (n-1)
a186253_list = filter ((== 0) . a261301) [1..]
-- Reinhard Zumkeller, Sep 07 2015

a = m = 1; Reap[For[n = 2, n <= 10^7, n++, a = Abs[a - GCD[a, m*n - 1]]; If[a == 0, Print[m*n + m - 1]; Sow[m*n + m - 1]]]][[2, 1]] (* Jean-François Alcover, Feb 05 2019, from PARI *)
nxt[{n_,a_}]:={n+1,Abs[a-GCD[a,n]]}; Position[NestList[nxt,{1,1},13*10^5][[All,2]],0]// Flatten (* The program generates the first 20 terms of the sequence. *) (* Harvey P. Dale, Oct 02 2022 *)

a=1;m=1;for(n=2,1e7,a=abs(a-gcd(a,m*n-1));if(a==0,print1(m*n+m-1,",")))

next_a(last_a) = {
  local(A=last_a,B=last_a,C=2*last_a+1);
  while(A>0,
    D=divisors(C);
    k1=10*D[2];
    for(j=2,#D, d=D[j];k=((A+1-B+d)/2)%d;
      if(k==0,k=d); if(k<=k1,k1=k;d1=d));
    if(k1-1+d1==A,B=B+1);
    A = max(A-(k1-1)-d1,0);
    B = B + k1;
    C = C - (d1 - 1);
  );
  return(B);
}
a=2
for(n=1,99,print1(a,", ");a=next_a(a)) \\ Jan Büthe and Moritz Firsching, Aug 04 2015

m=a=k=1; for(n=1, 30, while( a>d=vecmin(apply(p->a%p, factor(N=m*(k+a)+m-1)[,1])), a-=d+gcd(a-d,N); k+=1+d); k+=a+1; print1(a=N,",")) \\ M. F. Hasler, Aug 22 2015

Conjecture: a(n) is asymptotic to c*2^n with c = 1.1861...

Definition clarified by M. F. Hasler, Aug 14 2015

A186263 a(n) = 10*b_10(n) + 9, where b_10 lists the indices of zeros of the sequence A261310: u(n) = abs(u(n-1) - gcd(u(n-1), 10n-1)), u(1) = 1.

Original entry on oeis.org

29, 269, 2969, 32609, 357169, 3928669, 43213789, 475113649, 5226205969, 57488152069, 632360271769, 6955957188049, 76515529068529, 841670819753809, 9258379017291889, 101842168949117209, 1120263858440288929, 12322902442843176229, 135551926871245562989

Offset: 1

Benoit Cloitre, Feb 16 2011

nonn

For any fixed integer m>=1 define u(1)=1 and u(n)=abs(u(n-1)-gcd(u(n-1),m*n-1)). Then (b_m(k))_{k>=1} is the sequence of integers such that u(b_m(k))=0 and we conjecture that for k large enough m*b_m(k)+m-1 is a prime number. Here for m=10 it appears a(n) is prime for n>=1.

See A261310 for the sequence u relevant here (m=10). - M. F. Hasler, Aug 14 2015

Benoit Cloitre, 10 conjectures in additive number theory, preprint arXiv:2011.4274 [math.NT] , 2011.
M. F. Hasler, Rowland-Cloître type prime generating sequences, OEIS Wiki, August 2015.

Cf. A106108.

Cf. A261301 - A261310; A186253 - A186261.

a=1; m=10; for(n=2, 1e7, a=abs(a-gcd(a, m*n-1)); if(a==0, print1(m*n+m-1, ", ")))

m=10; a=k=1; for(n=1, 20, while( a>D=vecmin(apply(p->a%p, factor(N=m*(k+a)+m-1)[, 1])), a-=D+gcd(a-D, N); k+=1+D); k+=a+1; print1(a=N, ", ")) \\ M. F. Hasler, Aug 22 2015

We conjecture that a(n) is asymptotic to c*11^n with c>0.

See the wiki link for a sketch of a proof of this conjecture. We find c = 2.2163823215... - M. F. Hasler, Aug 22 2015

Edited by M. F. Hasler, Aug 14 2015

More terms from M. F. Hasler, Aug 22 2015

A261310 a(n+1) = abs(a(n) - gcd(a(n), 10n+9)), a(1) = 1.

Original entry on oeis.org

1, 0, 29, 28, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 269, 268, 267, 266, 265, 264, 263, 262, 261, 260, 259, 258, 257, 256, 255, 254, 253, 252, 251, 250, 249, 248, 247, 246, 245, 244, 243, 242, 241, 240, 239, 238, 237, 236, 235, 234, 233, 232, 231, 230, 229

Offset: 1

M. F. Hasler, Aug 14 2015

nonn

The absolute value is relevant only when a(n) = 0 in which case a(n+1) = gcd(a(n),10n+9) = 10n+9.

It is conjectured that for all n, a(n) = 0 implies that a(n+1) = 10n+9 is prime, cf. A186263.

			a(2) = a(1) - gcd(a(1),10+9) = 1 - 1 = 0.
a(3) = |a(2) - gcd(a(2),10*2+9)| = gcd(0,29) = 29 is prime.
a(4) = 28 and 10*4+9 = 49, thus a(5) = 28 - gcd(28,49) = 28 - 7 = 21. Note that for n = 4+32, 10n+9 = 329 is divisible by 7, but for n = 5+21 = 26, 10n+9 = 269 = a(27) is prime. Also, for n = 27+269 = 296, 10n+9 = 2969 = a(297) is prime again.

Cf. A261301 - A261309; A186253 - A186263.

print1(a=1);for(n=1,99,print1(",",a=abs(a-gcd(a,10*n+9))))

A261302 a(n+1) = abs(a(n) - gcd(a(n), 2n+1)), a(1) = 1.

Original entry on oeis.org

1, 0, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 53, 52, 51, 50, 49, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 149, 148, 147, 146, 145, 144, 143

Offset: 1

M. F. Hasler, Aug 14 2015

The absolute value is relevant only when a(n) = 0, in which case a(n+1) = gcd(a(n), 2n+1) = 2n+1.

It is conjectured that for all n, a(n) = 0 implies that 2n+1 = a(n+1) is prime, cf. A186254. (This is the sequence {u(n)} mentioned there.)

			For n = 1, a(n) = 1 therefore a(n+1) = a(n) - gcd(a(n),2*n+1) = 1 - 1 = 0. The same is true for n = 7.
a(2) = 0 therefore a(3) = gcd(0,2*2+1) = 5, which is prime.
a(3+5) = a(8) = 0 therefore a(9) = gcd(0,2*8+1) = 17, which is also prime.
a(9+17) = a(26) = 0 therefore a(27) = gcd(0,2*26+1) = 53, which is also prime.
a(31) = 49 and 2*31+1 = 63, therefore a(32) = 49 - gcd(49,63) = 49 - 7 = 42. Note that for n = 31+49 = 80, 2n+1 = 161 would not be prime, but for n = 32+42 = 74, 2n+1 = 149 = a(75) is prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A261301 - A261310, A186253 - A186263, A106108.

nxt[{n_,a_}]:={n+1,Abs[a-GCD[a,2n+1]]}; NestList[nxt,{1,1},80][[All,2]] (* Harvey P. Dale, Jan 17 2023 *)

print1(a=1);for(n=1,99,print1(",",a=abs(a-gcd(a,2*n+1))))

A186254 a(n) = 2b(n)+1, where b(n) lists the zeros of the sequence u(n)=abs(u(n-1)-gcd(u(n-1),2n-1)), u(1)=1.

Original entry on oeis.org

5, 17, 53, 149, 449, 1289, 3761, 11261, 33773, 101117, 302681, 907757, 2723069, 8169137, 24506597, 73519793, 220559369, 661677761, 1985001917, 5955003077, 17865008333, 53595020201, 160785060361, 482355180761, 1447065541373, 4341196624109, 13023589872329

Offset: 1

Benoit Cloitre, Feb 16 2011

nonn

For any fixed integer m>=1 define u(1)=1 and u(n)=abs(u(n-1)-gcd(u(n-1),m*n-1)). Then (b_m(k))_{k>=1} is the sequence of integers such that u(b_m(k))=0 and we conjecture that for all k large enough, m*b_m(k)+m-1 is a prime number. Here for m=2 it appears a(n) is prime for n>=1.

See A261302 for the sequence u relevant here (m=2). - M. F. Hasler, Aug 14 2015

B. Cloitre, 10 conjectures in additive number theory, arXiv:1101.4274 [math.NT], 2011.
M. F. Hasler, Rowland-Cloître type prime generating sequences, OEIS Wiki, August 2015.

Cf. A106108, A186253 - A186263.

Cf. A261301 - A261310.

a=1; m=2; for(n=2, 9e9, if(!a=abs(a-gcd(a, m*n-1)), print1(m*n+m-1, ", ")))

m=2; a=k=1; for(n=1, 30, while( a>D=vecmin(apply(p->a%p, factor(N=m*(k+a)+m-1)[, 1])), a-=D+gcd(a-D, N); k+=1+D); k+=a+1; print1(a=N, ", ")) \\ M. F. Hasler, Aug 22 2015

a(n+1) <= 3*a(n)+2 for all n.- See the wiki link for a sketch of a proof that a(n) ~ c*3^n with c = 1.7078779... - M. F. Hasler, Aug 22 2015

More terms from M. F. Hasler, Aug 22 2015

A186256 a(n) = 4b_4(n)+3, where b_4 lists the indices of zeros of the sequence A261304: u(n) = abs(u(n-1)-gcd(u(n-1),4n-1)), u(1) = 1.

Original entry on oeis.org

11, 59, 251, 1259, 6299, 31387, 152083, 758971, 3790651, 18953251, 94766251, 473831251, 2369156107, 11845755043, 59228775043, 296143874947, 1480718773123, 7403593861843, 37017965808931, 185089757395379, 925448786976163, 4627243883546971, 23136219387534283

Offset: 1

Benoit Cloitre, Feb 16 2011

nonn

For any fixed integer m>=1 define u(1)=1 and u(n)=abs(u(n-1)-gcd(u(n-1),m*n-1)). Then (b_m(k))_{k>=1} is the sequence of integers such that u(b_m(k))=0 and we conjecture that for k large enough m*b_m(k)+m-1 is a prime number. Here for m=4 it appears a(n) is prime for n>=1.

See A261304 for the sequence u relevant here (m=4). - M. F. Hasler, Aug 14 2015

B. Cloitre, 10 conjectures in additive number theory, preprint arxiv:2011.4274 (2011).
M. F. Hasler, Rowland-Cloître type prime generating sequences, OEIS Wiki, August 2015.

Cf. A106108.

Cf. A261301 - A261310; A186253 - A186263.

a=1; m=4; for(n=2, 1e7,a=abs(a-gcd(a, m*n-1)); if(a==0, print1(m*n+m-1, ", ")))

m=4; a=k=1; for(n=1, 25, while( a>D=vecmin(apply(p->a%p, factor(N=m*(k+a)+m-1)[, 1])), a-=D+gcd(a-D, N); k+=1+D); k+=a+1; print1(a=N, ", ")) \\ M. F. Hasler, Aug 22 2015

We conjecture that a(n) is asymptotic to c*5^n with c=1.9408...

See the wiki link for a sketch of a proof of this conjecture. We can give more decimals of c = 1.94080675... - M. F. Hasler, Aug 22 2015

More terms from M. F. Hasler, Aug 22 2015

A186261 a(n) = 9*b_9(n) + 8, where b_9 lists the indices of zeros of the sequence A261309: u(n) = abs(u(n-1) - gcd(u(n-1), 9n-1)), u(1) = 1.

Original entry on oeis.org

26, 269, 2699, 26423, 259829, 2595473, 25954289, 259491059, 2594910599, 25949104721, 259491047219, 2594905133453, 25949039883929, 259490398799609, 2594903521711517, 25949035214699921, 259490352146949701, 2594903520789157301, 25949035207891572929, 259490352078915446897

Offset: 1

Benoit Cloitre, Feb 16 2011

nonn

For any fixed integer m>=1 define u(1)=1 and u(n)=abs(u(n-1)-gcd(u(n-1),m*n-1)). Then (b_m(k))_{k>=1} is the sequence of integers such that u(b_m(k))=0 and we conjecture that for k large enough m*b_m(k)+m-1 is a prime number. Here for m=9 it appears a(n) is prime for n>=2.

See A261309 for the sequence u relevant here (m=9). - M. F. Hasler, Aug 14 2015

B. Cloitre, 10 conjectures in additive number theory, preprint arxiv:2011.4274 (2011).
M. F. Hasler, Rowland-Cloître type prime generating sequences, OEIS Wiki, August 2015.

Cf. A106108.

Cf. A261301 - A261310; A186253 - A186263.

a=1; m=9; for(n=2, 1e8, a=abs(a-gcd(a, m*n-1)); if(a==0, print1(m*n+m-1, ", ")))

m=9;a=0;k=2; for(n=1,20,while(1<#(f=factor(N=m*(k+a)+m-1)[,1]) && a, k+=1+D=vecmin(apply(p->a%p,f)); a-=D+gcd(a-D,N));k+=a+1;print1(a=N,",")) \\ M. F. Hasler, Aug 22 2015

We conjecture that a(n) is asymptotic to c*10^n with c>0.

See the wiki link for a sketch of a proof of this conjecture. We find c=2.59490352... - M. F. Hasler, Aug 22 2015

Edited by M. F. Hasler, Aug 14 2015

A261307 a(n+1) = abs(a(n) - gcd(a(n), 7*n+6)), a(1) = 1.

Original entry on oeis.org

1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 167, 166, 165, 164, 163, 162, 161, 160, 159, 158, 79, 78, 77, 76, 75, 74, 73, 72, 71, 70, 69, 68, 67, 66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35

Offset: 1

M. F. Hasler, Aug 14 2015

nonn

It is conjectured that for all n > 2, a(n) = 0 implies that 7n+6 = a(n+1) is prime, cf. A186259. (This is the sequence {u(n)} mentioned there.)

			a(2) = a(1) - gcd(a(1),7+6) = 1 - 1 = 0.
a(3) = |a(2) - gcd(a(2),7*2+6)| = gcd(0,17) = 17 is prime.
a(33) = 158, thus a(6) = 158 - gcd(158,7*33+6) = 158 - 79 = 79.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A261301 - A261310, A186253 - A186263, A106108.

nxt[{n_,a_}]:={n+1,Abs[a-GCD[a,7n+6]]}; NestList[nxt,{1,1},80][[All,2]] (* Harvey P. Dale, Apr 26 2017 *)

print1(a=1);for(n=1,99,print1(",",a=abs(a-gcd(a,7*n+6))))

A261309 a(n+1) = abs(a(n) - gcd(a(n), 9n+8)), u(1) = 1.

Original entry on oeis.org

1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 269, 268, 267, 266, 265, 264, 263, 262, 261, 260, 259, 258, 257, 256, 255, 254, 253, 252, 251, 250, 249, 248, 247, 246, 245, 244, 243, 242, 241, 240, 239, 238, 237, 236, 235, 234, 233, 232, 231

Offset: 1

M. F. Hasler, Aug 14 2015

nonn

It is conjectured that for all n > 2, u(n) = 0 implies that u(n+1) = 9n+8 is prime, cf. A186261. (This is the sequence {u(n)} mentioned there.)

			a(2) = a(1) - gcd(a(1),9+8) = 1 - 1 = 0.
a(3) = |a(2) - gcd(a(2),9*2+8)| = gcd(0,26) = 26.
a(3+26) = a(29) = 0 and a(29+1) = gcd(0,9*29+8) = 269 is prime.

Cf. A261301 - A261310, A186253 - A186263, A106108.

print1(a=1);for(n=1,99,print1(",",a=abs(a-gcd(a,9*n+8))))

A186255 a(n) = 3b_3(n)+2, where b_3 lists the zeros of the sequence A261303: u(n+1)=abs(u(n)-gcd(u(n),3n+2)), u(1)=1.

Original entry on oeis.org

8, 17, 71, 269, 1013, 4007, 15923, 63521, 253949, 1014317, 4056893, 16225589, 64902359, 259609439, 1038437759, 4153750883, 16614561281, 66458241569, 265832966279, 1063331407109, 4253325628439, 17013302513759, 68053207705097, 272212800371669, 1088851201483883

Offset: 1

Benoit Cloitre, Feb 16 2011

nonn

For any fixed integer m>=1 define u(1)=1 and u(n)=abs(u(n-1)-gcd(u(n-1),m*n-1)). Then (b_m(k))_{k>=1} is the sequence of integers such that u(b_m(k))=0 and we conjecture that for k large enough m*b_m(k)+m-1 is a prime number. Here for m=3 it appears a(n) is prime for n>=2.

See A261303 for the sequence u relevant here (m=3). - M. F. Hasler, Aug 14 2015

B. Cloitre, 10 conjectures in additive number theory, preprint arxiv:2011.4274
M. F. Hasler, Rowland-Cloître type prime generating sequences, OEIS Wiki, August 2015.

Cf. A106108.

Cf. A261301 - A261310; A186253 - A186263.

a=1; m=3; for(n=2, 10^7, a=abs(a-gcd(a, m*n-1)); if(a==0, print1(m*n+m-1, ", ")))

m=3; a=k=1; for(n=1, 25, while( a>D=vecmin(apply(p->a%p, factor(N=m*(k+a)+m-1)[, 1])), a-=D+gcd(a-D, N); k+=1+D); k+=a+1; print1(a=N, ", ")) \\ M. F. Hasler, Aug 22 2015

We conjecture that a(n) is asymptotic to c*4^n with c=0.96...

See the wiki link for a sketch of a proof that this is true. We can give more decimals of c = 0.967094... - M. F. Hasler, Aug 22 2015

More terms from M. F. Hasler, Aug 22 2015

cp's OEIS Frontend

A186253 Indices of zeros of the sequence u(n)=abs(u(n-1)-gcd(u(n-1),n-1)), u(1)=1.

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A186263 a(n) = 10*b_10(n) + 9, where b_10 lists the indices of zeros of the sequence A261310: u(n) = abs(u(n-1) - gcd(u(n-1), 10n-1)), u(1) = 1.

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A261310 a(n+1) = abs(a(n) - gcd(a(n), 10n+9)), a(1) = 1.

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A261302 a(n+1) = abs(a(n) - gcd(a(n), 2n+1)), a(1) = 1.

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A186254 a(n) = 2*b(n)+1, where b(n) lists the zeros of the sequence u(n)=abs(u(n-1)-gcd(u(n-1),2*n-1)), u(1)=1.

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A186256 a(n) = 4*b_4(n)+3, where b_4 lists the indices of zeros of the sequence A261304: u(n) = abs(u(n-1)-gcd(u(n-1),4*n-1)), u(1) = 1.

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A186261 a(n) = 9*b_9(n) + 8, where b_9 lists the indices of zeros of the sequence A261309: u(n) = abs(u(n-1) - gcd(u(n-1), 9n-1)), u(1) = 1.

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A186254 a(n) = 2b(n)+1, where b(n) lists the zeros of the sequence u(n)=abs(u(n-1)-gcd(u(n-1),2n-1)), u(1)=1.

A186256 a(n) = 4b_4(n)+3, where b_4 lists the indices of zeros of the sequence A261304: u(n) = abs(u(n-1)-gcd(u(n-1),4n-1)), u(1) = 1.

A186255 a(n) = 3b_3(n)+2, where b_3 lists the zeros of the sequence A261303: u(n+1)=abs(u(n)-gcd(u(n),3n+2)), u(1)=1.