A261301 -id:A261301 - cp's OEIS frontend

A186258 a(n) = 6b_6(n)+5, where b_6 lists the indices of zeros of the sequence A261306: u(n) = abs(u(n-1)-gcd(u(n-1),6n-1)), u(1) = 1.

17, 101, 461, 2801, 19553, 136649, 955841, 6684749, 46777229, 327440609, 2292083093, 16044575777, 112312028681, 786179138273, 5503253967269, 38522774910593, 269659405576049, 1887615818410877, 13213310659503893, 92493174561607361

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=6 it appears a(n) is prime for n>=1.

See A261306 for the sequence u relevant here (m=6). - 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=6; for(n=2, 1e7, a=abs(a-gcd(a, m*n-1)); if(a==0, print1(m*n+m-1, ", ")))

m=6; 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*7^n with c>0.

See the wiki link for a sketch of a proof of this conjecture. We find c = 1.15917467758687... - M. F. Hasler, Aug 22 2015

Edited by M. F. Hasler, Aug 14 2015

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

A186259 a(n) = 7*b_7(n) + 6, where b_7 lists the indices of zeros of the sequence u(n) = abs(u(n-1) - gcd(u(n-1), 7n-1)), u(1) = 1.

Original entry on oeis.org

20, 167, 797, 6299, 48817, 389437, 3114313, 24910031, 199280101, 1594149787, 12752862247, 102022886167, 816183074713, 6529464593329, 52235716720753, 417885733765933, 3343085868722137, 26744686949777089, 213957495598165381, 1711659964119801373

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=7 it appears a(n) is prime for n>=2.

See A261307 for the sequence u relevant here (m=7). - 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=7; for(n=2, 1e7, a=abs(a-gcd(a, m*n-1)); if(a==0, print1(m*n+m-1, ", ")))

m=7; 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*8^n with c>0.

See the wiki link for a sketch of a proof of this conjecture. We find c = 1.48462836... - M. F. Hasler, Aug 22 2015

Edited by M. F. Hasler, Aug 14 2015

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

A186260 a(n) = 8*b_8(n)+7, where b_8 lists the zeros of the sequence A261308: u(n+1)=|u(n)-gcd(u(n), 8n+7)|, u(1)=1.

Original entry on oeis.org

23, 167, 1511, 13463, 120167, 1076039, 9684359, 87158999, 784430279, 7059870119, 63537744791, 571838662007, 5146547952983, 46318929479831, 416870365318487, 3751833287866247, 33766499550040823, 303898495950141767, 2735086463015669687, 24615778167141027047

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=8 it appears a(n) is prime for n>=1.

See A261308 for the sequence u relevant here (m=8). - 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=8; for(n=2, 10^8, a=abs(a-gcd(a, m*n-1)); if(a==0, print1(m*n+m-1, ", ")))

m=8; 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*9^n with c>0.

See the wiki link for a sketch of a proof of this conjecture. We find c = 2.024712577430180... - M. F. Hasler, Aug 22 2015

Edited by M. F. Hasler, Aug 14 2015

More terms from M. F. Hasler, Aug 14 2015

A260940 a(n) is the smallest index j>n such that g(j)=0 for the sequence g defined (for indices > n) by g(n+1)=n and g(i) = g(i-1) - gcd(i,g(i-1)).

Original entry on oeis.org

3, 5, 7, 7, 11, 13, 13, 17, 19, 19, 23, 19, 21, 29, 31, 31, 31, 37, 37, 41, 43, 43, 47, 43, 43, 53, 43, 41, 59, 61, 61, 61, 67, 67, 71, 73, 73, 71, 79, 79, 83, 79, 79, 89, 79, 79, 79, 97, 97, 101, 103, 103, 107, 109, 109, 113, 109, 109, 113, 109

Offset: 1

Moritz Firsching, Aug 04 2015

nonn

a(n) is prime for all n<=10^10 except a(13)=21.
a(n) <= 2n + 1.
a(n) = 2n + 1 if and only if 2n + 1 is prime.
a(n) = 2n - 1 if and only if 2n - 1 is a prime and 2n - 1 = 1 mod 6.
a(n) = 2n - 3 if and only if 2n - 3 is a prime and 2n - 3 = 1 mod 30.

Moritz Firsching, Table of n, a(n) for n = 1..9999

Cf. A002476, A132230, A261301.

A186253(n) is a^n(2) where a^n denotes the n-th composition.

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,matsize(D)[2],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(n)={
my(A=n, B=n, C=2*n+1);
while(A>0,
my(k1=oo,d1);
fordiv(C,d,
if(d==1,next);
my(k=((A+1-B+d)/2)%d);
if(k==0, k=d);
if(k<=k1, k1=k; d1=d)
);
A -= k1 - 1 + d1;
B += k1 + (A==0);
C -= d1 - 1;
);
B;
} \\ Charles R Greathouse IV, Nov 04 2015

def a(n):
    g=n
    n+=1
    while(g!=0):
        g=g-gcd(n,g)
        n+=1
    return n

A261303 a(n+1) = abs(a(n) - gcd(a(n), 3n+2)), a(1) = 1.

Original entry on oeis.org

1, 0, 8, 7, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 71, 70, 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, 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

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), 3n+2) = 3n+2.

It is conjectured that a(n) = 0 implies that 3n+2 = a(n+1) is prime, for all n > 2, cf. A186255. (This is the sequence {u(n)} mentioned there.)

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

print1(a=1);for(n=1,199,print1(",",a=abs(a-gcd(a,3*n+2))))

A261304 a(n+1) = abs(a(n) - gcd(a(n), 4n+3)), a(1) = 1.

Original entry on oeis.org

1, 0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 59, 58, 57, 56, 55, 54, 53, 52, 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, 251, 250, 249, 248, 247, 246, 245, 244, 243, 242, 241, 240, 239, 238, 237, 236

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), 4n+3) = 4n+3.

It is conjectured that a(n) = 0 implies that 4n+3 = a(n+1) is prime, cf. A186256. (This is the sequence {u(n)} mentioned there.)

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

print1(a=1);for(n=1,199,print1(",",a=abs(a-gcd(a,4*n+3))))

A261305 a(n+1) = abs(a(n) - gcd(a(n), 5*n+4)), u(1) = 1.

Original entry on oeis.org

1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 89, 88, 77, 76, 75, 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, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14

Offset: 1

M. F. Hasler, Aug 14 2015

nonn

It is conjectured that a(n) = 0 implies that 5n+4 = a(n+1) is prime for n > 2, cf. A186256. (This is the sequence {u(n)} mentioned there.)

			a(2) = a(1) - gcd(a(1),5+4) = 1 - 1 = 0.
a(3) = |a(2) - gcd(a(2),5*2+4)| = 14.
a(19) = 88, thus a(20) = 88 - gcd(88, 5*19+4) = 88 - 11 = 77.

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

print1(a=1);for(n=1,199,print1(",",a=abs(a-gcd(a,5*n+4))))

A261306 a(n+1) = abs((n) - gcd(a(n), 6*n+5)), a(1) = 1.

Original entry on oeis.org

1, 0, 17, 16, 15, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 101, 100, 99, 98, 97, 96, 95, 94, 93, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 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, 461, 460, 459, 458, 457, 456

Offset: 1

M. F. Hasler, Aug 14 2015

nonn

It is conjectured that for all n, a(n) = 0 implies that a(n+1) = 6n+5 is prime, cf. A186258. (This is the sequence {u(n)} mentioned there.)

			a(2) = a(1) - gcd(a(1),6+5) = 1 - 1 = 0.
a(3) = |a(2) - gcd(a(2),6*2+5)| = gcd(0,17) = 17 is prime.
a(5) = 15, thus a(6) = 15 - gcd(15,6*5+5) = 15 - 5 = 10; similarly after a(25) = 93.

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

print1(a=1);for(n=1,199,print1(",",a=abs(a-gcd(a,6*n+5))))

A261308 a(n+1) = abs(a(n) - gcd(a(n), 8n+7)), a(1) = 1.

Original entry on oeis.org

1, 0, 23, 22, 21, 20, 15, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 167, 166, 165, 164, 163, 162, 161, 160, 159, 158, 157, 156, 155, 154, 153, 152, 151, 150, 149, 148, 147, 146, 145, 144, 143, 142, 141, 140, 139, 138, 137, 136, 135, 134, 133, 132, 131, 130, 129, 128, 127, 126, 125, 124, 123, 122

Offset: 1

M. F. Hasler, Aug 14 2015

nonn

It is conjectured that for all n, a(n) = 0 implies that 8n+7 = a(n+1) is prime, cf. A186260. (This is the sequence {u(n)} mentioned there.)

			a(2) = a(1) - gcd(a(1),8+7) = 1 - 1 = 0.
a(3) = |a(2) - gcd(a(2),8*2+7)| = gcd(0,23) = 23 (= A186260(1)) is prime.
a(6) = 20 and 8*6+7 = 55, thus a(7) = 20 - gcd(20,55) = 20 - 5 = 15.
a(8) = 15 - gcd(15,8*7+7) = 15 - 3 = 12. Note that for n = 8 + a(8) = 20, we have that 8n+7 = 167 = a(20+1) = A186260(2) is prime, while for n = 3 + a(3) = 26, 8n+7 = 215 was divisible by 5, and for n = 7 + a(7) = 22, 8n+7 = 183 was divisible by 3.

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

print1(a=1);for(n=1,99,print1(",",a=abs(a-gcd(a,8*n+7))))

A186257 a(n) = 5b_5(n)+4, where b_5 lists the indices of zeros of the sequence A261305: u(n) = abs(u(n-1)-gcd(u(n-1),5n-1)), u(1) = 1.

Original entry on oeis.org

14, 89, 479, 2879, 17099, 99839, 599009, 3592859, 21557099, 129336149, 775914479, 4655486369, 27932918219, 167597509319, 1005582321329, 6033492323549, 36200953941059, 217205705087639, 1303234230378959, 7819405361540219

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=5 it appears a(n) is prime for n>=2.

See A261305 for the sequence u relevant here (m=5). - 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=5; for(n=2, 1e7, a=abs(a-gcd(a, m*n-1)); if(a==0, print1(m*n+m-1, ", ")))

m=5; 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*6^n with c>0.

See the wiki link for a sketch of a proof of this conjecture. More precisely we find c = 1.15917467761... - M. F. Hasler, Aug 22 2015

Edited by M. F. Hasler, Aug 14 2015

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

cp's OEIS Frontend

A186258 a(n) = 6*b_6(n)+5, where b_6 lists the indices of zeros of the sequence A261306: u(n) = abs(u(n-1)-gcd(u(n-1),6*n-1)), u(1) = 1.

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A186259 a(n) = 7*b_7(n) + 6, where b_7 lists the indices of zeros of the sequence u(n) = abs(u(n-1) - gcd(u(n-1), 7n-1)), u(1) = 1.

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A186260 a(n) = 8*b_8(n)+7, where b_8 lists the zeros of the sequence A261308: u(n+1)=|u(n)-gcd(u(n), 8n+7)|, u(1)=1.

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A260940 a(n) is the smallest index j>n such that g(j)=0 for the sequence g defined (for indices > n) by g(n+1)=n and g(i) = g(i-1) - gcd(i,g(i-1)).

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A261303 a(n+1) = abs(a(n) - gcd(a(n), 3n+2)), a(1) = 1.

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A261304 a(n+1) = abs(a(n) - gcd(a(n), 4n+3)), a(1) = 1.

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A261305 a(n+1) = abs(a(n) - gcd(a(n), 5*n+4)), u(1) = 1.

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A261306 a(n+1) = abs((n) - gcd(a(n), 6*n+5)), a(1) = 1.

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A186258 a(n) = 6b_6(n)+5, where b_6 lists the indices of zeros of the sequence A261306: u(n) = abs(u(n-1)-gcd(u(n-1),6n-1)), u(1) = 1.

A186257 a(n) = 5b_5(n)+4, where b_5 lists the indices of zeros of the sequence A261305: u(n) = abs(u(n-1)-gcd(u(n-1),5n-1)), u(1) = 1.