A261309 -id:A261309 - cp's OEIS frontend

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.

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

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))))

cp's OEIS Frontend

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

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