A323075 The fixed point reached when map x -> 1+(x-(largest divisor d < x)) is iterated, starting from x = n.

1, 2, 3, 3, 5, 3, 7, 5, 7, 3, 11, 7, 13, 5, 11, 7, 17, 3, 19, 11, 11, 7, 23, 13, 11, 5, 19, 11, 29, 7, 31, 17, 23, 3, 29, 19, 37, 11, 19, 11, 41, 7, 43, 23, 31, 13, 47, 11, 43, 5, 29, 19, 53, 11, 31, 29, 19, 7, 59, 31, 61, 17, 43, 23, 53, 3, 67, 29, 47, 19, 71, 37, 73, 11, 29, 19, 67, 11, 79, 41, 31, 7, 83, 43, 47, 23, 59, 31, 89, 13

Offset: 1

Author

Antti Karttunen, Jan 04 2019

Keywords

• nonn

Comments

After a(1) = 1, the fixed point reached is always a prime. Question: Do all odd primes occur infinitely often?

Yes. All odd primes occur infinitely often. A060681(2*k) = k + 1 and so for each k > 1 there exists an integer m such that a(m) = p where p is an odd prime. - David A. Corneth, Jan 07 2019

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20669

Crossrefs

Cf. A000040, A000079, A000918, A060681, A323076, A323079, A323164, A323165 (ordinal transform).

Cf. also A039650, A039654.

Programs

Mathematica

{1}~Join~Array[FixedPoint[1 + (# - Divisors[#][[-2]]) &, #] &, 89, 2] (* Michael De Vlieger, Jan 04 2019 *)

PARI

A060681(n) = (n-if(1==n,n,n/vecmin(factor(n)[,1])));
A323075(n) = { my(nn = 1+A060681(n)); if(nn==n,n,A323075(nn)); };

Formula

If n == (1+A060681(n)), then a(n) = n, otherwise a(n) = a(1+A060681(n)).

a(2^k * p - 2^(k+1) + 2) = a(A000079(k) * p - A000918(k+1)) = p for k >= 0. - David A. Corneth, Jan 08 2019

a(1) = 1, and for n > 1, a(n) = A000040(A323164(n)). - Antti Karttunen, Jan 08 2019