A323165 -id:A323165 - cp's OEIS frontend

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

Antti Karttunen, Jan 04 2019

nonn

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

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

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

Cf. also A039650, A039654.

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

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)); };

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

A323164 a(n) = A000720(A323075(n)).

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 4, 3, 4, 2, 5, 4, 6, 3, 5, 4, 7, 2, 8, 5, 5, 4, 9, 6, 5, 3, 8, 5, 10, 4, 11, 7, 9, 2, 10, 8, 12, 5, 8, 5, 13, 4, 14, 9, 11, 6, 15, 5, 14, 3, 10, 8, 16, 5, 11, 10, 8, 4, 17, 11, 18, 7, 14, 9, 16, 2, 19, 10, 15, 8, 20, 12, 21, 5, 10, 8, 19, 5, 22, 13, 11, 4, 23, 14, 15, 9, 17, 11, 24, 6, 22, 15, 14, 5, 19, 14, 25, 3, 19, 10, 26, 8, 27, 16, 20

Offset: 1

Antti Karttunen, Jan 08 2019

nonn

One less than the restricted growth sequence transform of A323075.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000079, A000720, A000918, A323075, A323165 (ordinal transform).

A060681[n_] := n - If[1 == n, n, n/Min[FactorInteger[n][[All, 1]]]];
A323075[n_] := With[{nn = 1 + A060681[n]}, If[nn == n, n, A323075[nn]]];
a[n_] = PrimePi[A323075[n]];
Array[a, 105] (* Jean-François Alcover, Jan 10 2022, from PARI code *)

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)); };
A323164(n) = primepi(A323075(n));

a(n) = A000720(A323075(n)).

For all odd primes p, a(2^k * p - 2^(k+1) + 2) = a(A000079(k) * p - A000918(k+1)) = A000720(p) for k >= 0. - After David A. Corneth's similar observation for A323075.

cp's OEIS Frontend

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

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