A323079 -id:A323079 - cp's OEIS frontend

A322809 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = -1 if n is an odd prime, and f(n) = floor(n/2) for all other numbers.

1, 2, 3, 4, 3, 5, 3, 6, 6, 7, 3, 8, 3, 9, 9, 10, 3, 11, 3, 12, 12, 13, 3, 14, 14, 15, 15, 16, 3, 17, 3, 18, 18, 19, 19, 20, 3, 21, 21, 22, 3, 23, 3, 24, 24, 25, 3, 26, 26, 27, 27, 28, 3, 29, 29, 30, 30, 31, 3, 32, 3, 33, 33, 34, 34, 35, 3, 36, 36, 37, 3, 38, 3, 39, 39, 40, 40, 41, 3, 42, 42, 43, 3, 44, 44, 45, 45, 46, 3, 47, 47, 48, 48, 49, 49, 50, 3, 51, 51, 52, 3, 53, 3, 54, 54

Offset: 1

Antti Karttunen, Dec 26 2018

nonn

This sequence is a restricted growth sequence transform of a function f which is defined as f(n) = A004526(n), unless n is an odd prime, in which case f(n) = -1, which is a constant not in range of A004526. See the Crossrefs section for a list of similar sequences.
For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A039636(i) = A039636(j).
For all i, j: a(i) = a(j) <=> A323161(i+1) = A323161(j+1).
The shifted version of this filter, A323161, has a remarkable ability to find many sequences related to primes and prime chains. - Antti Karttunen, Jan 06 2019

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A004526, A039636, A305801, A323161.
A list of few similarly constructed sequences follows, where each sequence is an rgs-transform of such function f, for which the value of f(n) is the n-th term of the sequence whose A-number follows after a parenthesis, unless n is of the form ..., in which case f(n) is given a constant value outside of the range of that sequence:
  A322809 (A004526, unless an odd prime) [This sequence],
  A322589 (A007913, unless an odd prime),
  A322591 (A007947, unless an odd prime),
  A322805 (A252463, unless an odd prime),
  A323082 (A300840, unless an odd prime),
  A322822 (A300840, unless n > 2 and a Fermi-Dirac prime, A050376),
  A322988 (A322990, unless a prime power > 2),
  A323078 (A097246, unless an odd prime),
  A322808 (A097246, unless a squarefree number > 2),
  A322816 (A048675, unless an odd prime),
  A322807 (A285330, unless an odd prime),
  A322814 (A286621, unless an odd prime),
  A322824 (A242424, unless an odd prime),
  A322973 (A006370, unless an odd prime),
  A322974 (A049820, unless n > 1 and n is in A046642),
  A323079 (A060681, unless an odd prime),
  A322587 (A295887, unless an odd prime),
  A322588 (A291751, unless an odd prime),
  A322592 (A289625, unless an odd prime),
  A323369 (A323368, unless an odd prime),
  A323371 (A295886, unless an odd prime),
  A323374 (A323373, unless an odd prime),
  A323401 (A323372, unless an odd prime),
  A323405 (A323404, unless an odd prime).

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A322809aux(n) = if((n>2)&&isprime(n),-1,(n>>1));
v322809 = rgs_transform(vector(up_to,n,A322809aux(n)));
A322809(n) = v322809[n];

a(n) = A323161(n+1) - 1.

A323076 Number of iterations of map x -> 1+(x-(largest divisor d < x)), starting from x=n, needed to reach a fixed point, which is always either a prime or 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 1, 1, 3, 0, 1, 0, 2, 1, 2, 0, 4, 0, 1, 2, 2, 0, 1, 3, 3, 1, 2, 0, 3, 0, 1, 1, 5, 1, 1, 0, 2, 2, 3, 0, 3, 0, 1, 1, 2, 0, 4, 1, 4, 2, 2, 0, 3, 2, 1, 3, 4, 0, 1, 0, 2, 1, 2, 1, 6, 0, 2, 1, 2, 0, 1, 0, 3, 3, 3, 1, 4, 0, 1, 3, 4, 0, 1, 2, 2, 1, 2, 0, 3, 1, 1, 2, 5, 2, 2, 0, 5, 1, 3, 0, 3, 0, 1, 1

Offset: 1

Antti Karttunen, Jan 04 2019

nonn

Differs from A064918 at n = 25, 48, 51, 69, 75, 81, 85, 94, 95, 99, 100, 111, 115, 121, ...

Antti Karttunen, Table of n, a(n) for n = 1..20669
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A060681, A064918, A323075 (the fixed points reached), A323077, A323079.

Cf. also A039651.

{0}~Join~Array[-2 + Length@ NestWhileList[1 + (# - Divisors[#][[-2]]) &, #, UnsameQ, All] &, 104, 2] (* Michael De Vlieger, Jan 04 2019 *)

A060681(n) = (n-if(1==n,n,n/vecmin(factor(n)[,1])));
A323076(n) = { my(nn = 1+A060681(n)); if(nn==n,0,1+A323076(nn)); };

If n == (1+A060681(n)), then a(n) = 0, otherwise a(n) = 1 + a(1+A060681(n)).

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

Original entry on oeis.org

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

cp's OEIS Frontend

A322809 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = -1 if n is an odd prime, and f(n) = floor(n/2) for all other numbers.

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A323076 Number of iterations of map x -> 1+(x-(largest divisor d < x)), starting from x=n, needed to reach a fixed point, which is always either a prime or 1.

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

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