A376109 a(n) is the length of the longest arithmetic progression ending at n consisting of numbers with the same number of prime factors as n, counted with multiplicity.

1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 2, 3, 2, 1, 3, 2, 3, 2, 3, 3, 4, 2, 2, 3, 2, 3, 5, 2, 3, 1, 3, 3, 3, 2, 2, 3, 2, 2, 4, 3, 4, 3, 2, 4, 3, 2, 3, 2, 3, 3, 5, 2, 4, 3, 3, 5, 4, 2, 3, 3, 3, 1, 3, 3, 3, 3, 4, 3, 3, 2, 3, 3, 2, 3, 3, 3, 4, 2, 2, 4, 4, 3, 3, 4, 5, 3, 3, 2, 4, 4, 4, 3, 3, 2, 4, 3, 3

Offset: 1

Author

Robert Israel, Sep 10 2024

Keywords

• nonn

Comments

a(n) is the greatest k such that there exists d >= 1 with A001222(n-i*d) = A001222(n) for 0 <= i < k.

If m divides n, then a(n) >= a(m).

a(n) = 1 if and only if n is a power of 2.

Examples

			a(7) = 3 because 7 is prime and there is an arithmetic progression of 3 primes, namely 3, 5, 7, ending with 7 but no such arithmetic progression of 4 primes.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001222, A005115, A096003, A373887, A373888.

Programs

Maple

M:= Array(1..10):
for n from 2 to 100 do
  v:= numtheory:-bigomega(n);
  if M[v] = 0 then M[v]:= n else M[v]:= M[v],n fi;
od:
for i from 1 to 10 do M[i]:= [M[i]] od:
f:= proc(s) local n,i,m,d,v,j;
   m:= 1;
   v:= numtheory:-bigomega(s);
   member(s,M[v],n);
   for i from n-1 to 1 by -1 do
     d:= s - M[v][i];
     if s - m*d < M[v][1] then return m fi;
     for j from 2 while ListTools:-BinarySearch(M[v],s-j*d) <> 0 do od:
     m:= max(m,j);
   od;
  m;
end proc:
f(1):= 1:
map(f, [$1..100]);