A013634 -id:A013634 - cp's OEIS frontend

A368786 a(n) is the first prime p such that, if q are r are the next two primes, p + r, p + q, q + r and p + q + r all have n prime divisors, counted with multiplicity.

1559, 4073, 45863, 1369133, 82888913, 754681217, 118302786439

Offset: 3

Zak Seidov and Robert Israel, Jan 05 2024

a(n) is the first term p of A368785 such that A001222(A013634(p)) = n.

			a(5) = 45863 because 45863, 45869, 45887 are consecutive primes with
45863 + 45869 = 91732 = 2^2 * 17 * 19 * 71,
45863 + 45887 = 91750 = 2 * 5^3 * 367,
45869 + 45887 = 91756 = 2^2 * 7 * 29 * 113, and
45863 + 45869 + 45887 = 137619 = 3^4 * 1699
all have 5 prime divisors, counted with multiplicity, and 45763 is the least prime that works.

Cf. A001222, A013634, A368785.

V:= Array(3..8): count:= 0:
p:= 2: q:= 3: r:= 5: v:= numtheory:-bigomega(q+r);
while count < 6 do
  p:= q; q:= r; r:= nextprime(r);
  w:= numtheory:-bigomega(q+r);
  if (w > 7 or V[w] = 0) and w = v and numtheory:-bigomega(p+r) = v and numtheory:-bigomega(p+q+r) = v then
    V[w]:= p; count:= count+1;
    fi;
  v:= w;
od:
convert(V,list);

a(9) from Daniel Suteu, Jan 05 2024

A377403 For n >= 2, a(n) is the number of iterations needed for the map: x -> x / A085392(x) if A085392(x) > 1, otherwise x -> x + A151800(x), to (the first occurrence of) 2.

Original entry on oeis.org

0, 3, 1, 3, 1, 3, 2, 4, 1, 4, 2, 3, 1, 4, 3, 4, 2, 3, 2, 4, 1, 3, 3, 4, 1, 5, 2, 4, 2, 3, 4, 4, 1, 4, 3, 3, 1, 4, 3, 4, 2, 4, 2, 5, 1, 4, 4, 4, 2, 4, 2, 5, 3, 4, 3, 4, 1, 5, 3, 7, 1, 5, 5, 4, 2, 3, 2, 4, 2, 6, 4, 4, 1, 5, 2, 4, 2, 5, 4, 6, 1, 3, 3, 4, 1, 4, 3, 3, 3, 4, 2, 4, 1, 4, 5, 4, 2, 5, 3, 4, 2, 4, 3, 5, 1, 6, 4, 3, 2, 4, 4, 6, 2, 4, 2, 5, 1, 4, 4, 5

Offset: 2

Ctibor O. Zizka, Oct 27 2024

nonn

Also a(2*k + 1) = A001222(2*k + 1) + 2 + s, where s >= 1 for k = 5, 8, 14, 20, 21, 23, 26, 29, 30, 35, 36, 39, 48, 50, 51, ...

			n = 3: 3 -> 8 -> 4 -> 2, 3 iterations needed to reach 2, thus a(3) = 3.
n = 9: 9 -> 3 -> 8 -> 4 -> 2, 4 iterations needed to reach 2, thus a(9) = 4.
n = 11: 11 -> 24 -> 8 -> 4 - > 2, 4 iterations needed to reach 2, thus a(11) = 4.

Cf. A001222, A013634, A020639, A085392, A107286, A151800.

a[2] = 0; a[n_] := -1 + Length@ NestWhileList[If[CompositeQ[#], #/FactorInteger[#][[-1, 1]], # + NextPrime[#]] &, n, # > 2 &]; Array[a, 120, 2] (* Amiram Eldar, Oct 27 2024 *)

For n even: a(n) = A001222(n) - 1.

For n odd: a(n) = A001222(n) - 1 + A001222(A013634(A020639(n))).

cp's OEIS Frontend

A368786 a(n) is the first prime p such that, if q are r are the next two primes, p + r, p + q, q + r and p + q + r all have n prime divisors, counted with multiplicity.

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