A369139 -id:A369139 - cp's OEIS frontend

A369185 Numbers k such that Omega(k) = 1 + Omega(k+1) = 2 + Omega(k+2).

45, 104, 105, 165, 260, 261, 344, 345, 357, 440, 465, 476, 477, 561, 776, 860, 861, 884, 885, 981, 1016, 1017, 1112, 1160, 1185, 1269, 1281, 1395, 1424, 1544, 1572, 1624, 1644, 1652, 1683, 1808, 1812, 1827, 1905, 1917, 1989, 2037, 2060, 2061, 2097, 2145, 2277, 2444, 2445, 2805, 2817, 2852, 2877

Offset: 1

Zak Seidov and Robert Israel, Jan 15 2024

nonn

Numbers k such that both k and k+1 are in A369139.

			a(4) = 165 is a term because 165 = 3 * 5 * 11 has 3 prime factors (counted with multiplicity), 166 = 2 * 83 has 2 and 167 (which is prime) has 1.

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

Cf. A001222, A369139. Includes x-2 for x in A201147.

N:= 10^4: # for terms <= N
V:= map(numtheory:-bigomega, [$1..N+2]):
select(t -> V[t] = 1 + V[t+1] and V[t] = 2 + V[t+2], [$1..N]);

A369228 a(n) is the least k that starts a sequence of exactly n numbers on which i + Omega(i) is constant, where Omega = A001222.

Original entry on oeis.org

1, 4, 45, 104, 71874, 392274, 305778473, 24405534712

Offset: 1

Zak Seidov and Robert Israel, Jan 16 2024

a(n) is the least k such that Omega(k + j) = Omega(k) - j for 0 <= j <= n-1, but not for j = -1 or j = n.

Dickson's conjecture implies that a(n) exists for all n.

			Omega(1) = 0 while Omega(0) is undefined and Omega(2) = 1, so a(1) = 1.
Omega(4 .. 5) = (2, 1) while Omega(3) = 1 and Omega(6) = 2, so a(2) = 4.
Omega(45 .. 47) = (3, 2, 1) while Omega(44) = 3 and Omega(48) = 2  so a(3) = 45.
Omega(104 .. 107) = (4, 3, 2, 1) while Omega(103) = 1 and Omega(108) = 3 so a(4) = 104.
Omega(71874 .. 71878) = (7, 6, 5, 4, 3)  while Omega(71873) = 2 and Omega(71879) = 6 so a(5) = 71874.
Omega(392274 .. 392279) = (6, 5, 4, 3, 2, 1) while Omega(392273) = 2 and Omega(392280) = 5 so a(6) = 392274.
Omega(305778473 .. 305778479) = (7, 6, 5, 4, 3, 2, 1)  while Omega(305778472) = 4 and Omega(305778480) = 6 so a(7) = 305778473.

Cf. A001222, A369139, A369185.

V:= Vector(7): count:= 0: v:= 1 + numtheory:-bigomega(1); u:= 1;
for i from 2 while count < 7 do
    w:= i + numtheory:-bigomega(i);
    if w <> v then
      if V[i-u] = 0 then V[i-u]:= u; count:= count+1 fi;
      u:= i; v:= w;
    fi;
od:
convert(V,list);

a(8) from Daniel Suteu, Jan 18 2024

cp's OEIS Frontend

A369185 Numbers k such that Omega(k) = 1 + Omega(k+1) = 2 + Omega(k+2).

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