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Numbers k such that omega(k) = omega(k - omega(k)) where omega(k) is the number of distinct primes dividing k.
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%I #12 Nov 22 2023 22:30:09

%S 1,3,4,5,8,9,12,14,17,20,22,24,26,28,32,35,36,38,40,46,48,50,52,54,56,

%T 57,58,65,74,76,77,82,87,88,93,94,95,96,98,100,105,106,108,117,118,

%U 119,124,128,135,136,143,144,145,146,147,148,155,160,161,162,164

%N Numbers k such that omega(k) = omega(k - omega(k)) where omega(k) is the number of distinct primes dividing k.

%C omega is the function in A001221. If there are infinitely many primes p such that p and 2p-1 are primes (see A005382), then this sequence is infinite. Proof: the numbers of the form 4p are in a subsequence if p and 2p-1 are both prime, because from the property that omega(4p) = 2 and omega(p(2p-1)) = 2, if n = 4p then omega(n-omega(n)) = omega(4p - 2) = omega(2(2p-1)) = 2 = omega(n).

%H G. C. Greubel, <a href="/A207005/b207005.txt">Table of n, a(n) for n = 1..5000</a>

%e 12 is in the sequence because omega(12) = 2, omega(12 - 2) = omega(10) = 2.

%t Select[Range[10^4],PrimeNu[#]==PrimeNu[#-PrimeNu[#]]&]

%Y Cf. A001221, A005382.

%K nonn

%O 1,2

%A _Michel Lagneau_, Feb 14 2012