A339906 Numbers k for which A339812(2k) >= A339902(k).

1, 2, 4, 5, 8, 9, 10, 14, 16, 18, 32, 64, 65, 72, 84, 128, 129, 132, 136, 141, 145, 170, 256, 258, 261, 385, 448, 512, 516, 578, 642, 912, 1024, 1040, 1049, 1160, 1352, 2048, 4096, 4097, 4100, 4111, 4160, 4652, 4675, 4864, 5124, 5280, 8192, 8193, 8194, 8195, 8196, 8200, 8214, 8216, 8258, 8320, 8329, 8468, 8704

Offset: 1

Author

Antti Karttunen, Dec 21 2020

Keywords

• nonn

Comments

Terms of (1/2)*A048675(A339907(i)), for i >= 1, sorted into ascending order.

The first term not present in A339816 is 10, the second is 642; the first term of A339816 not present here is 12, the second is 21.

First terms with binary weights (A000120) w = 1..9 are: 1, 5, 14, 141, 4111, 25676, 41674, 1094530, 423297.

Examples

			10 ("1010" in binary) is present, because it encodes an odd squarefree number 5*11, for which phi(55) = 4*10 = 40, and bigomega(55-1) = 4 >= 4 = bigomega(40).
12 ("1100" in binary) is NOT present, because it encodes an odd squarefree number 7*11, for which phi(77) = 6*10 = 60, and bigomega(77-1) = 3 < 4 = bigomega(60).
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..514

Crossrefs

Cf. A000010, A001222, A019565, A048675, A339812, A339902, A339907.

Cf. A000079 (a subsequence).

Cf. also A339816.

Programs

PARI

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A339812(n) = bigomega(A019565(n)-1);
A339902(n) = { my(s=0, p=2); while(n>0, p = nextprime(1+p); if(n%2, s += bigomega(p-1)); n >>= 1); (s); };
isA339906(n) = (A339812(2*n) >= A339902(n));

PARI

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
isA339906(n) = { my(x=A019565(2*n)); (bigomega(eulerphi(x))<=bigomega(x-1)); };