A230070 -id:A230070 - cp's OEIS frontend

A230120 a(n) is the number of evil integers (A001969) not exceeding n and prime to n.

0, 0, 0, 1, 1, 1, 3, 2, 1, 2, 5, 1, 6, 3, 0, 4, 7, 2, 9, 3, 4, 5, 10, 3, 8, 6, 5, 7, 13, 3, 15, 8, 6, 8, 12, 4, 18, 9, 7, 8, 20, 4, 20, 10, 5, 11, 23, 5, 24, 9, 8, 12, 25, 6, 19, 14, 11, 14, 29, 5, 30, 15, 12, 16, 22, 7, 33, 15, 12, 12, 34, 8, 36, 18, 10, 18

Offset: 1

Vladimir Shevelev, Oct 10 2013

See comment in A230070, taking into account the equality a(n) = phi(n) - A230070(n), where phi(n) is Euler totient function (A000010).

Georg Fischer, Table of n, a(n) for n = 1..5000

Cf. A000010, A001969, A230070, A027697, A027699.

upTo[n_] := Block[{c, i, e = Select[Range[n], EvenQ@ DigitCount[#, 2, 1] &]}, Table[c = 0; i = 1; While[i <= Length@ e && e[[i]] < k, c += Boole@ CoprimeQ[e[[i]], k]; i++]; c, {k, n}]]; upTo[100] (* Giovanni Resta, Apr 14 2025 *)

a(n) = sum(k = 1, n, gcd(k, n) == 1 && !(hammingweight(k) % 2)); \\ Amiram Eldar, Nov 10 2024

For odd evil prime p (A027699), a(p) = (p-3)/2; for odd odious prime p (A027697), a(p) = (p-1)/2.

A256651 Numbers k for which both the number of evil numbers <= k and relatively prime to k, and the number of odious numbers <= k and relatively prime to k, are primes.

Original entry on oeis.org

7, 8, 10, 11, 14, 20, 22, 24, 27, 28, 30, 39, 45, 46, 47, 48, 59, 60, 66, 81, 94, 96, 107, 118, 124, 147, 166, 167, 172, 179, 189, 192, 200, 203, 210, 214, 225, 227, 231, 247, 259, 277, 285, 296, 304, 309, 315, 323, 324, 325, 327, 334, 358, 360, 407, 412, 436

Offset: 1

Vladimir Shevelev, Apr 06 2015

For odious-evil partition of positive integers, see A000069, A001969; for odious-evil partition of relatively prime to n numbers <=n, see A230070, A230120.

Since Euler totient sequence A000010 takes only even values, then for some of them we have a representations as a sum of two primes.

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Cf. A000010, A000069, A001969, A230070, A230120.

is(k) = {my(c1 = 0, c2 = 0, h); for(i = 1, k, if(gcd(i, k) == 1, h = hammingweight(i) % 2; c1 += h; c2 += !h)); isprime(c1) && isprime(c2);} \\ Amiram Eldar, Nov 10 2024

More terms from Peter J. C. Moses, Apr 06 2015

cp's OEIS Frontend

A230120 a(n) is the number of evil integers (A001969) not exceeding n and prime to n.

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