A095283 -id:A095283 - cp's OEIS frontend

A095293 Number of A095283-primes in range ]2^n,2^(n+1)].

0, 2, 1, 4, 4, 9, 14, 29, 52, 90, 167, 312, 577, 1072, 2026, 3776, 7177, 13585, 25726, 49125, 93569, 178735, 342381, 657180, 1262818, 2430501, 4684999, 9040931, 17470670, 33798206, 65455531, 126889351, 246216909

Offset: 1

Antti Karttunen, Jun 04 2004

nonn

As expected, the ratio a(n)/A095292(n) seems to approach 2: 0, 0, 1, 4, 1.33333, 2.25, 1.55556, 2.07143, 2.26087, 1.91489, 1.89773, 2.05263, 1.95593, 1.98519, 2.01793, 1.95344, 2.00924, 1.99633, 1.99287, 2.0083, 2.00075, 1.99746, 1.99841, 1.99971, 2.00034, 2.00001, 2.00018, 1.99977, 1.99971, 1.99997, 2.00004, 1.99995, 2.00003

A. Karttunen and J. Moyer: C-program for computing the initial terms of this sequence
Index entries for sequences related to occurrences of various subsets of primes in range 2^n,2^(n+1)

a(n) = A036378(n)-A095292(n). Cf. A095005.

a(n)=my(s);forprime(p=2^n+1,2^(n+1), s+=valuation(p+1, 2)%2); s \\ Charles R Greathouse IV, Oct 09 2013

A095282 Primes whose binary-expansion ends with an even number of 1's.

Original entry on oeis.org

2, 3, 11, 19, 43, 47, 59, 67, 79, 83, 107, 131, 139, 163, 179, 191, 211, 227, 239, 251, 271, 283, 307, 331, 347, 367, 379, 419, 431, 443, 463, 467, 491, 499, 523, 547, 563, 571, 587, 619, 643, 659, 683, 691, 719, 739, 751, 787, 811, 827, 859

Offset: 1

Antti Karttunen, Jun 04 2004

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Intersection of A000040 & (complement of A079523). Complement of A095283 in A000040. Cf. A027699, A095292.

q:= proc(n) local i, l, r; l, r:= convert(n, base, 2), 0;
      for i to nops(l) while l[i]=1 do r:=r+1 od; is(r, even)
    end:
select(q, [ithprime(i)$i=1..200])[];  # Alois P. Heinz, Dec 15 2019

been1Q[n_]:=Module[{c=Split[IntegerDigits[n,2]][[-1]]},c[[1]]==1&&EvenQ[ Length[ c]]]; Join[{2},Select[Prime[Range[150]],been1Q]] (* Harvey P. Dale, Dec 14 2019 *)

is(n)=valuation(n+1,2)%2==0 && isprime(n) \\ Charles R Greathouse IV, Oct 09 2013

A231271 Numbers k such that k and k+d are both odious (A000069) or both evil (A001969) for every divisor d of k.

Original entry on oeis.org

1, 5, 7, 9, 13, 17, 23, 29, 31, 37, 41, 49, 53, 61, 71, 73, 85, 89, 97, 101, 103, 109, 113, 119, 125, 127, 129, 133, 137, 149, 151, 157, 167, 173, 181, 193, 197, 199, 217, 223, 229, 233, 241, 249, 257, 263, 269, 277, 281, 293, 311, 313, 317, 321, 325, 337, 341

Offset: 1

Vladimir Shevelev, Nov 06 2013

A prime p is a term iff its binary expansion ends in odd number of 1's (A095283). All terms are in A079523.

			The odious number k = 341 has divisors {1, 11, 31, 341}. Since the numbers 341 + 1 = 342, 341 + 11 = 352, 341 + 31 = 372, 341 + 341 = 682 are all odious, then 341 is a term.

Peter J. C. Moses, Table of n, a(n) for n = 1..10000

Cf. A000069, A001969, A079523, A095283.

odiousQ[n_] := OddQ[DigitCount[n, 2][[1]]];selQ[n_] := Length[Union[Map[odiousQ, Flatten[{n, Map[n+#&, Divisors[n]]}]]]] == 1; Select[Range[200], selQ] (* Peter J. C. Moses, Nov 08 2013 *)

is(k) = {my(hw = hammingweight(k) % 2); fordiv(k, d, if(hammingweight(k+d) % 2 != hw, return(0))); 1;} \\ Amiram Eldar, Aug 12 2024

More terms from Peter J. C. Moses, Nov 08 2013

cp's OEIS Frontend

A095293 Number of A095283-primes in range ]2^n,2^(n+1)].

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A095282 Primes whose binary-expansion ends with an even number of 1's.

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A231271 Numbers k such that k and k+d are both odious (A000069) or both evil (A001969) for every divisor d of k.

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