A071538 -id:A071538 - cp's OEIS frontend

A089827 Number of subsets of {1,.., n} containing no twin prime pairs.

2, 4, 8, 16, 24, 48, 80, 160, 320, 640, 1280, 2560, 3840, 7680, 15360, 30720, 61440, 122880, 184320, 368640, 737280, 1474560, 2949120, 5898240, 11796480, 23592960, 47185920, 94371840, 188743680, 377487360, 566231040

Offset: 1

Reinhard Zumkeller, Nov 12 2003

nonn

a(n) = A000079(n) - A089828(n).

Cf. A071538, A001359, A006512.

a(n) = if n and n-2 are primes then (if n=7 then a(n-1)*5/3 else a(n-1)*3/2) else a(n-2)*2.

A089828 Number of subsets of {1,.., n} containing at least one twin prime pair.

Original entry on oeis.org

0, 0, 0, 0, 8, 16, 48, 96, 192, 384, 768, 1536, 4352, 8704, 17408, 34816, 69632, 139264, 339968, 679936, 1359872, 2719744, 5439488, 10878976, 21757952, 43515904, 87031808, 174063616, 348127232, 696254464, 1581252608

Offset: 1

Reinhard Zumkeller, Nov 12 2003

nonn

a(n) = A000079(n)-A089827(n).

Cf. A071538, A001359, A006512.

A071701 Number of twin prime pairs <= n of form (4k+1,4k+3), k>0.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 1

Reinhard Zumkeller, Jun 04 2002

nonn

As for A071538 the convention is followed that a twin prime pair is <= n if its smaller member is <= n.

			a(30)=3, since (29,31) is included along with (5,7) and (17,19).

a(n) = A071538(n) - A071702(n), cf. A071695, A071696.

A089881 Number of primes <= n that are members of twin prime pairs.

Original entry on oeis.org

0, 0, 0, 0, 2, 2, 3, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15

Offset: 1

Reinhard Zumkeller, Nov 13 2003

nonn

a(n) = if n<3 then 0 else (if n<7 then A071538(n-2)*2 else A071538(n-2)*2-1).

			a(30) = 7 = #{3,5,7,11,13,17,19}, as (3,5), (5,7), (11,13) and (17,19) are all twin prime pairs with members not greater than 30,
a(31) = 9 = #{3,5,7,11,13,17,19,29,31} with two more members for the next twin prime pair (29,31).

Cf. A001359, A006512, A089882.

A089882 Number of subsets of {1, ..., n} containing exactly one twin prime pair.

Original entry on oeis.org

0, 0, 0, 0, 8, 16, 32, 64, 128, 256, 512, 1024, 768, 1536, 3072, 6144, 12288, 24576, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 20971520, 41943040, 83886080, 167772160

Offset: 1

Reinhard Zumkeller, Nov 13 2003

nonn

Cf. A071538, A089827, A089828, A001359, A006512.

a(n) = ceiling(A089881(n)/2)*2^(n-A089881(n)).

A158288 The number of twin prime pairs in the n-th century.

Original entry on oeis.org

8, 7, 4, 2, 3, 2, 3, 0, 5, 0, 5, 1, 3, 2, 4, 0, 4, 2, 2, 3, 3, 3, 2, 3, 0, 2, 2, 3, 1, 1, 0, 2, 2, 4, 2, 4, 1, 1, 2, 2, 4, 2, 5, 1, 2, 2, 2, 2, 0, 2, 2, 0, 2, 0, 3, 2, 3, 1, 3, 0, 1, 2, 1, 1, 1, 2, 2, 4, 2, 2, 0, 1, 1, 3, 2, 3, 0, 1, 1, 1, 2, 0, 3, 1, 1, 2, 1, 0, 3, 1, 2, 0, 2, 1, 4, 0, 2, 2, 1, 1, 4, 1, 1, 2, 2

Offset: 1

Ki Punches, Mar 15 2009

nonn

The number of members of A001359 from 100*n-99 to 100*n-3. - Robert Israel, Jul 19 2017

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

Cf. A001359, A071538.

f:= proc(n) local P;
   P:= select(isprime, {seq(i,i=100*n-99..100*n-1,2)});
   nops(P intersect map(`+`,P,2))
end proc:
map(f, [$1..200]); # Robert Israel, Jul 19 2017

Table[Count[Prime@ Range[Boole[n == 1] + PrimePi[100 (n - 1) + 1], PrimePi[100 n] - 1], ?(PrimeQ[# + 2] &)], {n, 105}] (* _Michael De Vlieger, Jul 20 2017 *)

a(n)=my(s=0);forprime(p=100*n-99,100*n-2,if(isprime(p+2),s++));s \\ Charles R Greathouse IV, Feb 03 2011

from sympy import primerange, isprime
def a(n):
    s=0
    for p in primerange(100*n - 99, 100*n - 1):
        if isprime(p + 2):s+=1
    return s
print([a(n) for n in range(1, 201)]) # Indranil Ghosh, Jul 20 2017, after PARI code

a(n) = A071538(100*n-3) - A071538(100*n-100). - Robert Israel, Jul 19 2017

Corrected by Ray Chandler, May 23 2009

A113609 Number of prime powers q<=n such that also q+2 is a prime power.

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16

Offset: 1

Reinhard Zumkeller, Jan 14 2006

nonn

a(n) > A071538(n);

(25,27) is the smallest pair of prime powers (q,q+2) such that both q and q+2 are not primes, conjecture: there are more (but not < 10^6).

			a(10) = Card{(p^0,3), (2,2^2), (3,5), (5,7), (7,3^2), (3^2,11)} = 6.

Eric Weisstein's World of Mathematics, Prime Power
Index entries for sequences related to numbers of primes in various ranges

Cf. A000961, A071538.

ispp(n) = (n==1) || isprimepower(n);
a(n) = sum(k=1, n, ispp(k) && ispp(k+2)); \\ Michel Marcus, Jun 24 2019

A344117 Number of twin prime pairs in the range (6n + 1, 6(n + m) + 1], where m is the number of twin prime pairs, 6*k +- 1 for k = 1, 2, ..., n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 3, 3, 4, 4, 3, 3, 4, 4, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 4, 4, 4, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5

Offset: 1

Ya-Ping Lu, Jun 24 2021

nonn

Conjecture: a(n) >= 1.

Cf. A071538, A079629, A080356, A237840, A332086, A337788.

from sympy import isprime
def istwin(m): return 1 if isprime(6*m-1)*isprime(6*m+1) == 1 else 0
ct1 = 0
for n in range(1, 100):
    ct1 += istwin(n); ct = 0
    for m in range (n + 1, n + ct1 + 1): ct += istwin(m)
    print(ct)

cp's OEIS Frontend

A089827 Number of subsets of {1,.., n} containing no twin prime pairs.

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A089828 Number of subsets of {1,.., n} containing at least one twin prime pair.

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A071701 Number of twin prime pairs <= n of form (4*k+1,4*k+3), k>0.

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A089881 Number of primes <= n that are members of twin prime pairs.

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A089882 Number of subsets of {1, ..., n} containing exactly one twin prime pair.

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A158288 The number of twin prime pairs in the n-th century.

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A113609 Number of prime powers q<=n such that also q+2 is a prime power.

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PARI

A344117 Number of twin prime pairs in the range (6*n + 1, 6*(n + m) + 1], where m is the number of twin prime pairs, 6*k +- 1 for k = 1, 2, ..., n.

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Python

A071701 Number of twin prime pairs <= n of form (4k+1,4k+3), k>0.

A344117 Number of twin prime pairs in the range (6n + 1, 6(n + m) + 1], where m is the number of twin prime pairs, 6*k +- 1 for k = 1, 2, ..., n.