A067775 -id:A067775 - cp's OEIS frontend

A141427 a(n) = phi(A067775(n)).

1, 4, 10, 16, 22, 28, 30, 40, 46, 52, 58, 60, 70, 72, 82, 88, 100, 106, 112, 130, 136, 138, 148, 150, 156, 166, 172, 178, 180, 190, 196, 198, 210, 226, 232, 238, 240, 250, 256, 262, 268, 270, 280, 282, 292, 310, 316, 330, 336, 346, 352, 358, 366, 372, 382, 388

Offset: 1

Giovanni Teofilatto, Aug 06 2008

nonn

Count of numbers smaller than and coprime to p for primes p such that p + 4 is composite.

Subsequence of A006093.

Cf. A067775, A000010 (Euler totient function), A006093 (primes minus 1), A141426, A141515.

{forprime(p=2, 400, if(!isprime(p+4), print1(eulerphi(p), ",")))} \\ Klaus Brockhaus, Aug 31 2008

More terms from Robert G. Wilson v, Aug 18 2008

Edited by N. J. A. Sloane and Klaus Brockhaus, Aug 31 2008

A023200 Primes p such that p + 4 is also prime.

Original entry on oeis.org

3, 7, 13, 19, 37, 43, 67, 79, 97, 103, 109, 127, 163, 193, 223, 229, 277, 307, 313, 349, 379, 397, 439, 457, 463, 487, 499, 613, 643, 673, 739, 757, 769, 823, 853, 859, 877, 883, 907, 937, 967, 1009, 1087, 1093, 1213, 1279, 1297, 1303, 1423, 1429, 1447, 1483

Offset: 1

David W. Wilson

nonn

Smaller member p of cousin prime pairs (p, p+4).
A015913 contains the composite number 305635357, so it is different from both the present sequence and A029710. (305635357 is the only composite member of A015913 < 10^9.) - Jud McCranie, Jan 07 2001
Apart from the first term, all terms are of the form 6n + 1.
Complement of A067775 (primes p such that p + 4 is composite) with respect to A000040 (primes). With prime 2 also primes p such that q^2 + p is prime for some prime q (q = 3 if p = 2, q = 2 if p > 2). Subsequence of A232012. - Jaroslav Krizek, Nov 23 2013
Conjecture: The sequence is infinite and for every n, a(n+1) < a(n)^(1+1/n). Namely a(n)^(1/n) is a strictly decreasing function of n. - Jahangeer Kholdi and Farideh Firoozbakht, Nov 24 2014
From Alonso del Arte, Jan 12 2019: (Start)
If p splits in Z[sqrt(-2)], p + 4 is an inert prime in that domain. Likewise, if p splits in Z[sqrt(2)], p + 4 is an inert prime in that domain.
The only way for p or p + 4 to split in both domains is if it is congruent to 1 modulo 24, in which case the other prime is inert in both domains.
For example, 3 = (1 - sqrt(-2))*(1 + sqrt(-2)) but is inert in Z[sqrt(2)], while 7 = (3 - sqrt(2))*(3 + sqrt(2)) but is inert in Z[sqrt(-2)]. And also 11 = (3 - sqrt(-2))*(3 + sqrt(-2)) but 15 is composite in Z or any quadratic integer ring.
And 97 = (5 - 6*sqrt(-2))*(5 + 6*sqrt(-2)) = (1 - 7*sqrt(2))*(1 + 7*sqrt(2)), but 101 is inert in both Z[sqrt(-2)] and Z[sqrt(2)]. (End)

T. D. Noe, Table of n, a(n) for n = 1..10000
Andrew Granville and Greg Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
H. J. Weber, A Sieve for Cousin Primes, arXiv:1204.3795v1 [math.NT], 2012.
Eric Weisstein's World of Mathematics, Cousin Primes
Eric Weisstein's World of Mathematics, Twin Primes
Wikipedia, Cousin prime.
Index entries for primes, gaps between

Exactly the same as A029710 except for the exclusion of 3.

Cf. A000010, A003557, A007947, A046132, A098429, A000040, A010051.

a023200 n = a023200_list !! (n-1)
a023200_list = filter ((== 1) . a010051') $
               map (subtract 4) $ drop 2 a000040_list
-- Reinhard Zumkeller, Aug 01 2014

[p: p in PrimesUpTo(1500) | NextPrime(p)-p eq 4]; // Bruno Berselli, Apr 09 2013

A023200 := proc(n) option remember; if n = 1 then 3; else p := nextprime(procname(n-1)) ; while not isprime(p+4) do p := nextprime(p) ;  end do: p ; end if; end proc: # R. J. Mathar, Sep 03 2011

Select[Range[10^2], PrimeQ[#] && PrimeQ[# + 4] &] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)
Select[Prime[Range[250]],PrimeQ[#+4]&] (* Harvey P. Dale, Oct 09 2023 *)

print1(3);p=7;forprime(q=11,1e3,if(q-p==4,print1(", "p)); p=q) \\ Charles R Greathouse IV, Mar 20 2013

a(n) = A046132(n) - 4 = A087679(n) - 2.

a(n) >> n log^2 n via the Selberg sieve. - Charles R Greathouse IV, Nov 20 2016

Definition modified by Vincenzo Librandi, Aug 02 2009

Definition revised by N. J. A. Sloane, Mar 05 2010

A140555 Primes p such that p + 6 is not a prime.

Original entry on oeis.org

2, 3, 19, 29, 43, 59, 71, 79, 89, 109, 113, 127, 137, 139, 149, 163, 179, 181, 197, 199, 211, 229, 239, 241, 269, 281, 283, 293, 313, 317, 337, 349, 359, 379, 389, 397, 401, 409, 419, 421, 431, 439, 449, 463, 467, 479, 487, 491, 499, 509, 521, 523, 547, 569

Offset: 1

Juri-Stepan Gerasimov, Jul 03 2008

nonn

Note that if Goldbach's Conjecture (2n = p1 + p2 for all n>=2) is false and K is the smallest value of n for which it fails, then for 2(K-3) = p3 + p4, the primes p3 and p4 must be taken from this list. See also A067775. - Keith Backman, Apr 05 2012

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A000040, A023201, A067775.

Select[Prime[Range[200]],!PrimeQ[#+6]&] (* Harvey P. Dale, Dec 21 2016 *)

forprime(p=2,600,if(!isprime(p+6),print1(p,","))) \\ Klaus Brockhaus, Aug 12 2008

A000040 SET MINUS A023201. - R. J. Mathar, Aug 09 2008

Corrected by R. J. Mathar and Klaus Brockhaus, Aug 12 2008

A232009 a(n) = the smallest squarefree number (from A005117) of the form p*q with prime factors in a p^2 + n progression, or 0 if no such number exists.

Original entry on oeis.org

10, 33, 14, 39, 0, 155, 22, 51, 26, 57, 0, 185, 34, 69, 38, 205, 0, 215, 46, 87, 0, 93, 0, 511, 58, 0, 62, 111, 0, 553, 0, 123, 74, 129, 0, 305, 82, 141, 86, 623, 0, 335, 94, 159, 0, 355, 0, 365, 106, 177, 0, 183, 0, 395, 118, 0, 122, 201, 0, 763, 0, 213, 134

Offset: 1

Jaroslav Krizek, Nov 20 2013

nonn

a(n) = the smallest squarefree number m of the form p*q with prime factors p and q = p^2 + n, or 0 if no such number exists; m = p^3 + p*n.

a(n) = 0 for numbers n from A232010.

Cf. A005117, A232010, A067775, A232012, A023200.

a(4) = 39 because 39 = 3 * 13 = 3 * (3^2 + 4).

A232010 Numbers k such that p^2 + k is composite for all prime p.

Original entry on oeis.org

5, 11, 17, 21, 23, 26, 29, 31, 35, 41, 45, 47, 51, 53, 56, 59, 61, 65, 68, 71, 73, 77, 81, 83, 86, 87, 89, 91, 95, 101, 107, 110, 111, 113, 115, 116, 117, 119, 121, 125, 129, 131, 134, 137, 139, 141, 143, 146, 149, 151, 152, 155, 157, 161, 165, 167, 171, 173, 176

Offset: 1

Jaroslav Krizek, Nov 20 2013

nonn

Primes from this sequence are in A067775.

Complement of A232012.

			7 is not in the sequence because 2^2 + 7 = 11 (prime).

Cf. A232009, A067775, A232012, A023200.

A232009(a(n)) = 0.

cp's OEIS Frontend

A141427 a(n) = phi(A067775(n)).

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A023200 Primes p such that p + 4 is also prime.

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A140555 Primes p such that p + 6 is not a prime.

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A232009 a(n) = the smallest squarefree number (from A005117) of the form p*q with prime factors in a p^2 + n progression, or 0 if no such number exists.

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A232010 Numbers k such that p^2 + k is composite for all prime p.

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