Displaying 1-10 of 24 results found.
0, 0, 0, 1, 1, 0, 2, 2, 2, 1, 2, 2, 1, 3, 4, 3, 3, 2, 2, 3, 3, 3, 4, 0, 1, 5, 4, 4, 7, 7, 8, 7, 6, 8, 4, 4, 6, 4, 5, 5, 3, 4, 3, 6, 6, 9, 10, 9, 11, 11, 13, 8, 8, 8, 7, 8, 10, 9, 8, 10, 12, 12, 10, 11, 9, 8, 7, 8, 8, 5, 7, 6, 8, 7, 8, 9, 9, 13, 13, 11, 11, 12, 13, 11, 12, 13, 9, 11, 10, 11, 10, 6, 9
COMMENTS
Conjecture: a(n) > 0, except n=1,2,3,6,24, when a(n)=0.
This fact can be apparently explained by existence of twin primes. (End)
Primes of the form 6k-1.
(Formerly M3809)
+10
128
5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401, 419, 431, 443, 449, 461, 467, 479, 491, 503, 509, 521, 557, 563, 569, 587
COMMENTS
Also primes p such that p^q - 2 is not prime where q is an odd prime. These numbers cannot be prime because the binomial p^q = (6k-1)^q expands to 6h-1 some h. Then p^q-2 = 6h-1-2 is divisible by 3 thus not prime. - Cino Hilliard, Nov 12 2008 There exists a polygonal number P_s(3) = 3s - 3 = a(n) + 1. These are the only primes p with P_s(k) = p + 1, s >= 3, k >= 3, since P_s(k) - 1 is composite for k > 3. - Ralf Steiner, May 17 2018 A theorem due to Andrzej Mąkowski: every integer greater than 161 is the sum of distinct primes of the form 6k-1. Examples: 162 = 5 + 11 + 17 + 23 + 47 + 59; 163 = 17 + 23 + 29 + 41 + 53. (See Sierpiński and David Wells.)
Except for 2 and 3, all Sophie Germain primes are of the form 6k-1.
Except for 3, all the lesser of twin primes are also of the form 6k-1.
Dirichlet's theorem on arithmetic progressions states that this sequence is infinite. (End)
For all elements of this sequence p=6*k-1, there are no (x,y) positive integers such that k=6*x*y-x+y. - Pedro Caceres, Apr 06 2019
REFERENCES
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
A. Mąkowski, Partitions into unequal primes, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 8 (1960), 125-126.
Wacław Sierpiński, Elementary Theory of Numbers, p. 144, Warsaw, 1964.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, Revised edition, 1997, p. 127.
LINKS
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
FORMULA
Product_{k>=1} (1 - 1/a(k)^2) = 9* A175646/Pi^2 = 1/1.060548293.... =4/(3* A333240). Product_{k>=1} (1 + 1/a(k)^2) = A334482. Product_{k>=1} (1 - 1/a(k)^3) = A334480. Product_{k>=1} (1 + 1/a(k)^3) = A334479. (End) Legendre symbol (-3, a(n)) = -1 and (-3, A002476(n)) = +1, for n >= 1. For prime 3 one sets (-3, 3) = 0. - Wolfdieter Lang, Mar 03 2021
MAPLE
select(isprime, [seq(6*n-1, n=1..100)]); # Muniru A Asiru, May 19 2018
PROG
(Haskell)
a007528 n = a007528_list !! (n-1)
a007528_list = [x | k <- [0..], let x = 6 * k + 5, a010051' x == 1]
(GAP) Filtered(List([1..100], n->6*n-1), IsPrime); # Muniru A Asiru, May 19 2018
CROSSREFS
Prime sequences A# (k,r) of the form k*n+r with 0 <= r <= k-1 (i.e., primes == r (mod k), or primes p with p mod k = r) and gcd(r,k)=1: A000040 (1,0), A065091 (2,1), A002476 (3,1), A003627 (3,2), A002144 (4,1), A002145 (4,3), A030430 (5,1), A045380 (5,2), A030431 (5,3), A030433 (5,4), A002476 (6,1), this sequence (6,5), A140444 (7,1), A045392 (7,2), A045437 (7,3), A045471 (7,4), A045458 (7,5), A045473 (7,6), A007519 (8,1), A007520 (8,3), A007521 (8,5), A007522 (8,7), A061237 (9,1), A061238 (9,2), A061239 (9,4), A061240 (9,5), A061241 (9,7), A061242 (9,8), A030430 (10,1), A030431 (10,3), A030432 (10,7), A030433 (10,9), A141849 (11,1), A090187 (11,2), A141850 (11,3), A141851 (11,4), A141852 (11,5), A141853 (11,6), A141854 (11,7), A141855 (11,8), A141856 (11,9), A141857 (11,10), A068228 (12,1), A040117 (12,5), A068229 (12,7), A068231 (12,11). Cf. A034694 (smallest prime == 1 (mod n)). Cf. A038700 (smallest prime == n-1 (mod n)). Cf. A038026 (largest possible value of smallest prime == r (mod n)).
Numbers m such that 6m-1, 6m+1 are twin primes.
(Formerly M0641 N0235)
+10
91
1, 2, 3, 5, 7, 10, 12, 17, 18, 23, 25, 30, 32, 33, 38, 40, 45, 47, 52, 58, 70, 72, 77, 87, 95, 100, 103, 107, 110, 135, 137, 138, 143, 147, 170, 172, 175, 177, 182, 192, 205, 213, 215, 217, 220, 238, 242, 247, 248, 268, 270, 278, 283, 287, 298, 312, 313, 322, 325
COMMENTS
6m-1 and 6m+1 are twin primes iff m is not of the form 6ab +- a +- b. - Jon Perry, Feb 01 2002 Even terms correspond to twin primes of the form (4k - 1, 4k + 1), odd terms to twin primes of the form (4k + 1, 4k + 3). - Lekraj Beedassy, Apr 03 2002 Except for a(1)=1, all numbers in this sequence are congruent to (0, 2 or 3) mod 5.
It appears that when a(n)=6j, then j is also in the sequence (e.g., 138 = 6*23; 312 = 6*52). This also appears to hold for sequence A191626. If true, then it suggests that when seeking large twin primes, good candidates might be 36*a(n) +- 1, n >= 2. Conjecture: There is at least one number in the sequence in the interval [5k, 7k] inclusive, k >= 1. If true, then the twin prime conjecture also is true.
(End)
A counterexample to "It appears that ...": Take j = 63. Then 6j = 378 and 36j = 2268. Now 379, 2267, and 2269 are prime, but 377 = 13 * 29. The sequence of counterexamples is A263282. - Jason Kimberley, Oct 13 2015 Dinculescu calls all terms in the sequence "twin ranks", and all other positive integers "non-ranks", see links. Non-ranks are given by the formula kp +- round(p/6) for positive integers k and primes p > 4, while twin ranks (this sequence) cannot be represented as kp +- round(p/6) for any k, p > 4. Here round(p/6) is the nearest integer to p/6. - Alexei Kourbatov, Jan 03 2015 Number of terms less than 10^k: 0, 5, 25, 142, 810, 5330, 37915, ... - Muniru A Asiru, Jan 24 2018 6m-1 and 6m+1 are twin primes iff 36m^2-1 is semiprime. It is algebraically provable that 36m^2-1 having any factor of the form 6k+-1 is equivalent to the statement that m is congruent to +-k (mod (6k+-1)). Other than the trivial case m=k, the fact of such a congruence means 36m^2-1 has a factor other than 6m-1 and 6m+1, and is not semiprime. Thus, {a(n)} lists the numbers m such that for all k < m, m is not congruent to +-k modulo (6k+-1). This is an alternative formulation of the results of Dinculescu referenced above. - Keith Backman, Apr 25 2021 Other than a(1)=1, it is provable that a(n) is not a square unless it is a multiple of 5, and a(n) is not a cube unless it is a multiple of 7. Examples of the former include a(11)=5^2=25, a(26)=10^2=100, and a(166)=35^2=1225; examples of the latter are rarer, including a(1531)=28^3=21952 and a(4163)=42^3=74088. - Keith Backman, Jun 26 2021
REFERENCES
W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, p. 69.
W. Sierpiński, A Selection of Problems in the Theory of Numbers. Macmillan, NY, 1964, p. 120.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
MAPLE
select(n -> isprime(6*n-1) and isprime(6*n+1), [$1..1000]); # Robert Israel, Jan 11 2015
MATHEMATICA
Select[ Range[350], PrimeQ[6# - 1] && PrimeQ[6# + 1] & ]
Select[Range[400], AllTrue[6#+{1, -1}, PrimeQ]&] (* Harvey P. Dale, Jul 27 2022 *) #/6&/@Select[Range[6, 2500, 6], AllTrue[#+{1, -1}, PrimeQ]&] (* Harvey P. Dale, Mar 31 2023 *)
PROG
(Magma) [n: n in [1..200] | IsPrime(6*n+1) and IsPrime(6*n-1)] // Vincenzo Librandi, Nov 21 2010 (PARI) p=5; forprime(q=5, 1e4, if(q-p==2, print1((p+1)/6", ")); p=q); \\ Altug Alkan, Oct 13 2015 (PARI) list(lim)=my(v=List(), p=5); forprime(q=7, 6*lim+1, if(q-p==2, listput(v, q\6)); p=q); Vec(v) \\ Charles R Greathouse IV, Dec 03 2016 (Haskell)
a002822 n = a002822_list !! (n-1)
a002822_list = f a000040_list where
f (q:ps'@(p:ps)) | p > q + 2 || r > 0 = f ps'
| otherwise = y : f ps where (y, r) = divMod (q + 1) 6
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Mar 27 2001
Numbers k such that 6*k - 1 is composite.
+10
13
6, 11, 13, 16, 20, 21, 24, 26, 27, 31, 34, 35, 36, 37, 41, 46, 48, 50, 51, 54, 55, 56, 57, 61, 62, 63, 66, 68, 69, 71, 73, 76, 79, 81, 83, 86, 88, 89, 90, 91, 92, 96, 97, 101, 102, 104, 105, 106, 111, 112, 115, 116, 118, 119, 121, 122, 123, 125, 126, 128
COMMENTS
These numbers can be written as 6*x*y + x - y for x > 0, y > 0. - Ron R Spencer, Aug 01 2016
EXAMPLE
a(1)=6 because 6*6 - 1 = 35, which is composite.
MAPLE
remove(k-> isprime(6*k-1), [$1..130])[]; # Muniru A Asiru, Feb 22 2019
PROG
(Haskell)
a046953 n = a046953_list !! (n-1)
a046953_list = map (`div` 6) $
filter ((== 0) . a010051' . subtract 1) [6, 12..]
(Magma) [n: n in [1..200] | not IsPrime(6*n-1)]; // G. C. Greubel, Feb 21 2019 (Sage) [n for n in (1..200) if not is_prime(6*n-1)] # G. C. Greubel, Feb 21 2019 (GAP) Filtered([1..200], k-> not IsPrime(6*k-1)) # G. C. Greubel, Feb 21 2019
Numbers n such that 6n+1 and 6n+5 are both primes.
+10
12
1, 2, 3, 6, 7, 11, 13, 16, 17, 18, 21, 27, 32, 37, 38, 46, 51, 52, 58, 63, 66, 73, 76, 77, 81, 83, 102, 107, 112, 123, 126, 128, 137, 142, 143, 146, 147, 151, 156, 161, 168, 181, 182, 202, 213, 216, 217, 237, 238, 241, 247, 248, 258, 261, 263, 266, 268, 277, 282
COMMENTS
Note that if prime p>3 then p mod 6 = 1 or 5.
EXAMPLE
a(2)=2 since 6*2+1=13 and 6*2+5=17 are both prime.
MATHEMATICA
Select[Range[300], And @@ PrimeQ /@ ({1, 5} + 6#) &] (* Ray Chandler, Jun 29 2008 *)
PROG
(PARI) is(n)=isprime(n*6+1)&&isprime(n*6+5) \\ M. F. Hasler, Apr 05 2017
Numbers n such that 12n + 1 is prime.
+10
11
1, 3, 5, 6, 8, 9, 13, 15, 16, 19, 20, 23, 26, 28, 29, 31, 33, 34, 35, 36, 38, 45, 48, 50, 51, 55, 56, 59, 61, 63, 64, 69, 71, 73, 78, 83, 84, 85, 86, 89, 91, 93, 94, 96, 100, 101, 103, 104, 108, 110, 115, 119, 121, 124, 129, 133, 134, 135, 138, 139, 141, 145, 146, 148
EXAMPLE
If n=96 then 12*n + 1 = 1153 (prime).
PROG
(Magma) [ n: n in [1..150] | IsPrime(12*n+1) ]; // Klaus Brockhaus, Jan 02 2009
Primes p such that 6*p-1 is also prime.
+10
10
2, 3, 5, 7, 17, 19, 23, 29, 43, 47, 53, 59, 67, 103, 107, 109, 113, 127, 137, 157, 163, 197, 199, 227, 229, 239, 269, 283, 313, 317, 347, 359, 373, 379, 383, 389, 397, 439, 443, 449, 457, 463, 467, 523, 569, 577, 593, 599, 613, 617, 647, 653, 709, 733, 743, 773
PROG
(Magma) [p: p in PrimesUpTo(800) | IsPrime(6*p-1)]; // Vincenzo Librandi, Apr 14 2013
EXTENSIONS
Edited by the Associate Editors of the OEIS, Apr 22 2009
Numbers n such that 6*n-1 is prime while 6*n+1 is composite.
+10
7
4, 8, 9, 14, 15, 19, 22, 28, 29, 39, 42, 43, 44, 49, 53, 59, 60, 64, 65, 67, 74, 75, 78, 80, 82, 84, 85, 93, 94, 98, 99, 108, 109, 113, 114, 117, 120, 124, 127, 129, 133, 140, 144, 148, 152, 155, 157, 158, 159, 162, 163, 164, 169, 183, 184, 185, 194, 197, 198, 199
MATHEMATICA
Select[Range[200], PrimeQ[6# -1] && !PrimeQ[6# +1] &] (* Ray Chandler, Aug 22 2006 *)
PROG
(PARI) for(n=1, 250, if(isprime(6*n-1) && !isprime(6*n+1), print1(n", "))) \\ G. C. Greubel, Feb 20 2019 (Magma) [n: n in [1..250] | IsPrime(6*n-1) and not IsPrime(6*n+1)]; // G. C. Greubel, Feb 20 2019 (Sage)[n for n in (1..250) if is_prime(6*n-1) and not is_prime(6*n+1)] # G. C. Greubel, Feb 20 2019 (GAP) Filtered([1..250], k-> IsPrime(6*k-1) and not IsPrime(6*k+1)) # G. C. Greubel, Feb 20 2019
a(n) = 1 iff 6n-1 is prime.
+10
6
1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0
EXAMPLE
a(5) = 1 because 6*5-1 is prime, a(6) = 0 since 6*6-1 is composite.
PROG
(Magma) [IsPrime(6*n-1) select 1 else 0: n in[1..100]]; // Vincenzo Librandi, Jan 19 2019
a(n) is the smallest prime factor of 6*n-1 that is congruent to 5 modulo 6.
+10
5
5, 11, 17, 23, 29, 5, 41, 47, 53, 59, 5, 71, 11, 83, 89, 5, 101, 107, 113, 17, 5, 131, 137, 11, 149, 5, 23, 167, 173, 179, 5, 191, 197, 29, 11, 5, 17, 227, 233, 239, 5, 251, 257, 263, 269, 5, 281, 41, 293, 23, 5, 311, 317, 17, 47, 5, 11, 347, 353, 359, 5, 53, 29, 383, 389, 5
COMMENTS
a(n) = 5 if n == 1 (mod 5).
a(n) = 6*n - 1 if n is in A024898. (End)
REFERENCES
G. Pólya and G. Szegő, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part Eight, Chap. 2, Section 2, Problem 96.
EXAMPLE
For n = 13, 6*n - 1 = 77 = 7*11; 7 == 1 (mod 6), but 11 == 5 (mod 6), so a(13) = 11.
MAPLE
f:= n -> min(select(p -> p mod 6 = 5, numtheory:-factorset(6*n-1))):
PROG
(PARI) for(k=1, 60, my(f=factor(6*k-1)[, 1]); for(j=1, #f, if(f[j]%6==5, print1(f[j], ", "); break))) \\ Hugo Pfoertner, Dec 25 2019
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