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Search: a039704 -id:a039704
Displaying 1-10 of 12 results found. page 1 2
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A288915 Run lengths in A039704. +20
1
1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 1, 1, 3, 1, 1, 2, 1, 4, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 3, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,9
COMMENTS
Is the sequence bounded?
On Dickson's conjecture this sequence is unbounded. Records: a(1) = 1, a(9) = 2, a(13) = 3, a(39) = 4, a(180) = 6, a(1348) = 7, a(6698) = 8, a(8156) = 10, a(20230) = 11, a(79011) = 12, a(99250) = 13, a(710895) = 15, a(2421600) = 16, a(7128444) = 17, a(11898707) = 18, a(14368535) = 20, a(21943755) = 22, a(519775979) = 25, a(3111006505) = 27. - Charles R Greathouse IV, Jun 19 2017
LINKS
Zak Seidov, Table of n, a(n) for n = 1..20000
MATHEMATICA
Length /@ Split[Mod[Prime[Range[100]], 6]]
PROG
(PARI) t=1; p=2; forprime(q=3, 1e3, if((q-p)%6==0, t++, print1(t", "); t=1); p=q) \\ Charles R Greathouse IV, Jun 19 2017
CROSSREFS
Cf. A039704, A057620, A057622.
KEYWORD
nonn
AUTHOR
Zak Seidov, Jun 19 2017
EXTENSIONS
a(70) corrected by Charles R Greathouse IV, Jun 19 2017
STATUS
approved
A039703 a(n) = n-th prime modulo 5. +10
16
2, 3, 0, 2, 1, 3, 2, 4, 3, 4, 1, 2, 1, 3, 2, 3, 4, 1, 2, 1, 3, 4, 3, 4, 2, 1, 3, 2, 4, 3, 2, 1, 2, 4, 4, 1, 2, 3, 2, 3, 4, 1, 1, 3, 2, 4, 1, 3, 2, 4, 3, 4, 1, 1, 2, 3, 4, 1, 2, 1, 3, 3, 2, 1, 3, 2, 1, 2, 2, 4, 3, 4, 2, 3, 4, 3, 4, 2, 1, 4, 4, 1, 1, 3, 4, 3, 4, 2, 1, 3, 2, 4, 2, 1, 4, 3, 4, 1, 3, 1, 2, 2, 3, 4, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
a(A049084(A045356(n-1))) = even; a(A049084(A045429(n-1))) = odd. - Reinhard Zumkeller, Feb 25 2008
LINKS
R. Zumkeller, Table of n, a(n) for n = 1..1000
MAPLE
seq(ithprime(n) mod 5, n=1..105); # Nathaniel Johnston, Jun 29 2011
MATHEMATICA
Table[Mod[Prime[n], 5], {n, 105}] (* Nathaniel Johnston, Jun 29 2011 *)
Mod[Prime[Range[100]], 5] (* Vincenzo Librandi, May 06 2014 *)
PROG
(PARI) primes(105) % 5 \\ Zak Seidov, Apr 09 2013
(Magma) [p mod 5: p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014
CROSSREFS
Cf. A000040, A039701, A039702, A039704-A039706, A038194, A007652, A039709-A039715.
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling
STATUS
approved
A242119 Primes modulo 18. +10
14
2, 3, 5, 7, 11, 13, 17, 1, 5, 11, 13, 1, 5, 7, 11, 17, 5, 7, 13, 17, 1, 7, 11, 17, 7, 11, 13, 17, 1, 5, 1, 5, 11, 13, 5, 7, 13, 1, 5, 11, 17, 1, 11, 13, 17, 1, 13, 7, 11, 13, 17, 5, 7, 17, 5, 11, 17, 1, 7, 11, 13, 5, 1, 5, 7, 11, 7, 13, 5, 7, 11, 17, 7, 13, 1, 5 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
FORMULA
Sum_{i=1..n} a(i) ~ 9n. The derivation is the same as in the formula in A039715. - Jerzy R Borysowicz, Apr 27 2022
MATHEMATICA
Mod[Prime[Range[100]], 18]
PROG
(Magma) [p mod(18): p in PrimesUpTo(500)];
(Sage) [mod(p, 18) for p in primes(500)] # Bruno Berselli, May 05 2014
CROSSREFS
Cf. sequences of the type Primes mod k: A039701 (k=3), A039702 (k=4), A039703 (k=5), A039704 (k=6), A039705 (k=7), A039706 (k=8), A038194 (k=9), A007652 (k=10), A039709 (k=11), A039710 (k=12), A039711 (k=13), A039712 (k=14), A039713 (k=15), A039714 (k=16), A039715 (k=17), this sequence (k=18), A033633 (k=19), A242120(k=20), A242121 (k=21), A242122 (k=22), A229786 (k=23), A229787 (k=24), A242123 (k=25), A242124 (k=26), A242125 (k=27), A242126 (k=28), A242127 (k=29), A095959 (k=30), A110923 (k=100).
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, May 05 2014
STATUS
approved
A039705 a(n) = n-th prime modulo 7. +10
10
2, 3, 5, 0, 4, 6, 3, 5, 2, 1, 3, 2, 6, 1, 5, 4, 3, 5, 4, 1, 3, 2, 6, 5, 6, 3, 5, 2, 4, 1, 1, 5, 4, 6, 2, 4, 3, 2, 6, 5, 4, 6, 2, 4, 1, 3, 1, 6, 3, 5, 2, 1, 3, 6, 5, 4, 3, 5, 4, 1, 3, 6, 6, 3, 5, 2, 2, 1, 4, 6, 3, 2, 3, 2, 1, 5, 4, 5, 2, 3, 6, 1, 4, 6, 5, 2, 1, 2, 6, 1, 5, 3, 4, 1, 2, 6, 5, 3, 5, 2, 1, 4, 3, 2, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
a(A049084(A045370(n-1))) is even; a(A049084(A045415(n-1))) is odd. - Reinhard Zumkeller, Feb 25 2008
LINKS
R. Zumkeller, Table of n, a(n) for n = 1..1000
MAPLE
seq(ithprime(n) mod 7, n=1..105); # Nathaniel Johnston, Jun 29 2011
MATHEMATICA
Table[Mod[Prime[n], 7], {n, 105}] (* Nathaniel Johnston, Jun 29 2011 *)
Mod[Prime[Range[100]], 7] (* Vincenzo Librandi, May 06 2014 *)
PROG
(Magma) [p mod(7): p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014
(PARI) primes(100)%7 \\ Charles R Greathouse IV, May 06 2014
CROSSREFS
Cf. A039701-A039704, A039706, A038194, A007652, A039709-A039715.
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling
STATUS
approved
A247478 Primes p such that (p^4 + 5)/6 is prime. +10
5
7, 11, 17, 29, 53, 71, 101, 109, 127, 179, 227, 241, 281, 307, 349, 487, 587, 647, 683, 727, 829, 1009, 1061, 1109, 1289, 1487, 1511, 1523, 1567, 1621, 1627, 1709, 1847, 1987, 2017, 2027, 2087, 2099, 2297, 2311, 2393, 2437, 2447, 2521, 2531, 2617, 2729, 2887, 2909, 2969, 3167, 3221, 3301, 3319, 3329, 3347, 3413, 3527 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
(p^4+5)/6 is an integer for all primes p>3, because then p == (1 or 5) (mod 6) as in A039704, therefore p^2 == 1 (mod 6) and finally p^4 == 1 (mod 6).
LINKS
Table of n, a(n) for n=1..58.
EXAMPLE
(7^4+5)/6 = 401 prime, (11^4+5)/6 = 2441 prime.
MATHEMATICA
Select[Prime[Range[10^3]], PrimeQ[(#^4 + 5) / 6] &] (* Vincenzo Librandi, Jan 21 2015 *)
PROG
(PARI) lista(nn) = {forprime(p=4, nn, if (isprime((p^4 + 5)/6), print1(p, ", ")); ); } \\ Michel Marcus, Jan 20 2015
(Magma) [p: p in PrimesInInterval(3, 4000) | IsPrime((p^4+5) div 6)]; // Vincenzo Librandi, Jan 21 2015
CROSSREFS
Cf. A118915.
KEYWORD
nonn
AUTHOR
Zak Seidov, Jan 19 2015
STATUS
approved
A079950 Triangle of n-th prime modulo twice primes less n-th prime. +10
4
2, 3, 3, 1, 5, 5, 3, 1, 7, 7, 3, 5, 1, 11, 11, 1, 1, 3, 13, 13, 13, 1, 5, 7, 3, 17, 17, 17, 3, 1, 9, 5, 19, 19, 19, 19, 3, 5, 3, 9, 1, 23, 23, 23, 23, 1, 5, 9, 1, 7, 3, 29, 29, 29, 29, 3, 1, 1, 3, 9, 5, 31, 31, 31, 31, 31, 1, 1, 7, 9, 15, 11, 3, 37, 37, 37, 37, 37, 1, 5, 1, 13, 19, 15, 7, 3, 41 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The right border of the triangle are the primes: T(n,n)=A000040(n); T(n,1)=A039702(n), T(n,2)=A039704(n) for n>1, T(n,3)=A007652(n) for n>2, T(n,4)=A039712(n) for n>3;
LINKS
Table of n, a(n) for n=1..87.
FORMULA
T(n, k) = prime(n) mod 2*prime(k), 1<=k<=n.
EXAMPLE
Triangle begins:
2;
3, 3;
1, 5, 5;
3, 1, 7, 7;
3, 5, 1, 11, 11;
1, 1, 3, 13, 13, 13;
1, 5, 7, 3, 17, 17, 17;
...
MAPLE
A079950 := proc(n, k)
modp(ithprime(n), 2*ithprime(k)) ;
end proc:
seq(seq(A079950(n, k), k=1..n), n=1..12) ; # R. J. Mathar, Sep 28 2017
PROG
(PARI) T(n, k) = prime(n) % (2*prime(k));
tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Sep 21 2017
CROSSREFS
Cf. A001747, A079951, A079952, A079953.
KEYWORD
nonn,tabl
AUTHOR
Reinhard Zumkeller, Jan 19 2003
STATUS
approved
A099618 a(n) = 1 if the n-th prime == 1 mod 6, otherwise a(n) = 0. +10
4
0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Table of n, a(n) for n=1..105.
Index entries for characteristic functions
MATHEMATICA
Table[Mod[Mod[Mod[Mod[Prime[k], 6], 5], 3], 2], {k, 1, 120}]
PROG
(PARI) a(n) = (prime(n) % 6) == 1; \\ Michel Marcus, Jun 26 2019
CROSSREFS
Cf. A000040, A039704, A091178.
KEYWORD
nonn
AUTHOR
Joseph Biberstine (jrbibers(AT)indiana.edu), Nov 19 2004
STATUS
approved
A181938 Isolated primes = 1 mod 6: sandwiched by primes = 5 mod 6. +10
2
7, 13, 19, 43, 97, 103, 109, 127, 139, 181, 193, 229, 241, 283, 307, 313, 349, 397, 409, 421, 457, 463, 487, 499, 643, 691, 709, 769, 787, 811, 823, 829, 853, 859, 877, 883, 907, 919, 937, 967, 1021, 1051, 1093, 1153, 1171, 1279, 1303, 1423, 1429, 1447, 1483 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Primes p(m) = 1 mod 6 such that both p(m-1) and p(m+1) are congruent to 5 mod 6.
Corresponding indices m are 4, 6, 8, 14, 25, 27, 29, 31 (A181978).
Note that values of d = p(m+1) - p(m-1) are multiples of 6.
LINKS
Harvey P. Dale, Table of n, a(n) for n = 1..1000
EXAMPLE
7 = p(4) = 1 mod 6 and both p(3) = 5 and p(5) = 11 are congruent to 5 mod 6,
13 = p(6) = 1 mod 6 and both p(5) = 11 and p(7) = 17 are congruent to 5 mod 6,
43 = p(14) = 1 mod 6 and both p(13) = 41 and p(15) = 47 are congruent to 5 mod 6.
MATHEMATICA
Select[Prime[Range[2, 300]], Mod[#, 6] == 1 && Mod[NextPrime[#, -1], 6] == 5 && Mod[NextPrime[#, 1], 6] == 5 &] (* T. D. Noe, Apr 04 2012 *)
Transpose[Select[Partition[Prime[Range[250]], 3, 1], Mod[#[[1]], 6] == Mod[#[[3]], 6] == 5&&Mod[#[[2]], 6]==1&]][[2]] (* Harvey P. Dale, Sep 17 2012 *)
CROSSREFS
Cf. A002476, A039704, A055625, A181978, A210248.
KEYWORD
nonn
AUTHOR
Zak Seidov, Apr 03 2012
STATUS
approved
A141455 Irregular triangle showing the set of all possible values of primes modulo n in row n. +10
1
0, 1, 0, 1, 2, 1, 2, 3, 0, 1, 2, 3, 4, 1, 2, 3, 5, 0, 1, 2, 3, 4, 5, 6, 1, 2, 3, 5, 7, 1, 2, 3, 4, 5, 7, 8, 1, 2, 3, 5, 7, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 5, 7, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3, 5, 7, 9, 11, 13, 1, 2, 3, 4, 5, 7, 8, 11, 13, 14, 1, 2, 3, 5, 7, 9, 11, 13, 15 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,5
LINKS
Michael De Vlieger, Table of n, a(n) for n = 2..10209 (rows n = 2..180, flattened)
FORMULA
Row n = A027748(n) U A038566(n), writing n as 0 iff n is prime. - Michael De Vlieger, Apr 18 2022
EXAMPLE
Table begins:0, 1;
0, 1, 2;
1, 2, 3;
0, 1, 2, 3, 4;
1, 2, 3, 5;
0, 1, 2, 3, 4, 5, 6;
1, 2, 3, 5, 7;
1, 2, 3, 4, 5, 7, 8;
1, 2, 3, 5, 7, 9;
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10;
1, 2, 3, 5, 7, 11;
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12;
1, 2, 3, 5, 7, 9, 11, 13;
1, 2, 3, 4, 5, 7, 8, 11, 13, 14;
1, 2, 3, 5, 7, 9, 11, 13, 15;
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16;
1, 2, 3, 5, 7, 11, 13, 17;
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18;
1, 2, 3, 5, 7, 9, 11, 13, 17, 19;
MATHEMATICA
Table[Union[FactorInteger[n][[All, 1]] /. n -> 0, Select[Range[n - 1], CoprimeQ[n, #] &]], {n, 2, 15}] (* Michael De Vlieger, Apr 18 2022 *)
CROSSREFS
Cf. A057859 (row lengths), A039701 (row n=3), A039704 (row n=6), A027748, A038566.
KEYWORD
nonn,tabf
AUTHOR
Roger L. Bagula and Gary W. Adamson, Aug 07 2008
STATUS
approved
A263483 a(n) = prime(n)+(prime(n) modulo 6). +10
1
4, 6, 10, 8, 16, 14, 22, 20, 28, 34, 32, 38, 46, 44, 52, 58, 64, 62, 68, 76, 74, 80, 88, 94, 98, 106, 104, 112, 110, 118, 128, 136, 142, 140, 154, 152, 158, 164, 172, 178, 184, 182, 196, 194, 202, 200, 212, 224, 232, 230, 238, 244, 242, 256, 262, 268, 274, 272, 278, 286, 284, 298, 308, 316, 314 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
For n>2, a(n)-a(n+1)=2 iff prime(n) and prime(n+1) are twin primes; e.g., a(3)-a(4)=10-8=2 and prime(3)=5 and prime(4)=7 are twin primes.
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = A000040(n) + A039704(n). - Michel Marcus, Oct 27 2015
MAPLE
p:= 1:
for n from 1 to 100 do
p:= nextprime(p);
A[n]:= p + (p mod 6);
od:
seq(A[n], n=1..100); # Robert Israel, Jul 18 2018
MATHEMATICA
Table[(p=Prime[n])+Mod[p, 6], {n, 100}]
PROG
(PARI) a(n) = apply(x->(x + x%6), prime(n)); \\ Michel Marcus, Oct 27 2015
CROSSREFS
Cf. A000040, A039704.
KEYWORD
nonn
AUTHOR
Zak Seidov, Oct 19 2015
STATUS
approved
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Last modified August 30 00:57 EDT 2024. Contains 375520 sequences. (Running on oeis4.)