A014561 -id:A014561 - cp's OEIS frontend

A182393 Numbers n such that 210*n + {11,13,17,19,23,29,31,37} are 8 consecutive primes.

0, 75048, 122183, 445838, 868588, 1078331, 3152249, 4337790, 4962337, 5101537, 5572485, 6638215, 6948906, 8155956, 8298280, 9217084, 9752564, 11416369, 13331645, 13539754, 17782872, 19480161, 25473918, 25614474, 26299945, 27593165, 28335777, 28906807, 29231650

Offset: 1

Zak Seidov, Apr 27 2012

nonn

Subsequence of A182387: a(2) = 75048 = A182387(5) = A182282(7), a(3) = 122183 = A182387(8) = A182282(29).

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A014561, A022011, A022012, A022013, A182279, A182282, A182387.

a(n) = (A022011(n) - 11)/210. - Hugo Pfoertner, Nov 18 2022

A159910 Distance of prime quadruplets divided by 30, rounded towards the nearest integer.

Original entry on oeis.org

0, 3, 3, 21, 22, 13, 7, 39, 7, 73, 126, 119, 88, 3, 11, 66, 29, 17, 53, 42, 101, 214, 104, 298, 252, 133, 255, 141, 76, 91, 168, 81, 45, 56, 203, 301, 43, 66, 291, 223, 92, 97, 442, 290, 437, 281, 38, 144, 549, 241, 29, 192, 11, 518, 266, 490, 122, 130, 13, 329, 85, 209

Offset: 1

M. F. Hasler, May 04 2009

nonn

First differences of A007530, divided by 30 (and rounded to 0 for a(1)). The first prime quadruplet is the only one not starting at 11 (mod 30), and has no corresponding value in A014561. The "distance" can mean distance of starting points, or distance of barycenters, but also the distance in the strict sense (differing by 8 from the former), which gives the same value after rounding to the nearest integer.

All terms are of the form {0, 1, 3, 4, 6} mod 7. - Hugo Pfoertner, May 29 2020

			a(2) = A014561(2)-A014561(1) = 3-0, a(3) = A014561(3)-A014561(2) = 6-3, ...

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A007530, A014561, A047299.

A159910( n, list=0, s=5 )={ my(o,p,q,r); until(n--<0, o=s; until( p+8==s=nextprime(s+2), p=q; q=r; r=s); list & p>o & print1((s-o)\30,","););(s-o)\30}

a(n) = (A007530(n+1)-A007530(n))/30 = A014561(n)-A014561(n-1) for n>1.

A182450 Numbers n such that 210*n + {11,13,17,19,23,29,31,37,41} are 9 consecutive primes.

Original entry on oeis.org

0, 868588, 1078331, 4337790, 4962337, 6948906, 13539754, 30448177, 32218557, 39275297, 41670729, 52746284, 61193646, 81620584, 108499172, 118175956, 157531734, 198162240, 206181306, 208637331, 252388467, 258429278, 273526639, 305726202, 316425865, 383749862

Offset: 1

Zak Seidov, Apr 29 2012

nonn

Subsequence of A182393: a(2)=868588=A182393(5), a(3)=1078331=A182387(6).

Zak Seidov, Table of n, a(n) for n = 1..230

Cf. A014561, A182279, A182282, A182387, A182393.

A087771 Primes p giving prime quadruples (30p+11, 30p+13, 30p+17, 30p+19).

Original entry on oeis.org

3, 433, 521, 601, 647, 1459, 2099, 4073, 8123, 9491, 17293, 17881, 19001, 20887, 23057, 27457, 37253, 38923, 39119, 39409, 42409, 45293, 50287, 52391, 53861, 55103, 60661, 67213, 69481, 71483, 74941, 75211, 80447, 84503, 84731, 89459

Offset: 1

Zak Seidov, Oct 03 2003

			a(1)=3=A014561(1), a(2)=433=A014561(12), a(3)=521=A014561(13).

Zak Seidov, Table of n, a(n) for n = 1..1000

Subsequence of A014561.

A342814 Numbers k such that k - 1 and floor(k/5) are both prime.

Original entry on oeis.org

12, 14, 18, 38, 68, 98, 158, 308, 338, 368, 398, 488, 548, 758, 788, 908, 968, 998, 1118, 1568, 1658, 1748, 1868, 1988, 2288, 2438, 2618, 2708, 2858, 2888, 3038, 3068, 3218, 3308, 3458, 3548, 3638, 3698, 3848, 4058

Offset: 1

Claude H. R. Dequatre, Mar 22 2021

Except for a(1) and a(2), all terms == 8 (mod 10).
The first three absolute differences (d) between two consecutive floor(k/5) are respectively equal to 0, 1 and 4 and all the others to 6 or a multiple of 6.
Subsequence of A008864, by definition. - Michel Marcus, Mar 22 2021
For n >= 3, a(n) = 5*A023217(n-2) + 3. Higher terms also coincide with A265767 + 1. - Hugo Pfoertner, Mar 22 2021

			12 is a term because 12 - 1 = 11 and 11 is prime and 12/5 = 2.4 whose floor value is 2 and 2 is also prime.
97 is not a term because 97 - 1 = 96 and 96 is not prime although floor(97/5) = 19 is prime.
Initial terms, associated primes and d:
          k       k - 1     floor(k/5)     d
a(1)     12        11          2
a(2)     14        13          2           0
a(3)     18        17          3           1
a(4)     38        37          7           4
a(5)     68        67         13           6
a(6)     98        97         19           6
a(7)    158       157         31          12
a(8)    308       307         61          30
a(9)    338       337         67           6
a(10)   368       367         73           6

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

Cf. A000040, A007530, A007811, A014561, A259645.

Cf. A008864, A023217, A265767.

R:= NULL:
p:= 1: count:= 0:
while count < 100 do
  p:= nextprime(p);
  if isprime(floor((p+1)/5)) then
     R:= R,p+1; count:= count+1
  fi
od:
R; # Robert Israel, May 22 2024

Select[Range[2,5000,2],And@@PrimeQ[{#-1,Floor[#/5]}]&] (* Giorgos Kalogeropoulos, Apr 01 2021 *)

for(k = 1,10000,if(isprime(k - 1) && isprime(k\5),print1(k", ")))

A087772 Numbers n such that s=n^2 gives prime quadruples (30s+11, 30s+13, 30s+17, 30s+19).

Original entry on oeis.org

7, 602, 10269, 18690, 23597, 24906, 42574, 57246, 58639, 59647, 59927, 60634, 62384, 74697, 84525, 85491, 95291, 100401, 102844, 105917, 108843, 113701, 119364, 125713, 126539, 130389, 130774, 147959, 148470, 156100, 167755, 174391

Offset: 1

Zak Seidov, Oct 03 2003

Cf. A014561.

More terms from Ray Chandler, Oct 05 2003

A215473 Number of prime quadruples with smallest member < 2^n.

Original entry on oeis.org

0, 0, 1, 2, 2, 2, 3, 4, 4, 5, 7, 10, 11, 16, 23, 28, 43, 62, 106, 177, 309, 483, 795, 1305, 2105, 3525, 5923, 10096, 17259, 30004

Offset: 1

Alex Ratushnyak, Aug 12 2012

nonn

Prime quadruples (A007530) are numbers n such that n, n+2, n+6, n+8 are all prime.

			a(3) = 1 because there is only one prime quadruple below 2^3, namely {5, 7, 11, 13}.
a(4) = 2 because there are two prime quadruples below 2^4: the aforementioned and {11, 13, 17, 19}.

Cf. A050258, similar definition but with powers of 10 instead of 2.
Cf. A007530, A014561, A112540.
Cf. A033843, A007053.

(* First run program for A007530 *) Table[Length[Select[A007530, # < 2^n &]], {n, 14}] (* Alonso del Arte, Aug 12 2012 *)

A342809 Numbers k such that k-1 and round(k/5) are both prime.

Original entry on oeis.org

8, 12, 14, 24, 54, 84, 114, 234, 264, 294, 354, 444, 504, 564, 654, 684, 744, 864, 954, 984, 1164, 1194, 1284, 1554, 1584, 1734, 1914, 2004, 2154, 2214, 2244, 2334, 2394, 2544, 2844, 2964, 3084, 3204, 3414, 3594

Offset: 1

Claude H. R. Dequatre, Mar 22 2021

Except for a(1) and a(2), all terms == 4 (mod 10).
The first three absolute differences (d) between two consecutive rounded (k/5) are respectively equal to 0, 1 and 2 and all the others to 6 or a multiple of 6.
Subsequence of A008864, by definition. - Michel Marcus, Mar 22 2021
For n >= 3, a(n) = 5*A158318(n-2) - 1. - Hugo Pfoertner, Mar 22 2021

			8 is a term because 8 - 1 = 7 and 7 is prime and 8/5 = 1.6 which when rounded gives 2 and 2 is also prime.
235 is not a term because 235 - 1 = 234 and 234 is not a prime although 235/5 = 47 is prime.
Initial terms, associated primes and d:
         k     k - 1   round(k/5)    d
a(1)     8       7         2
a(2)    12      11         2         0
a(3)    14      13         3         1
a(4)    24      23         5         2
a(5)    54      53        11         6
a(6)    84      83        17         6
a(7)   114     113        23         6
a(8)   234     233        47        24
a(9)   264     263        53         6
a(10)  294     293        59         6

Cf. A000040, A007530, A007811, A014561, A259645.

Cf. A008864, A158318.

Select[Range[2,5000,2],And@@PrimeQ[{#-1,Round[#/5]}]&] (* Giorgos Kalogeropoulos, Apr 01 2021 *)

for(k = 1,10000,if(isprime(k - 1) && isprime(k\/5),print1(k", ")))

from sympy import isprime
A342809_list = [k for k in range(1,10**5) if isprime(k-1) and isprime(k//5+int(k % 5 > 2))] # Chai Wah Wu, Apr 08 2021

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A182393 Numbers n such that 210*n + {11,13,17,19,23,29,31,37} are 8 consecutive primes.

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A159910 Distance of prime quadruplets divided by 30, rounded towards the nearest integer.

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A182450 Numbers n such that 210*n + {11,13,17,19,23,29,31,37,41} are 9 consecutive primes.

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A087771 Primes p giving prime quadruples (30p+11, 30p+13, 30p+17, 30p+19).

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A342814 Numbers k such that k - 1 and floor(k/5) are both prime.

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A087772 Numbers n such that s=n^2 gives prime quadruples (30s+11, 30s+13, 30s+17, 30s+19).

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A215473 Number of prime quadruples with smallest member < 2^n.

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Mathematica

A342809 Numbers k such that k-1 and round(k/5) are both prime.

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Mathematica

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