A216292 -id:A216292 - cp's OEIS frontend

A007811 Numbers k for which 10k+1, 10k+3, 10k+7 and 10k+9 are primes.

1, 10, 19, 82, 148, 187, 208, 325, 346, 565, 943, 1300, 1564, 1573, 1606, 1804, 1891, 1942, 2101, 2227, 2530, 3172, 3484, 4378, 5134, 5533, 6298, 6721, 6949, 7222, 7726, 7969, 8104, 8272, 8881, 9784, 9913, 10111, 10984, 11653, 11929, 12220, 13546, 14416, 15727

Offset: 1

N. J. A. Sloane and J. H. Conway, Mar 15 1996

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A024912, A102338, A102342, A102700, A007530, A014561, A008471, A032352, A216292, A216293, A125855.

Cf. A010051, A245304, A245305.

a007811 n = a007811_list !! (n-1)
a007811_list = map (pred . head) $ filter (all (== 1) . map a010051') $
   iterate (zipWith (+) [10, 10, 10, 10]) [1, 3, 7, 9]
-- Reinhard Zumkeller, Jul 18 2014

[n: n in [0..10000] | forall{10*n+r: r in [1,3,7,9] | IsPrime(10*n+r)}]; // Bruno Berselli, Sep 04 2012

for n from 1 to 10000 do m := 10*n: if isprime(m+1) and isprime(m+3) and isprime(m+7) and isprime(m+9) then print(n); fi: od: quit

Select[ Range[ 1, 10000, 3 ], PrimeQ[ 10*#+1 ] && PrimeQ[ 10*#+3 ] && PrimeQ[ 10*#+7 ] && PrimeQ[ 10*#+9 ]& ]
Select[Range[15000], And @@ PrimeQ /@ ({1, 3, 7, 9} + 10#) &] (* Ray Chandler, Jan 12 2007 *)

p=2;q=3;r=5;forprime(s=7,1e5,if(s-p==8 && r-p==6 && q-p==2 && p%10==1, print1(p", ")); p=q;q=r;r=s) \\ Charles R Greathouse IV, Mar 21 2013

use ntheory ":all"; my @s = map { ($-1)/10 } sieve_prime_cluster(10,1e9, 2,6,8); say for @s; # _Dana Jacobsen, May 04 2017

a(n) = 3*A014561(n) + 1. - Zak Seidov, Sep 21 2009

A032352 Numbers k such that there is no prime between 10k and 10k+9.

Original entry on oeis.org

20, 32, 51, 53, 62, 84, 89, 107, 113, 114, 126, 133, 134, 135, 141, 146, 150, 164, 167, 168, 171, 176, 179, 185, 189, 192, 196, 204, 207, 210, 218, 219, 232, 236, 240, 248, 249, 251, 256, 258, 282, 294, 298, 305, 309, 314, 315, 317, 323, 324, 326, 328, 342

Offset: 1

Jeff Burch

Numbers k with property that appending any single decimal digit to k does not produce a prime.

A007920(n*10) > 10.

			m=32: 321=3*107, 323=17*19, 325=5*5*13, 327=3*109, 329=7*47, therefore 32 is a term.

M. I. Wilczynski, Table of n, a(n) for n = 1..10000

Cf. A124665 (subsequence), A010051, A007811, A216292, A216293.

a032352 n = a032352_list !! (n-1)
a032352_list = filter
   (\x -> all (== 0) $ map (a010051 . (10*x +)) [1..9]) [1..]
-- Reinhard Zumkeller, Oct 22 2011

[n: n in [1..350] | IsZero(#PrimesInInterval(10*n, 10*n+9))]; // Bruno Berselli, Sep 04 2012

a:=proc(n) if map(isprime,{seq(10*n+j,j=1..9)})={false} then n else fi end: seq(a(n),n=1..350); # Emeric Deutsch, Aug 01 2005

f[n_] := PrimePi[10n + 10] - PrimePi[10n]; Select[ Range[342], f[ # ] == 0 &] (* Robert G. Wilson v, Sep 24 2004 *)
Select[Range[342], NextPrime[10 # ] > 10 # + 9 &] (* Maciej Ireneusz Wilczynski, Jul 18 2010 *)
Flatten@Position[10*#+{1,3,7,9}&/@Range@4000,{?CompositeQ ..}] (* _Hans Rudolf Widmer, Jul 06 2024 *)

is(n)=!isprime(10*n+1) && !isprime(10*n+3) && !isprime(10*n+7) && !isprime(10*n+9) \\ Charles R Greathouse IV, Mar 29 2013

from sympy import isprime
def aupto(limit):
  alst = []
  for d in range(2, limit+1):
    td = [10*d + j for j in [1, 3, 7, 9]]
    if not any(isprime(t) for t in td): alst.append(d)
  return alst
print(aupto(342)) # Michael S. Branicky, May 30 2021

a(n) ~ n. - Charles R Greathouse IV, Mar 29 2013

More terms from Miklos Kristof, Aug 27 2002

A008471 Exactly 3 out of 10m+1, 10m+3, 10m+7, 10m+9 are primes.

Original entry on oeis.org

4, 7, 13, 22, 31, 43, 46, 61, 64, 85, 88, 103, 106, 109, 130, 142, 145, 160, 166, 169, 178, 199, 238, 268, 271, 316, 367, 376, 391, 400, 409, 415, 421, 451, 472, 478, 493, 523, 541, 544, 547, 550, 574

Offset: 1

Olivier Gérard

nonn

V. Raman, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)

Cf. A007811, A032352, A216292, A216293.

[n: n in [1..600] | #PrimesInInterval(10*n+1, 10*n+9) eq 3]; // Bruno Berselli, Sep 04 2012

m3Q[m_]:=Module[{c=10m},Count[{c+1,c+3,c+7,c+9},?PrimeQ]==3]; Select[ Range[ 600],m3Q] (* _Harvey P. Dale, Dec 24 2011 *)
Select[Range[600],Total[Boole[PrimeQ[10 #+{1,3,7,9}]]]==3&] (* Harvey P. Dale, Jul 27 2025 *)

A216293 Values of k such that there are exactly two primes between 10k and 10k + 9.

Original entry on oeis.org

2, 3, 5, 6, 8, 15, 16, 17, 23, 25, 26, 27, 28, 33, 34, 35, 37, 38, 40, 44, 49, 50, 52, 54, 56, 57, 59, 60, 65, 67, 70, 73, 75, 76, 91, 94, 97, 99, 101, 110, 112, 115, 118, 121, 122, 123, 127, 128, 129, 132, 136, 143, 149, 154, 155, 157, 161, 162, 172, 174

Offset: 1

V. Raman, Sep 03 2012

nonn

			23 is in the sequence because between 230 and 239 there are exactly two primes: 233 and 239. [_Bruno Berselli_, Sep 04 2012]

V. Raman, Table of n, a(n) for n = 1..10000

Cf. A032352, A007811, A078494, A216292.

[n: n in [1..200] | #PrimesInInterval(10*n, 10*n+9) eq 2]; // Bruno Berselli, Sep 04 2012

t = {}; Do[ps = Select[Range[10*n, 10*n + 9], PrimeQ]; If[Length[ps] == 2, AppendTo[t, n]], {n, 0, 229}]; t (* T. D. Noe, Sep 03 2012 *)
Select[Range[200],Count[Range[10#,10#+9],?PrimeQ]==2&] (* _Harvey P. Dale, Jan 19 2017 *)

a(n) >> n log^2 n. - Charles R Greathouse IV, Sep 07 2012

A356690 Product of the prime numbers that are between 10n and 10(n+1).

Original entry on oeis.org

210, 46189, 667, 1147, 82861, 3127, 4087, 409457, 7387, 97, 121330189, 113, 127, 2494633, 149, 23707, 27221, 30967, 181, 1445140189, 1, 211, 11592209, 55687, 241, 64507, 70747, 75067, 79523, 293, 307, 30857731, 1, 111547, 121103, 126727, 367, 141367, 148987, 397, 164009, 419, 421

Offset: 0

Hemjyoti Nath, Aug 23 2022

a(n) is prime iff n is in A216292. - Amiram Eldar, Aug 23 2022

For almost all n (in the sense of natural density), a(n) = 1. - Charles R Greathouse IV, Sep 30 2022

			210 = 2*3*5*7, 46189 = 11*13*17*19, 667 = 23*29, 1147 = 31*37, 82861 = 41*43*47.

Cf. A000040, A000720, A032352, A179816, A216292.

a[n_] := Times @@ Select[Range[10 n + 1, 10 n + 9], PrimeQ]; Array[a, 43, 0]

a(n) = vecprod(select(isprime, [10*n..10*(n+1)])); \\ Michel Marcus, Aug 24 2022

from math import prod
from sympy import primerange
def a(n): return prod(primerange(10*n, 10*(n+1)))
print([a(n) for n in range(43)]) # Michael S. Branicky, Aug 23 2022

from math import prod
from sympy import isprime
def A356690(n): return prod(m for i in (1,3,7,9) if isprime(m:=10*n+i)) if n else 210 # Chai Wah Wu, Sep 23 2022

Let m(n) = {isprime(10n-9) * (10n-9), isprime(10n-8) * (10n-8), isprime(10n-7) * (10n-7), isprime(10n-5) * (10n-5), isprime(10n-3) * (10n-3), isprime(10n-1) * (10n-1)}, where isprime = A010051; then a(n) = product of nonzero terms from m(n).
a(n) = 1 for n in A032352. - Michel Marcus, Aug 23 2022
a(n) = Product_{i=1+pi(10*n)..pi(10*(n+1))} prime(i). - Alois P. Heinz, Aug 23 2022

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A007811 Numbers k for which 10k+1, 10k+3, 10k+7 and 10k+9 are primes.

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A032352 Numbers k such that there is no prime between 10*k and 10*k+9.

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A008471 Exactly 3 out of 10m+1, 10m+3, 10m+7, 10m+9 are primes.

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A216293 Values of k such that there are exactly two primes between 10k and 10k + 9.

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A356690 Product of the prime numbers that are between 10*n and 10*(n+1).

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A032352 Numbers k such that there is no prime between 10k and 10k+9.

A356690 Product of the prime numbers that are between 10n and 10(n+1).