A032352 -id:A032352 - cp's OEIS frontend

A216312 The prime ending in 9 is the only prime in a decade.

149, 419, 479, 719, 809, 839, 929, 1009, 1049, 1249, 1259, 1319, 1399, 1409, 1709, 1889, 1949, 2039, 2099, 2129, 2179, 2309, 2459, 2579, 2609, 2729, 2789, 2819, 2879, 2939, 2999, 3079, 3109, 3119, 3299, 3359, 3389, 3449, 3659, 3719, 3779, 3989, 4049, 4229

Offset: 1

V. Raman, Sep 03 2012

Primes of the form 10n+9 such that 10n+1, 10n+3, and 10n+7 are composite. - Charles R Greathouse IV, Sep 06 2012

			149 is prime but 141, 143 and 147 are all composite (being 3 * 47, 11 * 13 and 3 * 7^2 respectively), thus 149 is in the sequence.

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

Subsequence of A030433. Cf. A032352, A007811, A078494, A030433.

[p: p in PrimesUpTo(4300) | p mod 10 eq 9 and IsOne(#PrimesInInterval(10*t+1, 10*t+9)) where t is Floor(p/10)]; // Bruno Berselli, Sep 14 2012

Select[Prime[Range[700]], Mod[#, 10] == 9 && Union[PrimeQ[{# - 8, # - 6, # - 2}]] == {False} &] (* Alonso del Arte, Sep 03 2012 *)
Select[Table[10n+{1,3,7,9},{n,450}],Boole[PrimeQ[#]]=={0,0,0,1}&][[;;,4]] (* Harvey P. Dale, Mar 08 2023 *)

a(n) ~ 4n log n. - Charles R Greathouse IV, Sep 06 2012

A316630 Numbers k such that 10k+1, 10k+3, 10k+7, and 10k+9 are all composite, and k == 1 (mod 3).

Original entry on oeis.org

133, 196, 232, 256, 298, 328, 397, 403, 406, 418, 430, 457, 484, 640, 643, 664, 709, 727, 742, 802, 847, 865, 898, 907, 970, 991, 1012, 1054, 1057, 1081, 1087, 1096, 1120, 1153, 1156, 1213, 1231, 1246, 1327, 1354, 1360, 1381, 1411, 1423, 1426, 1435, 1480, 1504

Offset: 1

Patrick A. Thomas, Jul 09 2018

The sequence contains all numbers of the form 50151*m + 42175. - Michael B. Porter, Jul 16 2018

If m is in the sequence, then so is m + 3*(10*m+1)*(10*m+3)*(10*m+7)*(10*m+9)*k for all k. - Robert Israel, Aug 08 2018

			1331 = 11^3, 1333 = 31*43, 1337 = 7*191, 1339 = 13*103, and 133 == 1 (mod 3), so 133 is a sequence member.

Robert Israel, Table of n, a(n) for n = 1..10000
C. K. Caldwell, The Prime Pages Top Twenty: Quadruplet, list of largest known prime quadruplets.

Cf. A007811, A032352.

m=1; for s=1:510 v=[30*s+11,30*s+13,30*s+17,30*s+19]; if isprime(v)==0  sol(m)=3*s+1; m=m+1;end; end; sol % Marius A. Burtea, Sep 17 2019

[3*s+1: s in [0..510] | forall{30*s+k: k in [11, 13, 17, 19] | not IsPrime(30*s+k)}]; // Marius A. Burtea, Sep 17 2019

remove(t -> ormap(isprime, [10*t+1,10*t+3,10*t+7,10*t+9]), [seq(k,k=1..2000,3)]); # Robert Israel, Aug 08 2018

Select[1 + 3 Range@510, Union[ PrimeQ[10 # + {1, 3, 7, 9}]] == {False} &] (* Robert G. Wilson v, Jul 16 2018 *)
Select[Range[1,2000,3],AllTrue[10#+{1,3,7,9},CompositeQ]&]  (* Harvey P. Dale, Feb 24 2024 *)

ok(k)={if(k%3==1, for(i=0, 4, if(isprime(10*k+2*i+1), return(0))); 1, 0)} \\ Andrew Howroyd, Jul 10 2018

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

A216289 Smallest k in which there are exactly n primes between 10k and 10k+9.

Original entry on oeis.org

20, 9, 2, 4, 0

Offset: 0

V. Raman, Sep 03 2012

Cf. A186311, A032352, A007811, A078494.

mx = 5; t = Table[-1, {mx}]; n = 0; found = 0; While[found < mx, ps = Select[Range[10*n, 10*n + 9], PrimeQ]; len = Length[ps]; If[t[[len + 1]] == -1, t[[len + 1]] = n; found++]; n++]; t (* T. D. Noe, Sep 03 2012 *)

A346979 Count of the prime decimal descendants of n.

Original entry on oeis.org

83, 63, 23, 22, 23, 11, 29, 23, 3, 4, 54, 1, 9, 14, 6, 7, 3, 4, 7, 40, 0, 4, 19, 15, 8, 7, 10, 14, 5, 6, 2, 7, 0, 16, 9, 11, 12, 13, 4, 1, 34, 1, 8, 14, 5, 1, 13, 5, 5, 16, 6, 0, 9, 0, 24, 4, 6, 19, 2, 9, 25, 16, 0, 7, 4, 4, 3, 11, 2, 7, 7, 4, 1, 15, 2, 8, 8

Offset: 0

Ya-Ping Lu, Aug 09 2021

The number of direct decimal descendants (i.e., decimal children) of n is A038800(n). The number of prime decimal descendants of the n-th prime is A214342(p_n). a(n) is the number of prime decimal descendants of n, which include the prime decimal children of n, the prime decimal children of the prime decimal children of n, and so on.
a(0) = Sum_{m=1..4} (A214342(m) + 1); a(1) = Sum_{m=5..8} (A214342(m) + 1).
a(A032352(m)) = 0; a(A119289(m)) = 0.
A214342 is a subset, as A214342(m) = a(prime(m)).
Conjecture 1: a(n) <= 83. Conjecture 2: lim_{n->oo} (n0/n) = 1, where n0 is the number of zero terms, a(k) = 0, for k <= n.

			a(4) = 23. The 23 prime decimal descendants of 4 are shown in the tree below.
       _____ 4__________________________
      /      |                          \
     41   ___43______________            47
    /    /   |               \             \
  419  431  433               439          479
            / \              /   \        /   \
        4337  4339         4391  4397   4793  4799
             /  |  \        |     |     /  \
        43391 43397 43399 43913 43973 47933 47939
                            |
                         439133
                            |
                        4391339

Cf. A030665, A032352, A038800, A119289, A122072, A214342, A218255, A256481.

Table[Length@Rest@Flatten[FixedPointList[(b=#;Select[Flatten[(a=#;FromDigits/@(Join[IntegerDigits@a,{#}]&/@If[b=={0},Range@9,{1,3,7,9}]))&/@b],PrimeQ])&,{n}]],{n,0,76}] (* Giorgos Kalogeropoulos, Aug 16 2021 *)

from sympy import isprime
def p_count(k):
    global ct; d = [2, 3, 5, 7] if k == 0 else [1, 3, 7, 9]
    for i in range(4):
        m = 10*k + d[i]
        if isprime(m): ct += 1; p_count(m)
    return ct
for n in range(100):
    ct = 0; print(p_count(n))

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A216312 The prime ending in 9 is the only prime in a decade.

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A316630 Numbers k such that 10k+1, 10k+3, 10k+7, and 10k+9 are all composite, and k == 1 (mod 3).

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

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A216289 Smallest k in which there are exactly n primes between 10*k and 10*k+9.

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A346979 Count of the prime decimal descendants of n.

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

A216289 Smallest k in which there are exactly n primes between 10k and 10k+9.