A023203 -id:A023203 - cp's OEIS frontend

A288024 Prime p1 of consecutive primes p1, p2, where p2 - p1 = 8, and p1, p2 are in different decades.

89, 359, 389, 449, 479, 683, 719, 743, 929, 983, 1109, 1163, 1193, 1373, 1439, 1523, 1559, 1733, 1823, 1979, 2003, 2153, 2213, 2243, 2273, 2459, 2609, 2663, 2699, 2843, 2879, 2909, 3209, 3449, 3623, 3719, 4289, 4349, 4583, 4943, 5189, 5399, 5573, 5693, 5783, 5813

Offset: 1

Hartmut F. W. Hoft, Jun 04 2017

nonn

The unit digits of the numbers in the sequence are 3's or 9's.

			89 is in the sequence since pair (89,97) is the first with difference 8 spanning a multiple of 10.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001359, A023201, A023203, A031924, A031925, A031928, A046117, A046132, A158277, A158861, A160370, A160440, A160500, A287049, A287050, A060229, A288021, A288022.

a288024[n_] := Map[Last, Select[Map[{NextPrime[#, 1], NextPrime[#, -1]}&, Range[10, n, 10]], First[#]-Last[#]==8&]]
a288024[6000] (* data *)
Select[Partition[Prime[Range[800]],2,1],#[[2]]-#[[1]]==8&&IntegerDigits[#[[1]]][[-2]]!= IntegerDigits[ #[[2]]][[-2]]&][[;;,1]] (* Harvey P. Dale, Jan 09 2024 *)

A140445 List of prime pairs of form p, p + 10.

Original entry on oeis.org

3, 13, 7, 17, 13, 23, 19, 29, 31, 41, 37, 47, 43, 53, 61, 71, 73, 83, 79, 89, 97, 107, 103, 113, 127, 137, 139, 149, 157, 167, 163, 173, 181, 191, 223, 233, 229, 239, 241, 251, 271, 281, 283, 293, 307, 317, 337, 347, 349, 359, 373, 383, 379, 389, 409, 419, 421

Offset: 1

Juri-Stepan Gerasimov, Jun 26 2008

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A023203 (1st bisection), A092146 (2nd bisection).

Cf. prime pairs of the form (p, p+k): A077800 (k=2), A094343 (k=4), A156274 (k=6), A156320 (k=8), this sequence (k=10), A156323 (k=12), A140446 (k=14), A272815 (k=16), A156328 (k=18), A272816 (k=20), A140447 (k=22).

i: 1: for k from 1 to 1200 do if isprim (k) and isprim (k+10) then a [ i ] : = k : a [ i + 1]: = k + 10 : i = i + 2 fi od : seq (a [ n ], n=1..i-1);

Flatten[{#,#+10}&/@Select[Prime[Range[100]],PrimeQ[#+10]&]]  (* Harvey P. Dale, Apr 11 2011 *)

Corrected by D. S. McNeil, Dec 10 2009

A287050 Square array read by antidiagonals upwards: M(n,k) is the initial occurrence of first prime p1 of consecutive primes p1, p2, where p2 - p1 = 2*k, and p1, p2 span a multiple of 10^n, n>=1, k>=1.

Original entry on oeis.org

29, 599, 7, 2999, 97, 47, 179999, 1999, 1097, 89, 23999999, 69997, 21997, 1193, 139, 23999999, 199999, 369997, 23993, 691, 199, 29999999, 19999999, 3199997, 149993, 10993, 199, 113, 17399999999, 19999999, 6999997, 1199999, 139999, 997, 293, 1831

Offset: 1

Hartmut F. W. Hoft, May 18 2017

The unit digits of the numbers in the matrix representation M(n,k) are 9's for column 1, 7's or 9's for column 2, 7's for column 3, 3's or 9's for column 4, and 1's, 3's, 7's or 9's for column 5.
The following matrix terms appear as first terms in sequence
A060229(1) = M(1,1)
A288021(1) = M(1,2)
A288022(1) = M(1,3)
A288024(1) = M(1,4)
A031928(1) = M(1,5)
A158277(1) = M(2,1)
A160440(1) = M(2,2)
A160370(1) = M(2,3)
A287049(1) = M(2,4)
A160500(1) = M(2,5)
A158861(1) = M(3,1).

			The matrix representation of the sequence with row n indicating the spanned power of 10 and column k indicating the difference of 2*k between the first pair of consecutive primes spanning a multiple of 10^n:
--------------------------------------------------------------------------
n\k   1             2             3             4            5
--------------------------------------------------------------------------
1 |   29            7             47            89           139
2 |   599           97            1097          1193         691
3 |   2999          1999          21997         23993        10993
4 |   179999        69997         369997        149993       139999
5 |   23999999      199999        3199997       1199999      1999993
6 |   23999999      19999999      6999997       38999993     1999993
7 |   29999999      19999999      159999997     659999999    379999999
8 |   17399999999   7699999999    9399999997    8999999993   499999993
9 |   92999999999   135999999997  85999999997   8999999993   28999999999
10|   569999999999  519999999997  369999999997  29999999993  819999999997
...
Every column in the matrix is nondecreasing.
For the first and fourth columns, ceiling(M[n,1]/10^n) and ceiling(M[n,4]/10^n) are divisible by 3, for all n>=1 (see A158277 and A287049).

Cf. A001359, A023201, A023203, A031924, A031925, A031928, A046117, A046132, A158277, A158861, A160370, A160440, A160500, A287049.

Cf. A060229, A288021, A288022, A288024.

M(n,k) = min( p_i : p_(i+1) - p_i = 2*k, p_i and p_(i+1) consecutive primes and p_i < m*10^n < p_(i+1) for some integer m) where p_j is the j-th prime, n>=1 and k>=1.

A220746 Numbers n such that n and n + 10 are prime and there is a power of two in the interval (n, n+10).

Original entry on oeis.org

3, 7, 13, 31, 61, 127, 1021, 1048573, 23945242826029513411849172299223580994042798784118781

Offset: 1

Brad Clardy, Feb 20 2013

nonn

Cf. A023203, A221211.

//Program finds primes separated by an even number (called gap) which
//have a power of two between them. Program starts with the smallest
//power of two above gap. Primes less than this starting point can be
//checked by inspection. In this example 3 also works.
gap:=10;
start:=Ilog2(gap)+1;
for i:= start to 1000 do
    powerof2:=2^i;
    for k:=powerof2-gap+1 to powerof2-1 by 2 do
        if (IsPrime(k) and IsPrime(k+gap)) then k;
        end if;
    end for;
end for;

Union[Flatten[Table[Select[Range[2^n - 9, 2^n - 1, 2], PrimeQ[#] && PrimeQ[# + 10] &], {n, 3, 200}]]] (* T. D. Noe, Feb 20 2013 *)
Union[Flatten[Table[Select[Thread[{Range[2^n-10,2^n],Range[ 2^n,2^n+10]}],AllTrue[ #,PrimeQ]&],{n,3,1000}],1][[;;,1]]] (* Harvey P. Dale, Feb 19 2023 *)

print1(3); for(n=4,500,forprime(p=2^n-9,2^n-1,if(isprime(p+10), print1(", "p)))) \\ Charles R Greathouse IV, Feb 20 2013

A271575 Primes p such that p+10, p+100 and p+1000 are all prime.

Original entry on oeis.org

13, 31, 97, 163, 181, 283, 409, 499, 709, 787, 811, 877, 1087, 1399, 1423, 1609, 1777, 1801, 1879, 2347, 2677, 2719, 3457, 3517, 3919, 4273, 4483, 5701, 6043, 6121, 6211, 6481, 6691, 7573, 8941, 9733, 9739, 10069, 10093, 10159, 10243, 10789, 11161, 11251, 11689, 12799, 12907

Offset: 1

Emre APARI, Apr 10 2016

nonn

Number of terms < 10^k: 0, 3, 12, 37, 159, 789, 3960, 21708, 129910, ..., . - Robert G. Wilson v, Jun 20 2018

			p=13; p+10=23 (is prime), p+100=113 (is prime), p+1000=1013 (is prime).

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

Cf. A000040, A002385, A015916, A023203.

select(t -> isprime(t+1000) and isprime(t+100) and isprime(t+10) and isprime(t), [seq(i,i=7..20000, 6)]); # Robert Israel, Jun 20 2018

Select[Prime[Range[10000]], PrimeQ[# + 10] && PrimeQ[# + 100] && PrimeQ[# + 1000] &] (* Robert Price, Apr 10 2016 *)

lista(nn) = forprime(p=2, nn, if (isprime(p+10) && isprime(p+100) && isprime(p+1000), print1(p, ", "))); \\ Michel Marcus, Apr 10 2016

A106061 Primes p such that 1p + 10 and 10p + 1 are primes.

Original entry on oeis.org

3, 7, 13, 19, 31, 43, 97, 103, 157, 181, 241, 271, 337, 349, 409, 421, 439, 547, 631, 691, 733, 811, 829, 937, 1009, 1021, 1039, 1249, 1279, 1291, 1483, 1489, 1597, 1609, 1723, 1867, 2203, 2287, 2383, 2437, 2467, 2539, 2689, 2719, 2887, 2953, 3001, 3109, 3319

Offset: 1

Zak Seidov, May 07 2005

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A023203.

[p: p in PrimesUpTo(100000)| IsPrime(p+10) and IsPrime(10*p+1)]; // Vincenzo Librandi, Nov 13 2010

Select[Prime[Range[220]], PrimeQ[10#+1]&&PrimeQ[1#+10]&]
Select[Prime[Range[500]],AllTrue[{10#+1,#+10},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jun 15 2017 *)

A120124 Smallest prime p such that p*10^n + 1 is a prime.

Original entry on oeis.org

3, 7, 3, 7, 7, 61, 3, 7, 7, 3, 19, 37, 109, 79, 97, 13, 37, 19, 73, 103, 97, 283, 157, 61, 19, 61, 1213, 3, 163, 691, 367, 163, 181, 157, 241, 3, 103, 733, 151, 283, 337, 193, 211, 163, 7, 73, 307, 61, 223, 1549, 31, 127, 13, 547, 103, 151, 193, 811, 337, 19, 1021, 151

Offset: 1

Alexander Adamchuk, Aug 15 2006

nonn

All terms belong to A007645. All terms also belong to A055664. Also many terms including the first 14 smallest primes from 3 to 139 {3,7,13,19,31,37,43,61,73,79,97,103,127,139} belong tpA023203. The smallest term that differs from A023203 is 151.

			a(1) = 3 because 31 = 3*10 + 1 is the smallest prime of form p*10 + 1, where p is a prime.
a(2) = 7 because 701 = 7*100 + 1 is the smallest prime of form p*100 + 1.

Robert Israel, Table of n, a(n) for n = 1..1000 (first 300 terms from _Vincenzo Librandi_)

Cf. A121172, A030430, A062800, A007645, A055664, A023203.

Primes:= select(isprime,[$1..10^5]):
for n from 1 to 1000 do
   for p in Primes do
      if isprime(p*10^n+1) then
        A[n]:= p
      fi
    od
od:
seq(A[n],n=1..1000); # Robert Israel, May 29 2014

prs=Prime[Range[2000]];Table[i=1;While[!PrimeQ[First[Take[prs,{i}]] 10^n+1],i++];Prime[i],{n,200}] (* Harvey P. Dale, May 15 2011 *)

A164623 Primes p such that p(p-1)/2-5 and p(p-1)/2+5 are also prime numbers.

Original entry on oeis.org

13, 157, 673, 1069, 1117, 1153, 1213, 1597, 2029, 2089, 2437, 2713, 2833, 3613, 4057, 4909, 5653, 6337, 6529, 7549, 7993, 8053, 9613, 10789, 11497, 11689, 12073, 12373, 13309, 13669, 13789, 14173, 15289, 15937, 16249, 18097, 18637, 19249, 19993

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 17 2009

Primes A000040(k) such that A008837(k)+-5 are also prime numbers.

			13 is in the sequence because 13*6-5=73 and 13*6+5=83 are both prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A008846, A020882, A068228, A068229, A082539, A086519, A107159, A158708, A139494, A164620, A164621, A164622, A023203.

Select[Prime[Range[2300]], PrimeQ[# (# - 1)/2 - 5] && PrimeQ[# (# - 1)/2 + 5] &]

forprime(p=2,10^6,my(b=binomial(p,2));if(isprime(b-5)&isprime(b+5),print1(p,", "))); /* Joerg Arndt, Apr 10 2013 */

Edited by R. J. Mathar, Aug 20 2009

Mathematica code adapted to the definition by Bruno Berselli, Apr 10 2013

A240986 Determinants of n X n matrices of sets of distinct primes selected by increasing prime gaps (see comments).

Original entry on oeis.org

3, 6, -36, -216, 1296, -5184, -145152, -3856896, -170325504, -6133211136, 1094593056768, 26742290558976, -497681937801216, -14357497419546624, 657148066947072000, 12008320398059765760, 1322255096225695531008, 70546799432003423698944, -6537119853797882157072384, -27940593871362459110473728

Offset: 1

Samuel J. Erickson, Aug 06 2014

sign

Let P = {3,5,7,11,...} be the sequence of odd primes and let P(k) = {prime in P: (prime+2k) is in P} (although set builder notation is used for P(k) we will still assume that P(k) is a sequence). Let M(n) be the n X n matrix where row 1 is the first n elements from P(1), row 2 is the first n elements from P(2), and in general row j is the first n elements from P(j). This sequence is the sequence of determinants for M(1), M(2), M(3), M(4), ..., M(9).

			For the first element of the sequence we find the determinant of the matrix [[3,5],[3,7]], where [3,5] is row 1 and [3,7] is row 2. These numbers are there because in row 1 we are looking at the primes where we can add 2 to get another prime; 3+2 is prime and so is 5+2, so they go in row 1. Similarly, for the second row we get [3,7] because these are the first primes such that when 4 is added we get a prime: 3+4 and 7+4 are both prime, so they go in row 2. For the second entry in the sequence we take the determinant of [[3,5,11],[3,7,13],[5,7,11]]; the reason we get [5,7,11] in the third row is because 5 is the first prime where 5+6 is prime, 7 is second prime where 7+6 is prime, and 11 is the third prime where 11+6 is prime.

Jinyuan Wang, Table of n, a(n) for n = 1..200
Samuel J. Erickson, Python Code
Sachin Joglekar, Determinant of matrix of any order (Python)

Cf. A001359, A023200, A023201, A023202, A023203.

a(n) = {my(m=matrix(n,n), j); for (i=1, n, j = 1; forprime(p=2, , if (isprime(p+2*i), m[i,j] = p; j++); if (j > n, break););); matdet(m);} \\ Michel Marcus, May 04 2019

Python
```
# See Erickson link.
```

Offset 1 and more terms from Michel Marcus, May 04 2019

A341044 Numbers k such that A318996(k) is prime.

Original entry on oeis.org

8, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 39, 42, 46, 57, 62, 65, 66, 69, 72, 74, 76, 80, 82, 87, 91, 92, 93, 94, 95, 100, 106, 111, 116, 119, 121, 122, 129, 133, 134, 145, 146, 159, 162, 166, 172, 176, 177, 183, 184, 190, 194, 202, 203, 206, 208, 213, 214, 215, 219, 232, 236, 237, 238, 240, 243

Offset: 1

J. M. Bergot and Robert Israel, Feb 03 2021

nonn

Contains 2*p for p in A023201 and 3*p for p in A023203.

			a(3) = 10 is a term because A318996(10) = 11 is prime.

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

Cf. A023201, A023203, A318996.

f:= proc(n) local s,D,t;
  D:= numtheory:-divisors(n);
  s:= convert(D,`+`);
  add(s mod t, t=D)
end proc:
select(t -> isprime(f(t)), [$1..1000]);

f(n) = my(sn = sigma(n)); sumdiv(n, d, sn % d); \\ A318996
isok(k) = isprime(f(k)); \\ Michel Marcus, Feb 04 2021

cp's OEIS Frontend

A288024 Prime p1 of consecutive primes p1, p2, where p2 - p1 = 8, and p1, p2 are in different decades.

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Mathematica

A140445 List of prime pairs of form p, p + 10.

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Maple

Mathematica

Extensions

A287050 Square array read by antidiagonals upwards: M(n,k) is the initial occurrence of first prime p1 of consecutive primes p1, p2, where p2 - p1 = 2*k, and p1, p2 span a multiple of 10^n, n>=1, k>=1.

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Formula

A220746 Numbers n such that n and n + 10 are prime and there is a power of two in the interval (n, n+10).

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Magma

Mathematica

PARI

A271575 Primes p such that p+10, p+100 and p+1000 are all prime.

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Author

Keywords

Comments

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Programs

Maple

Mathematica

PARI

A106061 Primes p such that 1*p + 10 and 10*p + 1 are primes.

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Magma

Mathematica

A120124 Smallest prime p such that p*10^n + 1 is a prime.

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Comments

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Maple

Mathematica

A164623 Primes p such that p*(p-1)/2-5 and p*(p-1)/2+5 are also prime numbers.

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Mathematica

A106061 Primes p such that 1p + 10 and 10p + 1 are primes.

A164623 Primes p such that p(p-1)/2-5 and p(p-1)/2+5 are also prime numbers.