A158861 -id:A158861 - cp's OEIS frontend

A060229 Smaller member of a twin prime pair whose mean is a multiple of A002110(3)=30.

29, 59, 149, 179, 239, 269, 419, 569, 599, 659, 809, 1019, 1049, 1229, 1289, 1319, 1619, 1949, 2129, 2309, 2339, 2549, 2729, 2789, 2969, 2999, 3119, 3299, 3329, 3359, 3389, 3539, 3929, 4019, 4049, 4229, 4259, 4649, 4799, 5009, 5099, 5279, 5519, 5639

Offset: 1

Labos Elemer, Mar 21 2001

nonn

Equivalently, smaller of twin prime pair with primes in different decades.
Primes p such that p and p+2 are prime factors of Fibonacci(p-1) and Fibonacci(p+1) respectively. - Michel Lagneau, Jul 13 2016
The union of this sequence and A282326 gives A132243. - Martin Renner, Feb 11 2017
The union of {3,5}, A282321, A282323 and this sequence gives A001359. - Martin Renner, Feb 11 2017
The union of {3,5,7}, A282321, A282322, A282323, A282324, this sequence and A282326 gives A001097. - Martin Renner, Feb 11 2017
Number of terms less than 10^k, k=2,3,4,...: 2, 11, 72, 407, 2697, 19507, 146516, ... - Muniru A Asiru, Jan 29 2018

			For the pair {149,151} (149 + 151)/2 = 5*30.

Muniru A Asiru, Table of n, a(n) for n = 1..1000

Cf. A001359, A002110, A060230, A060231, A158277, A158861.

Subset of A001097, A001359, A132236, A132243 and A132247.

Filtered(List([0..200], k -> 30*k-1), n -> IsPrime(n) and IsPrime(n+2));  # Muniru A Asiru, Feb 02 2018

[p: p in PrimesUpTo(7000) | IsPrime(p+2) and p mod 30 eq 29 ]; // Vincenzo Librandi, Feb 13 2017

isA060229 := proc(n)
    if modp(n+1,30) =0 and isprime(n) and isprime(n+2) then
        true;
    else
        false;
    end if;
end proc:
A060229 := proc(n)
    option remember;
    if n =1 then
        29;
    else
        for a from procname(n-1)+2 by 2 do
            if isA060229(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A060229(n),n=1..80) ; # R. J. Mathar, Feb 19 2017

Select[Prime@ Range[10^3], PrimeQ[# + 2] && Mod[# + 1, 30] == 0 &] (* Michael De Vlieger, Jul 14 2016 *)

isok(n) = isprime(n) && isprime(n+2) && !((n+1) % 30); \\ Michel Marcus, Dec 11 2013

Minor edits by Ray Chandler, Apr 02 2009

A158277 The lesser of twin prime pairs with each prime in a different century.

Original entry on oeis.org

599, 2999, 3299, 4799, 5099, 6299, 8999, 10499, 11699, 21599, 25799, 26699, 29399, 33599, 34499, 36899, 37199, 42899, 44699, 47699, 49199, 56099, 57899, 60899, 63599, 65099, 65699, 70199, 74099, 81899, 83399, 85199, 88799, 92399, 97499, 100799, 101999, 102299

Offset: 1

Ki Punches, Mar 15 2009

nonn

The sequence is conjecturally infinite; note that those ending in 999 straddle millenia: A158861.

Since any prime greater than 3 is congruent to 1 or 5 modulo 6, a(n)+1 is divisible by 300 (see A001359). - Hartmut F. W. Hoft, May 18 2017

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

Cf. A060229, A158861, A001359, A060229.

Select[Range[100,110000,100],AllTrue[#+{1,-1},PrimeQ]&]-1 (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 21 2016 *)
a158277[n_] := Map[Last, Select[Map[{NextPrime[#, 1], NextPrime[#, -1]}&, Range[100, n, 100]], First[#]-Last[#]==2&]]
a158277[105000] (* data *) (* Hartmut F. W. Hoft, May 18 2017 *)

Corrected by Ray Chandler and R. J. Mathar, Apr 03 2009

A173976 Numbers m such that the concatenation of m and 999 is the lesser of twin primes, i.e., a millennium twin prime couple.

Original entry on oeis.org

2, 8, 101, 164, 179, 230, 272, 293, 326, 389, 410, 419, 443, 512, 524, 536, 659, 662, 773, 788, 794, 800, 818, 890, 920, 932, 989, 1028, 1058, 1136, 1187, 1238, 1271, 1292, 1310, 1346, 1466, 1490, 1550, 1577, 1583, 1823, 1838, 1856, 1865, 1913, 2003, 2075

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Mar 04 2010

Necessarily, m == 2 (mod 3).

			2 is a term: 2999 = prime(430), 2999+2 = 3001 = prime(431).
8 is a term: 8999 = prime(1117), 8999+2 = 9001 = prime(1118).

Richard K. Guy: Unsolved Problems in Number Theory, New York, Springer-Verlag, 1994.
Theo Kempermann: Zahlentheoretische Kostproben, Harri Deutsch, 2. aktualisierte Auflage 2005.
Helmut Kracke, Mathe-musische Knobelisken, Duemmler Bonn, 2. Auflage 1983.

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

Cf. A001359, A158277, A158861, A173937.

tp999Q[n_]:=Module[{c=FromDigits[Join[IntegerDigits[n],{9,9,9}]]}, And @@ PrimeQ[c+{0,2}]]; Select[Range[2500],tp999Q] (* Harvey P. Dale, Oct 03 2013 *)
Select[3 Range[0,700]+2,AllTrue[1000#+{999,1001},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 05 2021 *)

isok(m) = my(x=eval(Str(m, 999))); isprime(x) && isprime(x+2); \\ Michel Marcus, Mar 08 2023

A288021 Prime p1 of consecutive primes p1, p2, where p2 - p1 = 4, and p1, p2 are in different decades.

Original entry on oeis.org

7, 19, 37, 67, 79, 97, 109, 127, 229, 277, 307, 349, 379, 397, 439, 457, 487, 499, 739, 757, 769, 859, 877, 907, 937, 967, 1009, 1087, 1279, 1297, 1429, 1447, 1489, 1549, 1567, 1579, 1597, 1609, 1867, 1999, 2137, 2239, 2269, 2347, 2377, 2389, 2437, 2539, 2617, 2659, 2689, 2707, 2749, 2797, 2857

Offset: 1

Hartmut F. W. Hoft, Jun 04 2017

nonn

The unit digits of the numbers in the sequence are 7's or 9's.

			7 is in this sequence since pair (7,11) is the first with difference 4 spanning a multiple of 10.

Cf. A001359, A023201, A023203, A031924, A031925, A031928, A046117, A046132, A158277, A158861, A160370, A160440, A160500, A287049, A287050, A060229, A288022, A288024.

a288021[n_] := Map[Last, Select[Map[{NextPrime[#, 1], NextPrime[#, -1]}&, Range[10, n, 10]], First[#]-Last[#]==4&]]
a288021[3000] (* data *)

A288022 Prime p1 of consecutive primes p1, p2, where p2 - p1 = 6, and p1, p2 are in different decades.

Original entry on oeis.org

47, 157, 167, 257, 367, 557, 587, 607, 647, 677, 727, 947, 977, 1097, 1117, 1187, 1217, 1367, 1657, 1747, 1777, 1907, 1987, 2207, 2287, 2417, 2467, 2677, 2837, 2897, 2957, 3307, 3407, 3607, 3617, 3637, 3727, 3797, 4007, 4357, 4457, 4507, 4597, 4657, 4937, 4987

Offset: 1

Hartmut F. W. Hoft, Jun 04 2017

nonn

The unit digits of the numbers in the sequence are 7's.

Number of terms < 10^k: 0, 0, 1, 13, 81, 565, 4027, 30422, 237715, ... - Muniru A Asiru, Jan 09 2018

			47 is in the sequence since pair (47,53) is the first with difference 6 spanning a multiple of 10.

Muniru A Asiru, Table of n, a(n) for n = 1..10000

Cf. A001359, A023201, A023203, A031924, A031925, A031928, A046117, A046132, A158277, A158861, A160370, A160440, A160500, A287049, A287050, A060229, A288021, A288024.

P:=Filtered([1..20000], IsPrime);
P1:=List(Filtered(Filtered(List([1..Length(P)-1],n->[P[n],P[n+1]]),i->i[2]-i[1]=6),j->j[1] mod 5=2),k->k[1]); # Muniru A Asiru, Jul 08 2017

for n from 1 to 2000 do if [ithprime(n+1)-ithprime(n), ithprime(n) mod 5] = [6,2] then print(ithprime(n)); fi; od; # Muniru A Asiru, Jan 19 2018

a288022[n_] := Map[Last, Select[Map[{NextPrime[#, 1], NextPrime[#, -1]}&, Range[10, n, 10]], First[#]-Last[#]==6&]]
a288022[3000] (* data *)

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

Original entry on oeis.org

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 *)

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.

cp's OEIS Frontend

A060229 Smaller member of a twin prime pair whose mean is a multiple of A002110(3)=30.

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A158277 The lesser of twin prime pairs with each prime in a different century.

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A173976 Numbers m such that the concatenation of m and 999 is the lesser of twin primes, i.e., a millennium twin prime couple.

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A288021 Prime p1 of consecutive primes p1, p2, where p2 - p1 = 4, and p1, p2 are in different decades.

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A288022 Prime p1 of consecutive primes p1, p2, where p2 - p1 = 6, and p1, p2 are in different decades.

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GAP

Maple

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

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Mathematica

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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