A023201 -id:A023201 - cp's OEIS frontend

A231608 Table whose n-th row consists of primes p such that p + 2n is also prime, read by antidiagonals.

3, 3, 5, 5, 7, 11, 3, 7, 13, 17, 3, 5, 11, 19, 29, 5, 7, 11, 13, 37, 41, 3, 7, 13, 23, 17, 43, 59, 3, 5, 11, 19, 29, 23, 67, 71, 5, 7, 17, 17, 31, 53, 31, 79, 101, 3, 11, 13, 23, 19, 37, 59, 37, 97, 107, 7, 11, 13, 31, 29, 29, 43, 71, 41, 103, 137

Offset: 1

T. D. Noe, Nov 26 2013

			The following sequences are read by antidiagonals
{3, 5, 11, 17, 29, 41, 59, 71, 101, 107,...}
{3, 7, 13, 19, 37, 43, 67, 79, 97, 103,...}
{5, 7, 11, 13, 17, 23, 31, 37, 41, 47,...}
{3, 5, 11, 23, 29, 53, 59, 71, 89, 101,...}
{3, 7, 13, 19, 31, 37, 43, 61, 73, 79,...}
{5, 7, 11, 17, 19, 29, 31, 41, 47, 59,...}
{3, 5, 17, 23, 29, 47, 53, 59, 83, 89,...}
{3, 7, 13, 31, 37, 43, 67, 73, 97, 151,...}
{5, 11, 13, 19, 23, 29, 41, 43, 53, 61,...}
{3, 11, 17, 23, 41, 47, 53, 59, 83, 89,...}
...

T. D. Noe, Rows n = 1..100 of triangle, flattened

Cf. A001359, A023200, A023201, A023202, A023203.
Cf. A046133, A153417, A049488, A153418, A153419.
Cf. A020483 (numbers in first column).
Cf. A086505 (numbers on the diagonal).

A231608 := proc(n,k)
    local j,p ;
    j := 0 ;
    p := 2;
    while j < k do
        if isprime(p+2*n ) then
            j := j+1 ;
        end if;
        if j = k then
            return p;
        end if;
        p := nextprime(p) ;
    end do:
end proc:
for n from 1 to 10 do
    for k from 1 to 10 do
        printf("%3d ",A231608(n,k)) ;
    end do;
    printf("\n") ;
end do: # R. J. Mathar, Nov 19 2014

nn = 10; t = Table[Select[Range[100*nn], PrimeQ[#] && PrimeQ[# + 2*n] &, nn], {n, nn}]; Table[t[[n-j+1, j]], {n, nn}, {j, n}]

A262084 Numbers m that satisfy the equation phi(m + 6) = phi(m) + 6 where phi(m) = A000010(m) is Euler's totient function.

Original entry on oeis.org

5, 7, 11, 13, 17, 21, 23, 31, 37, 40, 41, 47, 53, 56, 61, 67, 73, 83, 88, 97, 98, 101, 103, 107, 131, 136, 151, 152, 156, 157, 167, 173, 191, 193, 223, 227, 233, 237, 248, 251, 257, 263, 271, 277, 296, 307, 311, 328, 331, 347, 353, 367, 373, 376, 383, 433, 443

Offset: 1

Kevin J. Gomez, Sep 10 2015

The majority of solutions can be predicted by known properties of the equality. There are several solutions that do not fit these parameters.
An odd natural number m is a solution if m and m + 6 are both prime (sexy primes) (A023201).
Among the solutions for even natural numbers are all m = 8*p with odd primes p such that 4*p+3 is a prime number. Proof: From A000010 we can learn that the formula phi(p*2) = floor(((2 + p - 1) mod p)/(p - 1)) + p - 1 is known. If we define p = 4*q+3 and m = 8*q and insert, we will obtain phi(8*q+6) = 4*q+2. Also it is known that phi(8*q) = 4*q-4 if q is any odd prime. - Thomas Scheuerle, Dec 20 2024

			5 is a term since phi(5+6) = 10 = 6 + 4 = phi(5) + 6.

Peter Kagey, Table of n, a(n) for n = 1..10000
Wikipedia, Euler's totient function

Cf. A000010, A023201.

Cf. A001838 (k=2), A056772 (k=4), A262085 (k=8), A262086 (k=10).

[n: n in [1..500] | EulerPhi(n+6) eq EulerPhi(n)+6]; // Vincenzo Librandi, Sep 11 2015

Select[Range@500, EulerPhi@(# + 6)== EulerPhi[#] + 6 &] (* Vincenzo Librandi, Sep 11 2015 *)

is(n)=eulerphi(n + 6) == eulerphi(n) + 6 \\ Anders Hellström, Sep 11 2015

[n for n in [1..1000] if euler_phi(n+6)==euler_phi(n)+6] # Tom Edgar, Sep 10 2015

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 *)

A080841 Number of pairs (p,q) of (not necessarily consecutive) primes with q-p = 6 and q < 10^n.

Original entry on oeis.org

0, 15, 74, 411, 2447, 16386, 117207, 879908, 6849047, 54818296, 448725003, 3741217498

Offset: 1

Jason Earls, Mar 28 2003

nonn

Note that one has to be careful to distinguish between pairs of consecutive primes (p,q) with q-p = 6 (A031924), and pairs of primes (p,q) with q-p = 6 (A023201). Here we consider the latter, whereas A093738 considers the former. - N. J. A. Sloane, Mar 07 2021

A. Granville, G. Martin, Prime number races, Amer. Math. Monthly vol 113, no 1 (2006) p 1.
Eric Weisstein's World of Mathematics, Sexy Primes. [The definition in this webpage is unsatisfactory, because it defines a "sexy prime" as a pair of primes.- _N. J. A. Sloane_, Mar 07 2021].

Cf. A023201, A046117, A007508, A080840, A006512, A046132, A046117.

A104228 Primes one larger than the sum over a sexy prime pair.

Original entry on oeis.org

17, 29, 41, 53, 89, 101, 113, 173, 269, 353, 389, 461, 509, 521, 701, 773, 929, 1013, 1181, 1193, 1289, 1301, 1361, 1721, 1889, 1901, 1949, 2213, 2381, 2441, 2609, 2729, 2741, 2861, 2969, 3209, 3221, 3821, 4001, 4133, 4421, 4481, 4673, 4793, 4889, 5381, 5393, 5801, 5813, 6173, 6653, 7349, 7529

Offset: 1

Giovanni Teofilatto, Apr 02 2005

Primes of the form A023201(i)+A046117(i)+1 - R. J. Mathar, Nov 26 2008

			17=5+11+1 is prime and one larger than the sum 5+11 over the first sexy prime pair. - _R. J. Mathar_, Nov 26 2008

Cf. A023201, A046117, A104227, A104229.

Inserted 89 and extended beyond a(8). - R. J. Mathar, Nov 26 2008

A230261 Number of ways to write 2*n - 1 = p + q with p, p + 6 and q^4 + 1 all prime, where q is a positive integer.

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 4, 3, 3, 4, 4, 3, 3, 4, 1, 5, 4, 3, 5, 5, 5, 4, 6, 4, 5, 5, 3, 3, 5, 4, 4, 2, 6, 8, 5, 4, 6, 7, 5, 5, 7, 6, 5, 7, 4, 6, 6, 3, 6, 5, 7, 6, 4, 6, 7, 6, 2, 7, 6, 2, 5, 5, 3, 7, 7, 5, 7, 9, 6, 7, 4, 6, 6, 4, 3, 9, 7, 4, 9, 9, 6, 5, 10, 8, 5, 9, 6, 7, 8, 4

Offset: 1

Zhi-Wei Sun, Oct 14 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 3. Also, any odd number greater than 6 can be written as p + q (q > 0) with p, p + 6 and q^2 + 1 all prime.
(ii) Any integer n > 1 can be written as x + y (x, y > 0) with x^4 + 1 and y^2 + y + 1 both prime.
(iii) Each integer n > 2 can be expressed as x + y (x, y > 0) with 4*x^2 + 3 and 4*y^2 -3 both prime.
Either of parts (i) and (ii) implies that there are infinitely many primes of the form x^4 + 1.

			a(6) = 2 since 2*6-1 = 5 + 6 = 7 + 4, and 5, 5+6 = 11, 7, 7+6 = 13, 6^4+1 = 1297 and 4^4+1 = 257 are all prime.
a(25) = 1 since 2*25-1 = 47 + 2, and 47, 47+6 = 53, 2^4+1 = 17 are all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.

Cf. A000068, A023201, A037896, A219864, A227908, A227909, A230241, A230252, A230254.

a[n_]:=Sum[If[PrimeQ[Prime[i]+6]&&PrimeQ[(2n-1-Prime[i])^4+1],1,0],{i,1,PrimePi[2n-2]}]
Table[a[n],{n,1,100}]

a(n)=my(s,p=5,q=7);forprime(r=11,2*n+4,if(r-p==6&&isprime((2*n-1-p)^4+1),s++); if(r-q==6&&isprime((2*n-1-q)^4+1),s++); p=q;q=r);s \\ Charles R Greathouse IV, Oct 14 2013

A236508 a(n) = |{0 < k < n-2: p = 2*phi(k) + phi(n-k)/2 - 1, p + 2, p + 6 and prime(p) + 6 are all prime}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 2, 1, 1, 2, 1, 3, 2, 2, 0, 2, 3, 1, 2, 1, 3, 3, 2, 2, 1, 1, 1, 3, 0, 2, 3, 2, 1, 3, 0, 2, 0, 1, 1, 1, 1, 2, 0, 0, 0, 0, 2, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 2, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1

Offset: 1

Zhi-Wei Sun, Jan 27 2014

nonn

Conjecture: a(n) > 0 for all n > 146.
We have verified this for n up to 52000.
The conjecture implies that there are infinitely many prime triples {p, p + 2, p + 6} with {prime(p), prime(p) + 6} a sexy prime pair. See A236509 for such primes p.

			a(13) = 1 since 2*phi(3) + phi(10)/2 - 1 = 5, 5 + 2 = 7, 5 + 6 = 11 and prime(5) + 6 = 11 + 6 = 17 are all prime.
a(244) = 1 since 2*phi(153) + phi(244-153)/2 - 1 = 2*96 + 72/2 - 1 = 227, 227 + 2 = 229, 227 + 6 = 233 and prime(227) + 6 = 1433 + 6 = 1439 are all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000010, A000040, A022004, A023201, A046117, A236464, A236472, A236509.

p[n_]:=PrimeQ[n]&&PrimeQ[n+2]&&PrimeQ[n+6]&&PrimeQ[Prime[n]+6]
f[n_,k_]:=2*EulerPhi[k]+EulerPhi[n-k]/2-1
a[n_]:=Sum[If[p[f[n,k]],1,0],{k,1,n-3}]
Table[a[n],{n,1,100}]

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.

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A231608 Table whose n-th row consists of primes p such that p + 2n is also prime, read by antidiagonals.

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A262084 Numbers m that satisfy the equation phi(m + 6) = phi(m) + 6 where phi(m) = A000010(m) is Euler's totient function.

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

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Mathematica

A080841 Number of pairs (p,q) of (not necessarily consecutive) primes with q-p = 6 and q < 10^n.

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PARI

Extensions

A104228 Primes one larger than the sum over a sexy prime pair.

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A230261 Number of ways to write 2*n - 1 = p + q with p, p + 6 and q^4 + 1 all prime, where q is a positive integer.

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