A006512 -id:A006512 - cp's OEIS frontend

A218829 Number of ordered ways to write n = k + m with k > 0 and m > 0 such that prime(k) + 2 and prime(prime(m)) + 2 are both prime.

0, 0, 1, 2, 2, 3, 2, 3, 4, 2, 3, 2, 2, 3, 2, 4, 3, 2, 3, 3, 3, 1, 3, 3, 1, 4, 4, 2, 3, 4, 4, 4, 4, 5, 3, 4, 4, 1, 4, 4, 3, 5, 4, 3, 3, 4, 6, 3, 5, 5, 3, 3, 3, 2, 4, 5, 4, 5, 4, 2, 3, 4, 4, 5, 5, 7, 4, 5, 2, 6, 4, 5, 7, 3, 5, 6, 2, 4, 3, 2

Offset: 1

Zhi-Wei Sun, Feb 05 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 2, and a(n) = 1 only for n = 3, 22, 25, 38, 101, 273.
(ii) Each n = 2, 3, ... can be written as k + m with k > 0 and m > 0 such that 6*k - 1, 6*k + 1 and prime(prime(m)) + 2 are all prime.
(iii) Any integer n > 5 can be written as k + m with k > 0 and m > 0 such that phi(k) - 1, phi(k) + 1 and prime(prime(m)) + 2 are all prime, where phi(.) is Euler's totient function.
(iv) If n > 2 is neither 10 nor 31, then n can be written as k + m with k > 0 and m > 0 such that prime(k) + 2 and prime(prime(prime(m))) + 2 are both prime.
(v) If n > 1 is not equal to 133, then n can be written as k + m with k > 0 and m > 0 such that 6*k - 1, 6*k + 1 and prime(prime(prime(m))) + 2 are all prime.
Clearly, each part of the conjecture implies the twin prime conjecture.
We have verified part (i) for n up to 10^9. See the comments in A237348 for an extension of this part.

			a(3) = 1 since 3 = 2 + 1 with prime(2) + 2 = 3 + 2 = 5 and prime(prime(1)) + 2 = prime(2) + 2 = 5 both prime.
a(22) = 1 since 22 = 20 + 2 with prime(20) + 2 = 71 + 2 = 73 and prime(prime(2)) + 2 = prime(3) + 2 = 5 + 2 = 7 both prime.
a(25) = 1 since 25 = 2 + 23 with prime(2) + 2 = 3 + 2 = 5 and prime(prime(23)) + 2 = prime(83) + 2 = 431 + 2 = 433 both prime.
a(38) = 1 since 38 = 35 + 3 with prime(35) + 2 = 149 + 2 = 151 and prime(prime(3)) + 2 = prime(5) + 2 = 11 + 2 = 13 both prime.
a(101) = 1 since 101 = 98 + 3 with prime(98) + 2 = 521 + 2 = 523 and prime(prime(3)) + 2 = prime(5) + 2 = 11 + 2 = 13 both prime.
a(273) = 1 since 273 = 2 + 271 with prime(2) + 2 = 3 + 2 = 5 and prime(prime(271)) + 2 = prime(1741) + 2 = 14867 + 2 = 14869 both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Andrei-Lucian Dragoi, The "Vertical" Generalization of the Binary Goldbach's Conjecture as Applied on "Iterative" Primes with (Recursive) Prime Indexes (i-primeths), Journal of Advances in Mathematics and Computer Science (2017), Vol. 25, No. 2, pp. 1-32.
Zhi-Wei Sun, Unification of Goldbach's conjecture and the twin prime conjecture, a message to Number Theory List, Jan. 29, 2014.
Zhi-Wei Sun, Super Twin Prime Conjecture, a message to Number Theory List, Feb. 6, 2014.
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000010, A000040, A001359, A002822, A006512, A072281, A182662, A199920, A236531, A236566, A236831, A236968, A237127, A237130, A237168, A237253, A237259, A237260, A237348.

pq[n_]:=PrimeQ[Prime[n]+2]
PQ[n_]:=PrimeQ[Prime[Prime[n]]+2]
a[n_]:=Sum[If[pq[k]&&PQ[n-k],1,0],{k,1,n-1}]
Table[a[n],{n,1,80}]

A228615 Determinant of the n X n matrix with (i,j)-entry equal to 1 or 0 according as 2(i + j) - 1 and 2(i + j) + 1 are twin primes or not.

Original entry on oeis.org

1, -1, -1, -1, 0, 0, -1, 1, 1, 1, -1, 0, 0, 0, -1, -1, 1, 1, 1, -1, 2, 8, -18, -9, -1, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4096, -4096, 64, -20, -125, 5, -6, -216, 24, 152, 54872, -106742, 14045, 125, -21125, -274625, -274625, 10985, -16731, -970299, 1804275, 1312200, 373248, -691488, -192080

Offset: 1

Zhi-Wei Sun, Aug 27 2013

sign

Conjecture: a(n) is nonzero for any n > 35.

Clearly this conjecture implies the twin prime conjecture.

			a(1) = 1 since 2*(1 + 1) - 1 = 3 and 2*(1 + 1) + 1 = 5 are twin primes.

Harvey P. Dale, Table of n, a(n) for n = 1..500 (First 200 terms from Zhi-Wei Sun)

Cf. A001359, A006512, A069191, A228557, A228591.

a[n_]:=a[n]=Det[Table[If[PrimeQ[2(i+j)-1]&&PrimeQ[2(i+j)+1],1,0],{i,1,n},{j,1,n}]]
Table[a[n],{n,1,20}]
Table[Det[Table[If[AllTrue[2(i+j)+{1,-1},PrimeQ],1,0],{i,k},{j,k}]],{k,60}] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 21 2019 *)

A236119 Primes p with prime(p) - p - 1 and prime(p) - p + 1 both prime.

Original entry on oeis.org

5, 17, 23, 41, 71, 83, 173, 293, 337, 353, 563, 571, 719, 811, 911, 953, 1201, 1483, 1579, 1877, 2081, 2089, 2309, 2579, 2749, 2803, 3329, 3343, 3511, 3691, 3779, 3851, 3881, 3907, 4021, 4049, 4093, 4657, 4813, 5051, 5179, 5333, 5519, 5591, 6053, 6547, 6841, 7151, 7723, 8209

Offset: 1

Zhi-Wei Sun, Jan 19 2014

nonn

By the conjecture in A236097, this sequence should have infinitely many terms.

			a(1) = 5 since neither prime(2) - 2 - 1 = 0 nor prime(3) - 3 - 1 = 1 is prime, but prime(5) - 5 - 1 = 5 and prime(5) - 5 + 1 = 7 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000040, A001359, A006512, A014574, A234695, A235925, A236097.

p[n_]:=PrimeQ[Prime[n]-n-1]&&PrimeQ[Prime[n]-n+1]
n=0;Do[If[p[Prime[k]],n=n+1;Print[n," ",Prime[k]]],{k,1,1100}]

s=[]; forprime(p=2, 10000, if(isprime(prime(p)-p-1) && isprime(prime(p)-p+1), s=concat(s, p))); s \\ Colin Barker, Jan 19 2014

A373408 Minimum of the n-th maximal antirun of squarefree numbers differing by more than one.

Original entry on oeis.org

1, 2, 3, 6, 7, 11, 14, 15, 22, 23, 30, 31, 34, 35, 38, 39, 42, 43, 47, 58, 59, 62, 66, 67, 70, 71, 74, 78, 79, 83, 86, 87, 94, 95, 102, 103, 106, 107, 110, 111, 114, 115, 119, 123, 130, 131, 134, 138, 139, 142, 143, 146, 155, 158, 159, 166, 167, 174, 178, 179

Offset: 1

Gus Wiseman, Jun 05 2024

nonn

The maximum is given by A007674.
An antirun of a sequence (in this case A005117) is an interval of positions at which consecutive terms differ by more than one.
Consists of 1 and all squarefree numbers n such that n - 1 is also squarefree.

			Row-minima of:
   1
   2
   3   5
   6
   7  10
  11  13
  14
  15  17  19  21
  22
  23  26  29
  30
  31  33
  34
  35  37
  38
  39  41
  42
  43  46
  47  51  53  55  57

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers

Functional neighbors: A005381, A006512, A007674, A072284, A373127, A373410, A373411.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
Cf. A001223, A049093, A049094, A061398, A077641, A077643, A112925, A112926, A350842, A372683, A373123.

First/@Split[Select[Range[100],SquareFreeQ],#1+1!=#2&]//Most

a(1) = 1; a(n>1) = A007674(n-1) + 1.

A373822 Sum of the n-th maximal run of first differences of odd primes.

Original entry on oeis.org

4, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 12, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 12, 4, 12, 2, 10, 2, 4, 2, 24, 4, 2, 4, 6, 2, 10, 18, 2, 6, 4, 2, 10, 14, 4, 2, 4, 14, 6, 10, 2, 4, 6, 8, 12, 4, 6, 8, 4, 8, 10, 2, 10, 2, 6, 4, 6, 8, 4, 2, 4

Offset: 1

Gus Wiseman, Jun 22 2024

nonn

Run-sums of A001223. For run-lengths instead of run-sums we have A333254.

			The odd primes are
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, ...
with first differences
2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, ...
with runs
(2,2), (4), (2), (4), (2), (4), (6), (2), (6), (4), (2), (4), (6,6), ...
with sums a(n).

Run-sums of A001223.
For run-lengths we have A333254, run-lengths of run-lengths A373821.
Dividing by two gives A373823.
A000040 lists the primes.
A027833 gives antirun lengths of odd primes (partial sums A029707).
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
A071148 gives partial sums of odd primes.
A373820 gives run-lengths of antirun-lengths of odd primes.
For prime runs: A001359, A006512, A025584, A067774, A373405, A373406.
Cf. A037201, A038664, A054265, A176246, A251092, A373404, A373825.

Total/@Split[Differences[Select[Range[3,1000],PrimeQ]]]

A062721 Numbers k such that k is a product of two primes and k-2 is prime.

Original entry on oeis.org

4, 9, 15, 21, 25, 33, 39, 49, 55, 69, 85, 91, 111, 115, 129, 133, 141, 159, 169, 183, 201, 213, 235, 253, 259, 265, 295, 309, 319, 339, 355, 361, 381, 391, 403, 411, 445, 451, 469, 481, 489, 493, 501, 505, 511, 543, 559, 565, 573, 579, 589, 633, 649, 655, 679

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 14 2001

This sequence is a subsequence of A107986, which only requires k to be composite. The first term in that sequence which is not in this sequence is 45, a number with three prime factors. - Alonso del Arte, May 03 2014

Zak Seidov, Table of n, a(n) for n = 1..10000.

Cf. A006512, A001358, A043326.

Cf. A063637, A046315, A089268.

Select[ Range[ 2, 1500 ], Plus @@ Last@Transpose@FactorInteger[ # ] == 2 && PrimeQ[ # - 2 ] & ]
Select[Range[700], PrimeOmega[#] == 2 && PrimeQ[# - 2]&] (* Harvey P. Dale, Mar 25 2013 *)

{ n=0; for (m=1, 10^9, a=prime(m) + 2; f=factor(a)~; if ((length(f)==1 && f[2, 1]==2) || (length(f)==2 && f[2, 1]==1 && f[2, 2]==1), write("b062721.txt", n++, " ", a); if (n==10000, break)) ) } \\ Harry J. Smith, Aug 09 2009

A071696 Greater members of twin prime pairs of form (4k+1,4k+3), k>0.

Original entry on oeis.org

7, 19, 31, 43, 103, 139, 151, 199, 271, 283, 463, 523, 571, 619, 643, 811, 823, 859, 883, 1051, 1063, 1231, 1279, 1291, 1303, 1483, 1699, 1723, 1879, 1951, 1999, 2083, 2131, 2143, 2239, 2311, 2383, 2551, 2659, 2731, 2791, 2803, 2971

Offset: 1

Reinhard Zumkeller, Jun 04 2002

nonn

Corresponding lesser members: A071695(n).

A010051(a(n)) * A010051(a(n)-2) = 1. - Reinhard Zumkeller, Nov 10 2013

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A006512, A071699, A071697.

Cf. Subsequence of A002145.

a071696 n = a071696_list !! (n-1)
a071696_list = [p | p <- tail a002145_list, a010051' (p - 2) == 1]
-- Reinhard Zumkeller, Nov 10 2013

Select[Partition[Prime[Range[500]],2,1],#[[2]]-#[[1]]==2&&IntegerQ[ (#[[1]]-1)/4]&][[All,2]] (* Harvey P. Dale, Aug 27 2021 *)
Select[Table[4k+{1,3},{k,750}],AllTrue[#,PrimeQ]&][[;;,2]] (* Harvey P. Dale, Sep 10 2024 *)

A072281 Numbers n such that phi(n) + 1 and phi(n) - 1 are twin primes.

Original entry on oeis.org

5, 7, 8, 9, 10, 12, 13, 14, 18, 19, 21, 26, 27, 28, 31, 36, 38, 42, 43, 49, 54, 61, 62, 73, 77, 86, 91, 93, 95, 98, 99, 103, 109, 111, 117, 122, 124, 133, 135, 139, 146, 148, 151, 152, 154, 171, 181, 182, 186, 189, 190, 193, 198, 199, 206, 209, 216, 217, 218, 221, 222

Offset: 1

Joseph L. Pe, Jul 10 2002

Phi(n) is middle term between twin primes (A014574). Union of A006512 and A068019; intersection of A039698 and A078892. - Ray Chandler, May 26 2008

The positions of isolated nonprimes in A000010. - Juri-Stepan Gerasimov, Nov 10 2009

			phi(14) + 1 = 7 and phi(14) - 1 = 5, so 14 is a term of the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A000040, A006512, A014574, A039698, A068019, A078892.

Select[Range[10^3], PrimeQ[EulerPhi[ # ] + 1] && PrimeQ[EulerPhi[ # ] - 1] &]
Select[Range[300],And@@PrimeQ[EulerPhi[#]+{1,-1}]&] (* Harvey P. Dale, Apr 07 2012 *)

isok(n) = my(p); isprime(p=eulerphi(n)-1) && isprime(p+2); \\ Michel Marcus, Sep 29 2019

Extended by Ray Chandler, May 26 2008

A091182 Number of ways to write n = x + y (x >= y > 0) with xy - 1 and xy + 1 both prime.

Original entry on oeis.org

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

Offset: 1

Ray Chandler, Dec 27 2003

From Zhi-Wei Sun, Nov 27 2012: (Start)
Conjecture: a(n) > 0 for all n > 3120. This has been verified for n up to 5*10^7.
Note that if x >= y > 0 and x+y = n then n-1 = x+y-1 <= xy <= ((x+y)/2)^2 = n^2/4. So the conjecture implies that there are infinitely many twin primes.
For n=4,5,...,3120 we can write n = x+y (x >= y > 0) with xy-1 prime.
For each positive integer n <= 3120 different from 1,6,30,54, we can write n = x+y (x >= y > 0) with xy+1 prime.
More generally, we have the following conjecture: Let m be any positive integer. If n is sufficiently large and (m-1)n is even, then we can write n as x+y, where x and y are positive integers with xy-m and xy+m both prime. This general conjecture implies that for any positive even integer d there are infinitely many primes p and q with difference d. (End)
Sequence A090695 lists the 61 known values of n where a(n) = 0. - T. D. Noe, Nov 29 2012

			a(8)=1 since 8=6+2 with 6*2-1 and 6*2+1 both prime.
a(11)=2 since 11=6+5=9+2 with 6*5-1, 6*5+1, 9*2-1, 9*2+1 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..50000 (first 10000 terms from T. D. Noe)
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Cf. A001359, A006512, A014574, A090695, A091183, A219157.

with(numtheory); a:=n->sum( (pi((i)*(n-i)+1) - pi((i)*(n-i)))*(pi((i)*(n-i)-1) - pi((i)*(n-i) - 2)) , i=1..floor(n/2) ); seq(a(k),k=1..100); # Wesley Ivan Hurt, Jan 21 2013

Table[cnt = 0; Do[If[PrimeQ[k*(n - k) - 1] && PrimeQ[k*(n - k) + 1], cnt++], {k, n/2}]; cnt, {n, 100}] (* Zhi-Wei Sun, edited by T. D. Noe, Nov 29 2012 *)

Edited by N. J. A. Sloane, Nov 29 2012

A199920 Number of ways to write n = p+k with p, p+6, 6k-1 and 6k+1 all prime.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 1, 2, 0, 3, 1, 3, 2, 2, 2, 3, 2, 2, 1, 2, 3, 3, 3, 1, 1, 3, 2, 4, 1, 2, 2, 3, 3, 3, 2, 4, 2, 4, 3, 3, 5, 3, 3, 3, 3, 4, 5, 3, 3, 3, 3, 5, 4, 4, 3, 4, 3, 3, 2, 3, 6, 5, 4, 2, 1, 3, 5, 5, 5, 2, 2, 3, 5, 3, 5, 4, 5, 2, 3, 2, 5, 5, 6, 4, 2, 3, 3, 4, 3, 3, 5, 4, 3, 1, 1, 4, 5, 7

Offset: 1

Zhi-Wei Sun, Dec 22 2012

Conjecture: a(n)>0 for all n>11.
This implies that there are infinitely many twin primes and also infinitely many sexy primes. It has been verified for n up to 10^9. See also A199800 for a weaker version of this conjecture.
Zhi-Wei Sun also conjectured that any integer n>6 not equal to 319 can be written as p+k with p, p+6, 3k-2+(n mod 2) and 3k+2-(n mod 2) all prime.

			a(21)=1 since 21=11+10 with 11, 11+6, 6*10-1 and 6*10+1 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..50000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Cf. A001359, A006512, A023201, A199800, A219055, A219864, A219923, A002375.

a[n_]:=a[n]=Sum[If[PrimeQ[Prime[k]+6]==True&&PrimeQ[6(n-Prime[k])-1]==True&&PrimeQ[6(n-Prime[k])+1]==True,1,0],{k,1,PrimePi[n]}]
Do[Print[n," ",a[n]],{n,1,100}]
Table[Count[Table[{n-i,i},{i,n-1}],?(AllTrue[{#[[1]],#[[1]]+6,6#[[2]]-1,6#[[2]]+1},PrimeQ]&)],{n,100}] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale, May 19 2015 *)

a(n)=my(s,p=2,q=3); forprime(r=5,n+5, if(r-p==6 && isprime(6*n-6*p-1) && isprime(6*n-6*p+1), s++); if(r-q==6 && isprime(6*n-6*q-1) && isprime(6*n-6*q+1), s++); p=q; q=r); s \\ Charles R Greathouse IV, Jul 31 2016

cp's OEIS Frontend

A218829 Number of ordered ways to write n = k + m with k > 0 and m > 0 such that prime(k) + 2 and prime(prime(m)) + 2 are both prime.

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A228615 Determinant of the n X n matrix with (i,j)-entry equal to 1 or 0 according as 2*(i + j) - 1 and 2*(i + j) + 1 are twin primes or not.

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A236119 Primes p with prime(p) - p - 1 and prime(p) - p + 1 both prime.

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PARI

A373408 Minimum of the n-th maximal antirun of squarefree numbers differing by more than one.

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Formula

A373822 Sum of the n-th maximal run of first differences of odd primes.

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Mathematica

A062721 Numbers k such that k is a product of two primes and k-2 is prime.

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PARI

A071696 Greater members of twin prime pairs of form (4*k+1,4*k+3), k>0.

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Mathematica

A072281 Numbers n such that phi(n) + 1 and phi(n) - 1 are twin primes.

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A228615 Determinant of the n X n matrix with (i,j)-entry equal to 1 or 0 according as 2(i + j) - 1 and 2(i + j) + 1 are twin primes or not.

A071696 Greater members of twin prime pairs of form (4k+1,4k+3), k>0.