A078611 -id:A078611 - cp's OEIS frontend

A082467 Least k>0 such that n-k and n+k are both primes.

1, 2, 1, 4, 3, 2, 3, 6, 1, 6, 3, 2, 3, 6, 1, 12, 3, 2, 9, 6, 5, 6, 3, 4, 9, 12, 1, 12, 9, 4, 3, 6, 5, 6, 9, 2, 3, 12, 1, 24, 3, 2, 15, 6, 5, 12, 3, 8, 9, 6, 7, 12, 3, 4, 15, 12, 1, 18, 9, 4, 3, 6, 5, 6, 15, 2, 3, 12, 1, 6, 15, 4, 3, 6, 5, 18, 9, 2, 15, 24, 5, 12, 3, 14, 9, 18, 7, 12, 9, 4, 15, 6, 7, 30, 9

Offset: 4

Benoit Cloitre, Apr 27 2003

nonn

The existence of k>0 for all n >= 4 is equivalent to the strong Goldbach Conjecture that every even number >= 8 is the sum of two distinct primes.
n and k are coprime, because otherwise n + k would be composite. So the rational sequence r(n) = a(n)/n = k/n is injective. - Jason Kimberley, Sep 21 2011
Because there are arbitrarily many composites from m!+2 to m!+m, there are also arbitrarily large a(n) but they increase very slowly. The twin prime conjecture implies that infinitely many a(n) are 1. - Juhani Heino, Apr 09 2020

			n=10: k=3 because 10-3 and 10+3 are both prime and 3 is the smallest k such that n +/- k are both prime.

Klaus Brockhaus, Table of n, a(n) for n = 4..5000
OEIS (Plot 2), log_10(A082467(n)/n) vs n
J. S. Kimberley, A082467

Cf. A087695, A087696, A087697, A087678, A087679, A087680, A087681, A087682, A087683, A087711.
Cf. A129301 (records), A129302 (where records occur).
Cf. A047160 (allows k=0).
Cf. A078611 (subset for prime n).

A082467 := func; [A082467(n):n in [4..98]]; // Jason Kimberley, Sep 03 2011

A082467 := proc(n) local k; k := 1+irem(n,2);
while n > k do if isprime(n-k) then if isprime(n+k)
then RETURN(k) fi fi; k := k+2 od; print("Goldbach erred!") end:
seq(A082467(i),i=4..90); # Peter Luschny, Sep 21 2011

f[n_] := Block[{k}, If[OddQ[n], k = 2, k = 1]; While[ !PrimeQ[n - k] || !PrimeQ[n + k], k += 2]; k]; Table[ f[n], {n, 4, 98}] (* Robert G. Wilson v, Mar 28 2005 *)

a(n)=if(n<0,0,k=1; while(isprime(n-k)*isprime(n+k) == 0,k++); k)

A078496(n)-a(n) = A078587(n)+a(n) = n.

Entries checked by Klaus Brockhaus, Apr 08 2007

A047160 For n >= 2, a(n) = smallest number m >= 0 such that n-m and n+m are both primes, or -1 if no such m exists.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 3, 2, 3, 0, 1, 0, 3, 2, 3, 0, 1, 0, 3, 2, 9, 0, 5, 6, 3, 4, 9, 0, 1, 0, 9, 4, 3, 6, 5, 0, 9, 2, 3, 0, 1, 0, 3, 2, 15, 0, 5, 12, 3, 8, 9, 0, 7, 12, 3, 4, 15, 0, 1, 0, 9, 4, 3, 6, 5, 0, 15, 2, 3, 0, 1, 0, 15, 4, 3, 6, 5, 0, 9, 2, 15, 0, 5, 12, 3, 14, 9, 0, 7, 12, 9, 4, 15, 6, 7, 0, 9, 2, 3

Offset: 2

Lior Manor

I have confirmed there are no -1 entries through integers to 4.29*10^9 using PARI. - Bill McEachen, Jul 07 2008
From Daniel Forgues, Jul 02 2009: (Start)
Goldbach's Conjecture: for all n >= 2, there are primes (distinct or not) p and q s.t. p+q = 2n. The primes p and q must be equidistant (distance m >= 0) from n: p = n-m and q = n+m, hence p+q = (n-m)+(n+m) = 2n.
Equivalent to Goldbach's Conjecture: for all n >= 2, there are primes p and q equidistant (distance >= 0) from n, where p and q are n when n is prime.
If this conjecture is true, then a(n) will never be set to -1.
Twin Primes Conjecture: there is an infinity of twin primes.
If this conjecture is true, then a(n) will be 1 infinitely often (for which each twin primes pair is (n-1, n+1)).
Since there is an infinity of primes, a(n) = 0 infinitely often (for which n is prime).
(End)
If n is composite, then n and a(n) are coprime, because otherwise n + a(n) would be composite. - Jason Kimberley, Sep 03 2011
From Jianglin Luo, Sep 22 2023: (Start)
a(n) < primepi(n)+sigma(n,0);
a(n) < primepi(primepi(n)+n);
a(n) < primepi(n), for n>344;
a(n) = o(primepi(n)), as n->+oo. (End)
If -1 < a(n) < n-3, then a(n) is divisible by 3 if and only if n is not divisible by 3, and odd if and only if n is even. - Robert Israel, Oct 05 2023

			16-3=13 and 16+3=19 are primes, so a(16)=3.

T. D. Noe, Table of n, a(n) for n = 2..10000
Jason Kimberley, Symmetrical plot of A047160

Cf. A001031, A002092, A002372, A002373, A002374, A002375, A014092, A025583, A035026, A047949, A071406, A082467, A102084, A103147, A112823, A155764, A155765, A177461, A078611, A010051, A045917, A325142.

a047160 n = if null ms then -1 else head ms
            where ms = [m | m <- [0 .. n - 1],
                            a010051' (n - m) == 1, a010051' (n + m) == 1]
-- Reinhard Zumkeller, Aug 10 2014

A047160:=func;[A047160(n):n in[2..100]]; // Jason Kimberley, Sep 02 2011

Table[k = 0; While[k < n && (! PrimeQ[n - k] || ! PrimeQ[n + k]), k++]; If[k == n, -1, k], {n, 2, 100}]
smm[n_]:=Module[{m=0},While[AnyTrue[n+{m,-m},CompositeQ],m++];m]; Array[smm,100,2] (* Harvey P. Dale, Nov 16 2024 *)

a(n)=forprime(p=n,2*n, if(isprime(2*n-p), return(p-n))); -1 \\ Charles R Greathouse IV, Jun 23 2017

10 N=2// 20 M=0// 30 if and{prmdiv(N-M)=N-M,prmdiv(N+M)=N+M} then print M;:goto 50// 40 inc M:goto 30// 50 inc N: if N>130 then stop// 60 goto 20

a(n) = n - A112823(n).

a(n) = A082467(n) * A005171(n), for n > 3. - Jason Kimberley, Jun 25 2012

More terms from Patrick De Geest, May 15 1999
Deleted a comment. - T. D. Noe, Jan 22 2009
Comment corrected and definition edited by Daniel Forgues, Jul 08 2009

A078497 The member r of a triple of primes (p,q,r) in arithmetic progression which sum to 3*prime(n) = A001748(n) = p + q + r.

Original entry on oeis.org

7, 11, 17, 19, 23, 31, 29, 41, 43, 43, 53, 67, 53, 59, 71, 79, 73, 83, 79, 97, 107, 107, 127, 113, 109, 113, 139, 137, 151, 149, 167, 151, 167, 163, 163, 199, 197, 179, 191, 199, 233, 223, 227, 241, 223, 283, 257, 277, 239, 251, 271, 263, 263, 269, 281, 313

Offset: 3

Serhat Sevki Dincer (sevki(AT)ug.bilkent.edu.tr), Nov 27 2002

nonn

In case more than one triple of primes p, q=p+d and r=p+2*d exists, we take r=a(n) from the triple with the smallest d. This shows the difference from A092940, which would take the maximum r over all triples. - R. J. Mathar, May 19 2007

			a(1) = 7 because 3+5+7 = 15;
a(2) = 11 because 3+7+11 = 21;
a(3) = 17 because 5+11+17= 33.

Cf. A078496, A078498, A001748, A092940, A071681, A078611.

A078497 := proc(n) local p3, i,d,r,p; p3 := ithprime(n) ; i := n+1 ; while true do r := ithprime(i) ; d := r-p3 ; p := p3-d ; if isprime(p) then RETURN(r) ; fi ; i := i+1 ; od ; RETURN(-1) ; end: for n from 3 to 60 do printf("%d, ",A078497(n)) ; od ; # R. J. Mathar, May 19 2007

f[n_] := Block[{p = Prime[n], k}, k = p + 1; While[ !PrimeQ[k] || !PrimeQ[2p - k], k++ ]; k]; Table[ f[n], {n, 3, 60}]

Edited and extended by Robert G. Wilson v, Nov 29 2002

Further edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar

A094709 Smallest k such that prime(n)# - k and prime(n)# + k are primes, where prime(n)# = A002110(n).

Original entry on oeis.org

0, 1, 1, 13, 1, 17, 59, 23, 79, 101, 83, 239, 71, 149, 367, 73, 911, 313, 373, 523, 313, 331, 197, 101, 1493, 523, 293, 577, 2699, 1481, 1453, 5647, 647, 419, 757, 4253, 509, 239, 10499, 191, 4013, 2659, 617, 6733, 1297, 971

Offset: 1

Reinhard Zumkeller, May 21 2004

nonn

a(n) = A002110(n) - A094710(n) = A094711(n) - A002110(n),

Goldbach's conjecture implies that a(n) is defined for all n. - David Wasserman, May 31 2007

			a(4)=13 because prime(4)=7, 7# = 2*3*5*7 = 210, and 210 - 13 and 210 + 13 are primes.

David Wasserman, Table of n, a(n) for n = 1..250

Cf. A078611, A002110, A094710, A094711.

pc[n_]:=Module[{x=0,i=0},Do[If[PrimeQ[n-i]&&PrimeQ[n+i],x=i;Break[]],{i,9!}];x]; r=2;lst={};Do[p=Prime[n];r*=p;AppendTo[lst,pc[r]],{n,2,2*4!}];lst (* Vladimir Joseph Stephan Orlovsky, Jun 14 2009 *)
sk[n_]:=Module[{k=0},While[!PrimeQ[n+k]||!PrimeQ[n-k],k++];k]; sk/@ FoldList[ Times,Prime[Range[50]]] (* Harvey P. Dale, Apr 03 2022 *)

from sympy import isprime, prime, primerange
def aupton(terms):
  phash, alst = 2, [0]
  for p in primerange(3, prime(terms)+1):
    phash *= p
    for k in range(1, phash//2):
      if isprime(phash-k) and isprime(phash+k): alst.append(k); break
  return alst
print(aupton(46)) # Michael S. Branicky, May 29 2021

More terms from Don Reble, May 27 2004

A127329 Semiprimes equal to the sum of three primes in arithmetic progression.

Original entry on oeis.org

15, 21, 33, 39, 51, 57, 69, 87, 93, 111, 123, 129, 141, 159, 177, 183, 201, 213, 219, 237, 249, 267, 291, 303, 309, 321, 327, 339, 381, 393, 411, 417, 447, 453, 471, 489, 501, 519, 537, 543, 573, 579, 591, 597, 633, 669, 681, 687, 699, 717, 723, 753, 771, 789

Offset: 1

Giovanni Teofilatto, Mar 30 2007

			a(1) = 15 because 15 = 3 + 5 + 7;
a(2) = 21 because 21 = 3 + 7 + 11.

Cf. A000040, A001358, A078611.

[3*NthPrime(n+2): n in [1..60]]; // Vincenzo Librandi, Jun 28 2015, assuming the conjecture holds

Conjecture: a(n) = 3*A000040(n+2). - Zak Seidov, Jun 28 2015

Every member of the sequence is 3 times a prime; it is believed that every prime >= 5 arises in this way. This is related to Goldbach's conjecture: see comments to A078611. - Robert Israel and Michel Marcus, Jun 28 2015

Corrected (57 = 7+19+31, 87 = 5+29+53, etc. inserted) by R. J. Mathar, Apr 22 2010

A209266 a(n) is the number of 3-prime arithmetic progression prime chains surrounding the n-th prime number with 5-smooth intervals.

Original entry on oeis.org

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

Offset: 1

Lei Zhou, Feb 07 2013

Based on the conjecture in A211376, a(n) > 0.
Last appearance of positive integers in a(n) at n<220000
a(11)=1 (a(n) > 1 for 11a(46)=2; a(10680)=3; a(32293)=4; a(212493)=5

			n=3: prime(3)=5, 3,5,7 form a 3-prime arithmetic progression prime chain with the interval of 2, a 5-smooth number.  And this is the only case.  So a(3)=1;
...
n=43: prime(43)=191, the following 3-prime arithmetic progression prime chains exists:
  149,191,233 (gap 42=2*3*7, not 5-smooth)
  131,191,251 (gap 60=2^2*3*5, 5-smooth)
  113,191,269 (gap 78=2*3*13, not 5-smooth)
  101,191,281 (gap 90=2*3^2*5, 5-smooth)
  89,191,293  (gap 102=2*3*17, not 5-smooth)
  71,191,311  (gap 120=2^3*3*5, 5-smooth)
  29,191,353  (gap 162=2*3^4, 5-smooth)
  23,191,359  (gap 168=2^3*3*7, not 5-smooth)
  3,191,379   (gap 188=2^2*47, not 5-smooth)
Among these groups, there are 4 5-smooth gaps.  So, a(43)=4.

Lei Zhou, Table of n, a(n) for n = 1..10000

Cf. A078611, A051037, A001031, A211376

Table[p = Prime[i]; ct = 0; Do[If[(PrimeQ[p - j]) && (PrimeQ[p + j]),
   f = Last[FactorInteger[j]][[1]]; If[f <= 5, ct++]], {j, 2, p,
   2}]; ct, {i, 3, 89}]

A211376 a(n) is the smallest 5-smooth number k such that both prime(n) - k and prime(n) + k are prime.

Original entry on oeis.org

2, 4, 6, 6, 6, 12, 6, 12, 12, 6, 12, 24, 6, 6, 12, 18, 6, 12, 6, 18, 24, 18, 30, 12, 6, 6, 30, 24, 24, 18, 30, 12, 18, 12, 6, 36, 30, 6, 12, 18, 60, 30, 30, 72, 12, 60, 30, 48, 6, 12, 30, 12, 6, 6, 12, 60, 6, 12, 54, 24, 24, 48, 36, 36, 18, 30, 36, 18, 6, 90

Offset: 3

Lei Zhou, Feb 07 2013

The three numbers prime(n) - k, prime(n), prime(n) + k are an arithmetic progression of primes.
Conjecture: a(n) is defined for all integers n > 2.
Conjecture confirmed true up to n = 300000, no exceptions.
Note that if (p1, n, p2) is an arithmetic progression where p1 and p2 are prime, then 2n = p1 + p2 is a Goldbach pair. There are numbers n such that no such sequence (p1, n, p2) exists for which the common difference n - p1 = p2 - n is 5-smooth. The first such number is 90. The first such odd number is 1845.
a(n) is defined for 3 <= n <= 10^7. - David A. Corneth, Jul 10 2021

			Let n = 43. The 43rd prime is 191, and 191-42 = 149 and 191+42 = 233 are both prime. However, 42 = 2*3*7 is not a 5-smooth number, so a(43) != 42. But 191-60 = 31 and 191+60 = 251 are both prime numbers, and 60 = 2^2*3*5 is the smallest such 5-smooth number. So a(43) = 60.

Lei Zhou, Table of n, a(n) for n = 3..10000

Cf. A078611, A051037, A001031.

Table[p=Prime[i];j=0;While[j=j+2;If[(PrimeQ[p-j])&&(PrimeQ[p+j]), f=Last[FactorInteger[j]][[1]],f=p];f>5];j,{i,3,72}]

A215642 Primes p such that there is no D such that p+D, p-D, p+2D, p-2D are all primes.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 19, 23, 31, 37, 41, 43, 47, 53, 59, 61, 73, 79, 83, 103, 107, 109, 113, 127, 137, 139, 149, 151, 157, 179, 181, 199, 223, 227, 229, 239, 251, 271, 277, 281, 293, 311, 331, 349, 353, 359, 367, 379, 383, 389, 397, 401, 409, 421, 431, 439, 487, 499, 541

Offset: 1

Alex Ratushnyak, Aug 18 2012

nonn

Conjecture: a(243)=34613 is the last term.

			17 doesn't occur in the sequence, because there is D=6: 17-12, 17-6, 17+6 and 17+12 are all primes: 5, 11, 23, 29.

Joerg Arndt, Table of n, a(n) for n = 1..243

Cf. A078611.

fQ[p_] := Module[{d = 1}, While[4d < p && !(PrimeQ[p-4d] && PrimeQ[p-2d] && PrimeQ[p+2d] && PrimeQ[p+4d]), d++]; 4d > p]; Select[Prime[Range[4000]], fQ] (* T. D. Noe, Aug 20 2012 *)

N=10^9;
default(primelimit,N);
print1(2,", ");
{ forprime (p=3, N,
    D=2;  D2 = D << 1;
    t = 1;
    while ( p > D2,
        if ( isprime(p+D) & isprime(p-D) &
             isprime(p+D2) & isprime(p-D2)
        , /* then */
            t=0; break()
        );
        D += 2;  D2 += 4;
    );
    if ( t==1, print1(p,", ") );
); }
/* Joerg Arndt, Aug 20 2012 */

A129758 Smallest prime p such that there are primes q and r with the property that p, q and r form an arithmetic progression and their sum is the same as three times the (n+2)-nd prime number.

Original entry on oeis.org

3, 3, 5, 7, 11, 7, 17, 17, 19, 31, 29, 19, 41, 47, 47, 43, 61, 59, 67, 61, 59, 71, 67, 89, 97, 101, 79, 89, 103, 113, 107, 127, 131, 139, 151, 127, 137, 167, 167, 163, 149, 163, 167, 157, 199, 163, 197, 181, 227, 227, 211, 239, 251, 257, 257, 229, 271, 269

Offset: 1

Giovanni Teofilatto, May 15 2007

The same selection rule as in A078497 applies: if there is more than one prime triple (p,q=p+d,r=q+d) with p+q+r=A001748(n), take p from the triple with minimum d. - R. J. Mathar, May 19 2007

			3 + 5 + 7 = 15, which is three times the (1+2)th prime number. Thus a(1) = 3.

Cf. A078497, A071681, A078611.

A129758 := proc(n) local p3, i,d,r,p; p3 := ithprime(n) ; i := n+1 ; while true do r := ithprime(i) ; d := r-p3 ; p := p3-d ; if isprime(p) then RETURN(p) ; fi ; i := i+1 ; od ; RETURN(-1) ; end: for n from 3 to 60 do printf("%d, ",A129758(n)) ; od ; # R. J. Mathar, May 19 2007

a[n_]:=Module[{},k=1; While[Not[PrimeQ[Prime[n+1]-k] && PrimeQ[Prime[n+1]+k]], k++ ]; Prime[n + 1] - k]; Table[a[n], {n, 2, 60}]

A078497(n)-prime(n)=prime(n)-a(n)=d. - R. J. Mathar, May 19 2007

Conjecture: Limit_{N->oo} (Sum_{n=1..N} a(n)) / (Sum_{n=1..N} prime(n+2)) = 1. - Alain Rocchelli, May 01 2024

Edited and extended by R. J. Mathar, Giovanni Teofilatto and Stefan Steinerberger, May 19 2007

A177463 The smallest k such that Catalan(n)+k and Catalan(n)-k are both prime.

Original entry on oeis.org

0, 3, 1, 5, 10, 3, 69, 33, 45, 45, 9, 47, 86, 97, 97, 41, 19, 49, 191, 11, 101, 283, 1, 69, 597, 549, 1341, 243, 552, 121, 157, 115, 1341, 1905, 165, 597, 27, 87, 31, 731, 1093, 449, 127, 37, 37, 157, 1145, 587, 317, 659, 1523, 487, 865, 4879, 463, 1351, 4471

Offset: 3

Vladimir Joseph Stephan Orlovsky, May 09 2010

nonn

			5 +- 0 -> primes, 14 +- 3 -> primes, 42 +- 1 -> primes, 132 +- 5 -> primes, ...

Cf. A000108, A078611.

A000108 := proc(n) binomial(2*n,n)/(n+1) ; end proc:
A047160 := proc(n) for k from 0 to n-1 do if isprime(n-k) and isprime(n+k) then return k; end if; end do: return -1 ; end proc:
A177463 := proc(n) A047160(A000108(n)) ; end proc:
seq(A177463(n),n=3..40) ; # R. J. Mathar, Jan 23 2011

g[n_]:=(2n)!/n!/(n+1)!; f[n_]:=Block[{k},If[OddQ[n],k=0,k=1];While[ !PrimeQ[n-k]||!PrimeQ[n+k],k+=2];k]; Table[f[g[n]],{n,3,4*4!}]

a(n) = A047160(A000108(n)). - R. J. Mathar, Jan 23 2011

Showing 1-10 of 14 results. Next

cp's OEIS Frontend

A082467 Least k>0 such that n-k and n+k are both primes.

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A047160 For n >= 2, a(n) = smallest number m >= 0 such that n-m and n+m are both primes, or -1 if no such m exists.

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A078497 The member r of a triple of primes (p,q,r) in arithmetic progression which sum to 3*prime(n) = A001748(n) = p + q + r.

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A094709 Smallest k such that prime(n)# - k and prime(n)# + k are primes, where prime(n)# = A002110(n).

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A127329 Semiprimes equal to the sum of three primes in arithmetic progression.

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A209266 a(n) is the number of 3-prime arithmetic progression prime chains surrounding the n-th prime number with 5-smooth intervals.

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A211376 a(n) is the smallest 5-smooth number k such that both prime(n) - k and prime(n) + k are prime.

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A215642 Primes p such that there is no D such that p+D, p-D, p+2D, p-2D are all primes.