A129919 -id:A129919 - cp's OEIS frontend

A163981 a(n) is the smallest prime of the form prime(n+1)*k - prime(n), k >= 1, where prime(n) is the n-th prime.

7, 2, 2, 37, 2, 89, 2, 73, 151, 2, 43, 127, 2, 239, 59, 419, 2, 73, 359, 2, 401, 419, 1163, 881, 307, 2, 967, 2, 569, 3697, 397, 691, 2, 457, 2, 163, 821, 839, 179, 1259, 2, 2111, 2, 1777, 2, 223, 3803, 3863, 2, 3499, 1201, 2, 2269, 263, 269, 1889, 2, 283, 1409, 2, 2647

Offset: 1

Leroy Quet, Aug 07 2009

nonn

a(n) = 2 if and only if n is in A029707. - Robert Israel, Jan 16 2019

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

Cf. A029707, A129919.

Contains A085704.

a := proc (n) local k: for k while isprime(ithprime(n+1)*k-ithprime(n)) = false do end do: ithprime(n+1)*k-ithprime(n) end proc: seq(a(n), n = 1 .. 65); # Emeric Deutsch, Aug 10 2009

a[n_] := Module[{p, q, r}, For[p = Prime[n]; q = Prime[n + 1]; k = 1, True, k++, If[PrimeQ[r = q k - p], Return[r]]]];
Array[a, 100] (* Jean-François Alcover, Aug 28 2020 *)

a(n) = my(k=1); while (!isprime(p=prime(n+1)*k - prime(n)), k++); p; \\ Michel Marcus, Jul 02 2021

from sympy import isprime, nextprime, prime
def a(n):
    pn = prime(n); pn1 = nextprime(pn); k = 1
    while not isprime(pn1*k - pn): k += 1
    return pn1*k - pn
print([a(n) for n in range(1, 62)]) # Michael S. Branicky, Jul 02 2021

Extended by Emeric Deutsch, Aug 10 2009

A287615 Let r = prime(n). Then a(n) is the smallest prime p such that there is a prime q with p > q > r and p mod q = r.

Original entry on oeis.org

5, 13, 19, 29, 37, 47, 79, 101, 97, 103, 113, 131, 127, 137, 181, 199, 181, 227, 233, 229, 239, 257, 277, 283, 311, 307, 317, 409, 383, 367, 389, 409, 439, 521, 463, 509, 491, 509, 613, 571, 541, 563, 577, 587, 619, 653, 677, 677, 709, 743, 787, 853, 743, 877

Offset: 1

Ophir Spector, May 27 2017

Prime p such that p = k * q + r, r < q < p primes; k even multiples such that p is minimal.

			a(1) = 5, as r = 2, q = 3, p = 5, is the smallest prime such that 5 = 2 mod 3.
a(9) = 97, as r = 23, q = 37, p = 97.  97 = 2 * 37 + 23 is smaller than 139 = 4 * 29 + 23 (A129919).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Ophir Spector, C++ code and Excel file

Cf. A129919.

f:= proc(n) local p,q,r;
  r:= ithprime(n);
  p:= r+1;
  do
   p:= nextprime(p);
   q:= max(numtheory:-factorset(p-r));
   if q > r then return p fi
  od:
end proc:
map(f, [$1..100]); # Robert Israel, Jun 05 2017

a[n_] := Module[{p, q, r}, r = Prime[n]; p = r+1; While[True, p = NextPrime[p]; q = Max[FactorInteger[p-r][[All, 1]]]; If[q>r, Return[p]]] ];
Array[a, 100] (* Jean-François Alcover, Oct 06 2020, after Robert Israel *)

findfirstTerms(n)=my(t:small=0,a:vec=[]);forprime(r=2,,forprime(p=r+2,,forprime(q=r+2,p-2,if(p%q==r,a=concat(a,[p]);if(t++==n,a[1]-=2;return(a),break(2)))))) \\ R. J. Cano, Jun 06 2017

first(n)=my(v=vector(n),best,k=1); v[1]=5; forprime(r=3,prime(n), best=oo; forprime(q=r+2,, if(q>=best, v[k++]=best; next(2)); forstep(p=r+2*q,best,2*q, if(isprime(p), best=p; break)))); v \\ Charles R Greathouse IV, Jun 07 2017

A381532 Smallest integer k>0 such that prime(n) + k*prime(n+1) is prime.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 6, 6, 4, 8, 4, 6, 2, 2, 10, 10, 2, 6, 12, 6, 4, 6, 4, 2, 14, 2, 2, 6, 6, 2, 2, 6, 6, 6, 20, 6, 4, 8, 4, 16, 2, 2, 2, 2, 8, 10, 4, 2, 6, 6, 6, 14, 2, 4, 10, 6, 2, 6, 2, 6, 18, 2, 2, 2, 2, 12, 10, 2, 6, 6, 4, 2, 22, 4, 6, 10, 12, 6, 8, 8, 12, 2

Offset: 1

Jean-Marc Rebert, Mar 07 2025

nonn

			a(1)= 1, because 2 and 3 are consecutive primes and 2 + 1*3 = 5 is prime, and no lesser number has this property.
 p + k*q, where p and q are consecutive primes
 2 + 1* 3 =   5 is prime;
 3 + 2* 5 =  13 is prime;
 5 + 2* 7 =  19 is prime;
 7 + 2*11 =  29 is prime;

Cf. A129919 (resulting primes), A175914 (primes for which k=2), A368691 (primes for which k=4).

Do[k=0;Until[PrimeQ[Prime[n]+k*Prime[n+1]],k++];a[n]=k,{n,82}];Array[a,82] (* James C. McMahon, Mar 28 2025 *)

a(n) = my(p=prime(n), q=nextprime(p+1), k=1); while (!isprime(p+k*q), k++); k; \\ Michel Marcus, Mar 09 2025

cp's OEIS Frontend

A163981 a(n) is the smallest prime of the form prime(n+1)*k - prime(n), k >= 1, where prime(n) is the n-th prime.

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A287615 Let r = prime(n). Then a(n) is the smallest prime p such that there is a prime q with p > q > r and p mod q = r.

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Maple

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A381532 Smallest integer k>0 such that prime(n) + k*prime(n+1) is prime.

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Author

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Examples

Crossrefs

Programs

Mathematica

PARI