A177017 -id:A177017 - cp's OEIS frontend

A072667 Consider m such that prime(m) + prime(m+1) = prime(k) + 1 for some k; sequence gives values of prime(m).

3, 5, 7, 11, 13, 19, 29, 31, 41, 43, 61, 67, 73, 97, 113, 127, 151, 179, 191, 193, 199, 211, 223, 229, 239, 241, 271, 277, 281, 293, 307, 317, 337, 349, 367, 373, 379, 397, 401, 409, 419, 431, 439, 463, 487, 523, 541, 547, 577, 613, 619, 641, 643, 659, 683, 701, 709, 727

Offset: 1

Herman H. Rosenfeld (herm3(AT)pacbell.net), Aug 12 2002

nonn

Vincenzo Librandi and Harvey P. Dale, Table of n, a(n) for n = 1..10000 ( first 1000 terms from Vincenzo Librandi)

Cf. A072666, A072669, A177017.

[NthPrime(n): n in [1..200] | IsPrime(NthPrime(n) + NthPrime(n+1)-1)] // Vincenzo Librandi, Aug 26 2012

Transpose[Select[Partition[Prime[Range[130]], 2, 1], PrimeQ[Total[#]-1]&]] [[1]] (* Harvey P. Dale, Feb 29 2012 *)
Select[Prime[Range[200]],PrimeQ[#+NextPrime[#]-1]&] (* Vincenzo Librandi, Aug 26 2012 *)

a(n) = prime(A072666(n)) = A000040(A072666(n)). - Zak Seidov, Dec 08 2014

A072666 Numbers n such that prime(n) + prime(n+1) - 1 is prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 10, 11, 13, 14, 18, 19, 21, 25, 30, 31, 36, 41, 43, 44, 46, 47, 48, 50, 52, 53, 58, 59, 60, 62, 63, 66, 68, 70, 73, 74, 75, 78, 79, 80, 81, 83, 85, 90, 93, 99, 100, 101, 106, 112, 114, 116, 117, 120, 124, 126, 127, 129, 130, 131, 132, 137, 138, 140, 145, 147, 149

Offset: 1

Herman H. Rosenfeld (herm3(AT)pacbell.net), Aug 12 2002

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000720, A072667, A072669, A177017.

[n: n in [1..200] | IsPrime(NthPrime(n) + NthPrime(n+1)-1)] // Vincenzo Librandi, Aug 26 2012

N:= 10^4: # to get all terms n such that prime(n+1) <= N
Primes:= select(isprime,[2,seq(2*i+1,i=1..floor(N/2))]):
PP:= Primes[1..-2]+Primes[2..-1]:
select(t -> isprime(PP[t]-1), [$1..nops(PP)]); # Robert Israel, Dec 11 2014

Select[Range[200], PrimeQ[Prime[#]+Prime[#+1]-1] &] (* Harvey P. Dale, Dec 16 2010 *)

a(n) = pi(A072667(n)) = A000720(A072667(n)). - Zak Seidov, Dec 08 2014

Definition clarified by Robert Israel, Dec 11 2014

A098084 a(n) satisfies P(n) + P(n+1) + a(n) = least prime >= P(n) + P(n+1), where P(i)=i-th prime.

Original entry on oeis.org

0, 3, 1, 1, 5, 1, 1, 1, 1, 1, 3, 1, 5, 7, 1, 1, 7, 3, 1, 5, 5, 1, 1, 5, 1, 7, 1, 7, 1, 1, 5, 1, 1, 5, 7, 3, 11, 1, 7, 1, 7, 1, 5, 7, 1, 9, 5, 7, 1, 1, 7, 7, 7, 1, 1, 9, 1, 9, 5, 5, 1, 1, 1, 7, 1, 5, 5, 7, 5, 7, 7, 1, 3, 5, 7, 1, 1, 11, 1, 1, 13, 1, 13, 5, 1, 15, 1, 1, 5, 7, 1, 1, 5, 1, 7, 1, 1, 5, 5, 3, 5, 3, 19

Offset: 1

Pierre CAMI, Sep 13 2004

a(n) = 1 iff prime(n) is in A177017. - Robert Israel, Feb 04 2020

			P(1) + P(2) = 2 + 3 = 5; least prime >= 5 = 5, so a(1)=0.
P(2) + P(3) = 3 + 5 = 8; least prime > 8 = 11, so a(2) = 11 - 8 = 3.
P(3) + P(4) = 5 + 7 = 12; least prime > 12 = 13, so a(3) = 13 - 12 = 1.

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

The primes are in A098085.

Cf. A177017.

P:= [seq(ithprime(i),i=1..200)]:
map(t -> nextprime(t-1)-t,P[1..-2]+P[2..-1]); # Robert Israel, Feb 04 2020

f[n_] := Block[{k = 0, p = Prime[n] + Prime[n + 1]}, While[ !PrimeQ[p + k], k++ ]; k]; Table[ f[n], {n, 103}] (* Robert G. Wilson v, Sep 24 2004 *)

More terms from Robert G. Wilson v, Sep 25 2004

A224789 Primes p such that both p + nextprime(p) + 1 and p*nextprime(p) + 2 are primes.

Original entry on oeis.org

5, 7, 13, 19, 67, 229, 269, 307, 313, 401, 439, 613, 643, 863, 1051, 1693, 1783, 1999, 2143, 2239, 2309, 2423, 2549, 2753, 2819, 3037, 3079, 3089, 3361, 3373, 3389, 3677, 3863, 3877, 4139, 4259, 4519, 4663, 4909, 4933, 5323, 5527, 5581, 5849, 6359, 6577

Offset: 1

Jayanta Basu, Apr 17 2013

nonn

Intersection of A051507 and A177017.

			5 is a member since 5 + 7 + 1 = 13 and 5 * 7 + 2 = 37 are both primes.

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

Cf. A051507, A177017.

Select[Prime[Range[900]], PrimeQ[# + NextPrime[#] + 1] && PrimeQ[#*NextPrime[#] + 2] &]
npQ[n_]:=Module[{np=NextPrime[n]},AllTrue[{n+np+1,n*np+2},PrimeQ]]; Select[ Prime[ Range[900]],npQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 04 2017 *)

A244095 Primes of the form (p + q)^2 + 1, where p and q are consecutive primes.

Original entry on oeis.org

577, 1297, 7057, 8101, 14401, 41617, 44101, 57601, 90001, 115601, 147457, 156817, 331777, 484417, 547601, 746497, 820837, 864901, 894917, 933157, 1299601, 1664101, 1742401, 1822501, 1887877, 1988101, 2131601, 2232037, 2702737, 2944657, 3218437

Offset: 1

K. D. Bajpai, Jun 20 2014

nonn

Subsequence of A002496.

Also, primes of form p^2 + 2pq + q^2 + 1; p and q are consecutive primes.

			577 is in the sequence because (11 + 13)^2 + 1 = 577, which is prime.
1297 is in the sequence because (17 + 19)^2 + 1 = 1297, which is prime.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A000040, A002496, A177017.

[t: p in PrimesUpTo(1000) | IsPrime(t) where t is (p+NextPrime(p))^2+1]; // Bruno Berselli, Jun 24 2014

with(numtheory):A244095:= proc() local k,p,q; p:=ithprime(n); q:=ithprime(n+1); k:=(p+q)^2 + 1; if isprime(k) then RETURN (k); fi; end: seq(A244095 (), n=1..300);

A244095 = {}; Do[k = (Prime[n] + Prime[n + 1])^2 + 1; If[PrimeQ[k], AppendTo[A244095, k]], {n, 2, 300}]; A244095

A356745 a(n) is the first prime that starts a string of exactly n consecutive primes where the prime + the next prime + 1 is prime.

Original entry on oeis.org

37, 5, 283, 929, 13, 696607, 531901, 408079937, 17028422981

Offset: 1

J. M. Bergot and Robert Israel, Sep 17 2022

a(n) is the first prime p(k) such that p(k+i)+p(k+i+1)+1 is prime for i from 0 to n-1, but not for i=-1 or n.

			a(5) = 13 because 13+17+1 = 31, 17+19+1 = 37, 19+23+1 = 43, 23+29+1 = 53, and 29+31+1 = 61 are prime, but 11+13+1 = 25 and 31+37+1 = 69 are not, and 13 is the first prime that works.

Cf. A177017.

P:= select(isprime, [seq(i,i=3..10^6,2)]):
V:= Vector(7):
state:= 0:
for i from 1 to nops(P)-1 do
if isprime(P[i]+P[i+1]+1) then
  state:= state+1
else
  if state > 0 and V[state] = 0 then V[state]:= P[i-state] fi;
  state:= 0
fi
od:
convert(V,list);

from itertools import count, islice
from sympy import isprime, nextprime
def f(p):
    c, p0, p1 = 0, p, nextprime(p)
    while isprime(p0+p1+1):
        c, p0, p1 = c+1, p1, nextprime(p1)
    return c, p1
def agen():
    n, adict, pk = 1, dict(), 2
    for k in count(1):
        fk, pk2 = f(pk)
        if fk not in adict: adict[fk] = pk
        while n in adict: yield adict[n]; n += 1
        pk = pk2
print(list(islice(agen(), 7))) # Michael S. Branicky, Sep 18 2022

a(8)-a(9) from Michael S. Branicky, Sep 18 2022

cp's OEIS Frontend

A072667 Consider m such that prime(m) + prime(m+1) = prime(k) + 1 for some k; sequence gives values of prime(m).

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A072666 Numbers n such that prime(n) + prime(n+1) - 1 is prime.

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A098084 a(n) satisfies P(n) + P(n+1) + a(n) = least prime >= P(n) + P(n+1), where P(i)=i-th prime.

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A224789 Primes p such that both p + nextprime(p) + 1 and p*nextprime(p) + 2 are primes.

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Mathematica

A244095 Primes of the form (p + q)^2 + 1, where p and q are consecutive primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

A356745 a(n) is the first prime that starts a string of exactly n consecutive primes where the prime + the next prime + 1 is prime.

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Python

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