A118134 -id:A118134 - cp's OEIS frontend

A163487 Primes p such that 6*p is the sum of two consecutive primes.

2, 3, 5, 7, 13, 23, 31, 37, 43, 103, 127, 131, 151, 163, 167, 229, 241, 257, 293, 311, 313, 337, 389, 433, 509, 521, 523, 613, 647, 661, 719, 739, 743, 757, 797, 821, 887, 937, 953, 971, 1013, 1033, 1063, 1151, 1153, 1217, 1283, 1303, 1307, 1319, 1373, 1451

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 28 2009

nonn

			2*6=12=5+7, 3*6=18=7+11, 5*6=30=13+17, ..

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

Cf. A001043, A118134.

Primes:= select(isprime, [seq(i,i=3..10^4, 2)]):
select(t -> t::integer and isprime(t), (Primes[1..-2]+Primes[2..-1])/6); # Robert Israel, Jun 19 2018

Select[ListConvolve[{1,1},Prime[Range[1000]]]/6,PrimeQ] (* Paolo Xausa, Nov 03 2023 *)

Edited by N. J. A. Sloane, Aug 08 2009

A166685 Odd numbers that are the sum of two consecutive nonprimes.

Original entry on oeis.org

1, 5, 17, 19, 29, 31, 41, 43, 49, 51, 53, 55, 65, 67, 69, 71, 77, 79, 89, 91, 97, 99, 101, 103, 109, 111, 113, 115, 125, 127, 129, 131, 137, 139, 149, 151, 153, 155, 161, 163, 169, 171, 173, 175, 181, 183, 185, 187, 189, 191, 197, 199, 209, 211, 221, 223, 229

Offset: 1

Juri-Stepan Gerasimov, Oct 18 2009

nonn

			a(1)=0(1st nonprime)+1(2nd nonprime)=1(odd);
a(2)=1(2nd nonprime)+4(3rd nonprime)=5(odd).

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

Cf. A118134, A141468, A163467.

Select[Total/@Partition[Select[Range[0,200],!PrimeQ[#]&],2,1],OddQ] (* Harvey P. Dale, Mar 07 2020 *)

Entries checked by R. J. Mathar, Apr 24 2010

A164132 Primes which are an eighth of the sum of two consecutive primes.

Original entry on oeis.org

3, 19, 59, 89, 109, 149, 151, 317, 331, 359, 389, 401, 439, 571, 599, 829, 941, 953, 1019, 1153, 1249, 1279, 1319, 1373, 1381, 1451, 1657, 1669, 1733, 1741, 1867, 1871, 1973, 2131, 2161, 2179, 2251, 2459, 2819, 3119, 3539, 3659, 3967, 4001, 4099, 4231, 4261

Offset: 1

Juri-Stepan Gerasimov, Aug 11 2009

nonn

Primes of the form A001043(k)/8.

			19 is there because it is prime and 19=(73+79)/8.

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

Cf. A000040, A118134, A163487.

p:= 2: R:= NULL: count:= 0:
while count < 100 do
 q:= p; p:= nextprime(p);
 v:= (q+p)/8;
 if v::integer and isprime(v) then
   R:= R,v; count:= count+1;
 fi;
od:
R; # Robert Israel, Dec 08 2024

Select[Total[#]/8&/@Partition[Prime[Range[2500]],2,1],PrimeQ]  (* Harvey P. Dale, Apr 22 2011 *)

Extended by R. J. Mathar, Aug 27 2009

A164134 Primes p such that 12*p is the sum of two consecutive primes.

Original entry on oeis.org

2, 3, 5, 7, 17, 23, 31, 41, 47, 71, 97, 103, 107, 137, 139, 193, 283, 313, 337, 347, 349, 373, 397, 421, 443, 467, 487, 491, 577, 587, 593, 619, 631, 643, 653, 673, 691, 701, 773, 787, 811, 827, 907, 991, 1021, 1033, 1051, 1093, 1117, 1217, 1249, 1259, 1289

Offset: 1

Juri-Stepan Gerasimov, Aug 11 2009

nonn

			p=17 is there because it is prime and 12*17=101+103 = A001043(26) .

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A000040, A001043, A118134, A163487.

Select[ListConvolve[{1,1},Prime[Range[2000]]]/12,PrimeQ] (* Paolo Xausa, Nov 03 2023 *)

233 replaced by 283 and extended by R. J. Mathar, Aug 21 2009

A203836 Smallest sum s of two consecutive primes such that s = 0 mod prime(n).

Original entry on oeis.org

8, 12, 5, 42, 198, 52, 68, 152, 138, 696, 186, 222, 410, 172, 564, 1272, 472, 1220, 268, 852, 1460, 2212, 1494, 712, 1164, 1818, 618, 1284, 872, 2486, 508, 786, 548, 1668, 1192, 906, 3768, 978, 668, 6228, 3222, 6516, 3820, 772, 4728, 3980, 6330, 892, 5448, 1374

Offset: 1

Zak Seidov, Jan 06 2012

nonn

Besides a(3)=5, all terms are even and >=4. - Zak Seidov, Nov 29 2014

			a(1) = 8 = 3 + 5 is the least sum of two consecutive primes that is a multiple of prime(1) = 2.
a(3) = 5 = 2 + 3 is the least sum of two consecutive primes that is a multiple of prime(3) = 5.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001043, A062703, A111163, A247245, A247252, A188815 (the smaller prime), A118134.

N := 100: # for a(1)..a(N)
M := ithprime(N):
V := Vector(M):
count:= 0:
for i from 1 while count < N do
  x:= ithprime(i)+ithprime(i+1);
  Q:= convert(select(t -> t <= M and V[t]=0, numtheory:-factorset(x)), list);
  V[Q]:= x;
  count:= count + nops(Q);
od:
seq(V[ithprime(i)], i=1..N); # Robert Israel, May 25 2020

pr=Prime[Range[1000]];rm=Rest[pr]+Most[pr];Table[Select[rm,Mod[#,pr[[n]]]==0&][[1]],{n,50}]
s = Total /@ Partition[Prime@ Range[10^4], 2, 1]; Table[SelectFirst[s, Divisible[#, Prime@ n] &], {n, 52}] (* Michael De Vlieger, Jul 04 2017 *)

a(n)=p = 2; pn = prime(n); forprime(q=3, , if (((s=p+q) % pn) == 0, return (s)); p = q;); \\ Michel Marcus, Jul 04 2017

isA001043(n)=precprime((n-1)/2)+nextprime(n/2)==n&&n>2
a(n,p=prime(n))=if(p==5, return(5)); my(k=2); while(!isA001043(k*p), k+=2); k*p \\ Charles R Greathouse IV, Jul 05 2017

a(n) = 4*prime(n) if prime(n) is in A118134. - Robert Israel, May 25 2020

A339414 Primes p such that (p+q)/4 is prime, where q is the next prime after p.

Original entry on oeis.org

3, 5, 23, 31, 83, 131, 251, 271, 331, 383, 443, 563, 971, 1123, 1223, 1231, 1283, 1291, 1543, 2063, 2371, 2383, 2551, 2851, 2903, 2963, 3083, 3323, 3691, 3889, 4051, 4283, 4591, 4733, 4831, 4871, 4951, 5003, 5209, 5351, 5683, 5711, 5851, 6229, 6271, 6323, 6491, 6863, 6911, 7393, 7451, 7583, 7643

Offset: 1

Robert Israel, Dec 03 2020

nonn

After the initial 2 terms, a(n)=2*A118134(n)-3. - Hugo Pfoertner, Dec 03 2020

			a(5)=83 is in the sequence because it is prime, the next prime is 89, and (83+89)/4 = 43 is prime.

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

Subset of A098058.

Cf. A118134.

P:= select(isprime, [seq(i,i=3..10000,2)]):
R:= (P[1..-2]+P[2..-1])/4:
P[select(i-> R[i]::integer and isprime(R[i]), [$1..nops(R)])];

isok(p) = isprime(p) && iferr(isprime((p+nextprime(p+1))/4),E,0); \\ Michel Marcus, Dec 04 2020

A098016 Indices x such that (1/4)(prime(x+1) + prime(x)) is prime.

Original entry on oeis.org

2, 3, 9, 11, 23, 32, 54, 58, 67, 76, 86, 103, 164, 188, 200, 202, 208, 210, 243, 311, 351, 354, 374, 414, 420, 427, 441, 468, 515, 539, 559, 588, 621, 639, 650, 652, 662, 670, 693, 708, 748, 752, 769, 811, 816, 823, 842, 883, 889, 939, 943, 963, 970, 1006, 1009

Offset: 1

Cino Hilliard, Sep 09 2004

Conjecture: (1/2)(prime(x+1) + prime(x)) is not prime for all x.

This is obvious: (prime(x+1)+prime(x))/2 is strictly between prime(x) and prime(x+1), so if it were prime, prime(x+1) wouldn't be the next prime after prime(x). - Robert Israel, Feb 04 2019

			Prime(2+1) + prime(2) = 5+3 = 8. 1/4(8) = 2. 2 is the first entry.

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

Cf. A000720, A118134.

filter:= proc(t) local v; v:= (ithprime(t)+ithprime(t+1))/4; v::integer and isprime(v) end proc:
select(filter, [$1..2000]); # Robert Israel, Feb 04 2019

Transpose[Select[Table[{i,Prime[i],Prime[i+1]}, {i,1200}], PrimeQ[Total[Rest[#]]/4]&]][[1]](* Harvey P. Dale, Mar 24 2011 *)
Position[Partition[Prime[Range[1100]],2,1],?(PrimeQ[Total[#]/4]&)]//Flatten (* _Harvey P. Dale, Sep 11 2022 *)

f(n) = for(x=1,n,y=prime(x+1)+prime(x);if(y%4==0 && isprime(y\4),print1(x",")))

a(n) = A000720(2*A118134(n)-1). - Robert Israel, Feb 04 2019

A164133 Primes p such that 4p and 6p are each the sum of two consecutive primes.

Original entry on oeis.org

2, 3, 13, 43, 127, 167, 613, 647, 1033, 1483, 1543, 2297, 2927, 3701, 3823, 4463, 5101, 5417, 5657, 6133, 8081, 9227, 11273, 11833, 12511, 13291, 13873, 17627, 19853, 20011, 21313, 21727, 22193, 23041, 23059, 23081, 23159, 24443, 26347, 26947, 27407, 27527

Offset: 1

Juri-Stepan Gerasimov, Aug 11 2009

nonn

			p=13 is in the sequence because 4*13 = 52 = A001043(9) and 6*13 = 78 = A001043(12) are both in A001043.

Robert Israel, Table of n, a(n) for n = 1..3000

Cf. A000040, A118134, A163487.

P:= select(isprime, [2,seq(i,i=3..10^6)]):
PS:= P[1..-2] + P[2..-1]:
convert(P,set) intersect convert(1/4 * PS, set) intersect convert(1/6*PS,set); # Robert Israel, Dec 08 2024

stcpQ[n_]:=Module[{a=4n,b=6n},a==NextPrime[a/2]+NextPrime[a/2,-1]&&b== NextPrime[b/2]+NextPrime[b/2,-1]]; Select[Prime[Range[3100]],stcpQ] (* Harvey P. Dale, May 01 2019 *)

A163487 INTERSECT A118134.

Extended by R. J. Mathar, Aug 27 2009

A289270 Primes p such that 10*p is the sum of two consecutive primes.

Original entry on oeis.org

3, 41, 103, 293, 359, 379, 421, 653, 701, 733, 821, 883, 907, 911, 937, 1009, 1237, 1423, 1567, 1627, 1637, 1931, 1973, 2017, 2129, 2203, 2281, 2417, 2459, 2477, 2647, 2879, 3209, 3271, 3347, 3413, 3539, 3593, 3659

Offset: 1

Zak Seidov, Jun 30 2017

nonn

			10*3 = 30 = prime(6) + prime(7) = 13+17;
10*41 = 410 = prime(46) + prime(47) = 199+211.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Primes p such that m*p is the sum of two consecutive primes: A118134 (m=4), A163487 (m=6), A164132 (m=8), this sequence (m=10), A164134 (m=12).

Select[Map[Total, Partition[Prime@ Range@ 2200, 2, 1]]/10, PrimeQ] (* Michael De Vlieger, Jun 30 2017 *)

is(n)=isprime(n) && precprime(5*n)+nextprime(5*n)==10*n \\ Charles R Greathouse IV, Jul 02 2017

A350736 Lesser twin primes p such that 4*p is the sum of two consecutive primes.

Original entry on oeis.org

3, 17, 137, 617, 1277, 1427, 1949, 2027, 3119, 4157, 5417, 5657, 10139, 13217, 13691, 13709, 16187, 17657, 17837, 18911, 19379, 20507, 20807, 24371, 25577, 27407, 27527, 29207, 31391, 31847, 32117, 32909, 33767, 34847, 36467, 39839, 40037, 47057, 47387, 47657, 48311, 49199, 49367, 49739, 49787

Offset: 1

J. M. Bergot and Robert Israel, Jan 12 2022

nonn

			a(3) = 137 is a term because 137 and 139 are primes and 4*137 = 548 is the sum of consecutive primes 271 and 277.

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

Intersection of A001359 and A118134.

P:= select(isprime,{seq(i,i=3..100000,2)}):
T:= P intersect map(`-`,P,2):
sort(convert(select(r -> prevprime(2*r)+nextprime(2*r)=4*r,T),list));

Select[Plus @@@ Partition[Select[Range[10^5], PrimeQ], 2, 1]/4, And @@ PrimeQ[# + {0, 2}] &] (* Amiram Eldar, Jan 13 2022 *)

cp's OEIS Frontend

A163487 Primes p such that 6*p is the sum of two consecutive primes.

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A166685 Odd numbers that are the sum of two consecutive nonprimes.

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A164132 Primes which are an eighth of the sum of two consecutive primes.

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A164134 Primes p such that 12*p is the sum of two consecutive primes.

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A203836 Smallest sum s of two consecutive primes such that s = 0 mod prime(n).

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A339414 Primes p such that (p+q)/4 is prime, where q is the next prime after p.

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A098016 Indices x such that (1/4)(prime(x+1) + prime(x)) is prime.

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A164133 Primes p such that 4p and 6p are each the sum of two consecutive primes.