A034961 -id:A034961 - cp's OEIS frontend

A234973 Primes that are 6 plus the sum of three consecutive primes.

29, 37, 47, 89, 103, 127, 137, 149, 179, 193, 229, 241, 257, 293, 307, 317, 401, 431, 463, 509, 557, 571, 587, 613, 719, 809, 823, 863, 937, 947, 967, 991, 1021, 1039, 1193, 1213, 1277, 1291, 1321, 1373, 1439, 1483, 1499, 1559, 1709, 1723, 1741, 1873, 1949, 1979

Offset: 1

Vincenzo Librandi, Jan 02 2014

nonn

Primes of the form 6+A034961(k).

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

Cf. A034961, A165982, A165985.

Select[Total[#] + 6&/@Partition[Prime[Range[200]], 3, 1], PrimeQ]

A244186 Primes which are the concatenation of five consecutive primes p, q, r, s, t while the sum (p + q + r + s + t) is another prime.

Original entry on oeis.org

711131719, 5359616771, 6771737983, 149151157163167, 401409419421431, 479487491499503, 757761769773787, 14091423142714291433, 18111823183118471861, 21132129213121372141, 26892693269927072711, 27192729273127412749, 36133617362336313637, 37613767376937793793

Offset: 1

K. D. Bajpai, Jun 21 2014

Subsequence of A086041.

Numbers: Concatenation of 5 consecutive primes at A132905.

			711131719 is in the sequence because the concatenation of [7, 11, 13, 17, 19] = 711131719 which is prime. The sum [7 + 11 + 13 + 17 + 19] = 67 is another prime.
5359616771 is in the sequence because the concatenation of [53, 59, 61, 67, 71] = 5359616771 which is prime. The sum [53 + 59 + 61 + 67 + 71] = 311 is another prime.

K. D. Bajpai, Table of n, a(n) for n = 1..1703

Cf. A000040, A086041, A034961, A034962, A132903, A244163.

FromDigits[Flatten[IntegerDigits/@#]]&/@Select[Partition[Prime[Range[ 1000]],5,1],AllTrue[{Total[#],FromDigits[Flatten[ IntegerDigits/@ #]]}, PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 24 2014 *)

A270639 Fermat pseudoprimes (A001567) that are the sum of three consecutive primes.

Original entry on oeis.org

13741, 16705, 150851, 208465, 249841, 252601, 258511, 410041, 486737, 635401, 1052503, 1082401, 1457773, 1507963, 1579249, 1615681, 2113921, 2184571, 3090091, 3375487, 3726541, 4682833, 4895065, 5044033, 5133201, 6233977, 6255341, 6350941, 6474691, 6912079, 7259161

Offset: 1

Altug Alkan, Mar 20 2016

nonn

In other words, Fermat pseudoprimes to base 2 of the form p + q + r where p, q and r are consecutive primes.
If a Fermat pseudoprime is the sum of n consecutive primes, it is so obvious that the minimum value of n is 3.
Intersection of A001567 and A034961.

			4567, 4583 and 4591 are consecutive primes and their sum is 13741, a Fermat pseudoprime.
84191, 84199 and 84211 are consecutive primes and their sum is 252601, a Fermat pseudoprime.

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

Cf. A001567, A034961, A270267.

isA001567(n) = {Mod(2, n)^n==2 && !ispseudoprime(n) && n > 1}
a034961(n) = my(p=prime(n), q=nextprime(p+1)); p+q+nextprime(q+1);
for(n=1, 200000, if(isA001567(a034961(n)), print1(a034961(n), ", ")));

A281960 Primes that are the sum of three consecutive odd semiprimes.

Original entry on oeis.org

61, 79, 107, 139, 163, 191, 211, 263, 271, 373, 443, 617, 719, 733, 761, 971, 991, 1097, 1129, 1231, 1259, 1373, 1439, 1531, 1543, 1597, 1663, 1697, 1733, 1753, 1777, 1831, 2053, 2081, 2099, 2137, 2161, 2213, 2383, 2423, 2543, 2677, 2687, 2719, 2777, 2843, 2917

Offset: 1

K. D. Bajpai, Feb 03 2017

nonn

			a(1) = 61 is a prime and 61 = 15 + 21 + 25; the sum of three consecutive odd semiprimes.
a(2) = 79 is a prime and 79 = 21 + 25 + 33; the sum of three consecutive odd semiprimes.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (1..3263 from K. D. Bajpai)

Cf. A000040, A001358, A034961, A034962, A281824.

Select[Total /@ Partition[Select[Range[2000], Plus @@ Last /@ FactorInteger[#] == 2 && OddQ[#] &], 3, 1], PrimeQ]

list(lim)=my(v=List(),u=v,t,L=lim+10); forprime(p=3,L\3, forprime(q=3,min(p,L\p), listput(u,p*q))); u=Set(u); for(i=3,#u, if(isprime(t=u[i-2]+u[i-1]+u[i]), listput(v,t))); while((t=u[#u-1]+u[#u]+L++)lim, break); if(isprime(t), listput(v,t)))); Vec(v) \\ Charles R Greathouse IV, Feb 03 2017

A289361 Least sum s of three consecutive primes such that s is a multiple of the n-th prime.

Original entry on oeis.org

10, 15, 10, 49, 121, 143, 187, 551, 23, 319, 31, 407, 41, 301, 235, 159, 59, 1891, 1943, 71, 803, 395, 83, 2759, 97, 1717, 3193, 749, 109, 565, 3175, 131, 2329, 1807, 7301, 6493, 471, 1793, 1169, 173, 1611, 5611, 2101, 3281, 985, 199, 211, 223, 1135, 4351, 5359, 11233, 2651

Offset: 1

Zak Seidov, Jul 04 2017

nonn

Are all terms distinct? Is a(1)=a(3)=10 the only case of equality?

			a(1)=10=A034961(1), a(2)=15=A034961(2), a(3)=319=A034961(27).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Zak Seidov, Graph of first 50 terms.

Cf. A034961, A034962 (subsequence).

Table[Function[p, k = 1; While[! Divisible[Set[s, Total@ Prime@ Range[k, k + 2]], p], k++]; s]@ Prime@ n, {n, 53}] (* or *)
s = Total /@ Partition[Prime@ Range[10^4], 3, 1]; Table[SelectFirst[s, Divisible[#, Prime@ n] &], {n, 53}] (* Michael De Vlieger, Jul 04 2017 *)

a(n)=p = 2; q = 3; pn = prime(n); forprime(r=5,,if (((s=p+q+r) % pn) == 0, return (s)); p = q; q = r;); \\ Michel Marcus, Jul 04 2017

isA034961(n)=my(p=precprime(n\3),q=nextprime(n\3+1),r=n-p-q); if(r>q, r==nextprime(q+2), r==precprime(p-1) && r)
a(n,p=prime(n))=if(p==5, return(10)); my(k=1); while(!isA034961(p*k), k+=2); p*k \\ Charles R Greathouse IV, Jul 05 2017

A335969 Sphenic numbers that are also the sum of three consecutive primes.

Original entry on oeis.org

1015, 1533, 1645, 2233, 2737, 2915, 3219, 3515, 3745, 3815, 4301, 4503, 4565, 4623, 4697, 4921, 5289, 5621, 6055, 6095, 6213, 6251, 6409, 7055, 7347, 7657, 7847, 8099, 8455, 8569, 8687, 8729, 9499, 9581, 9955, 10105, 10153, 10295, 10735, 11155, 11297, 11315, 11803, 12665, 12805, 12845

Offset: 1

Zak Seidov, Jul 04 2020

nonn

Intersection of A007304 and A034961.

Includes 15*p where p, 5*p-14, 5*p-2 and 5*p+16 are consecutive primes. Dickson's conjecture implies there are infinitely many such terms. - Robert Israel, Nov 24 2022

			1015 = A007304(140) = A034961(67), 1533 = A007304(226) = A034961(96).

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

Cf. A007304, A034961.

P:= select(isprime, [seq(i,i=3..10^4,2)]):
P3:= P[1..-3] + P[2..-2] + P[3..-1]:
filter:= proc(t) local F; F:= ifactors(t)[2]; nops(F) = 3 and F[1,2]=1 and F[2,2] = 1 and F[3,2]=1 end proc:
select(filter, P3); # Robert Israel, Nov 24 2022

Intersection[ Select[Range[105, 40000,2], 3 == PrimeOmega[#] == PrimeNu[#] &], Total /@ Partition[Prime[Range[40000]], 3, 1]]

A351579 Primes p such that the sum of p and the next two primes is the product of two consecutive primes.

Original entry on oeis.org

3, 43, 3671, 51473, 53051, 64811, 71143, 121591, 137383, 154111, 161459, 228521, 284573, 344053, 433141, 544403, 679709, 702743, 767071, 995303, 1158139, 1267481, 1301507, 1320023, 1342667, 1512293, 1682987, 1839221, 1982891, 2022101, 2174287, 2198153, 2370943, 2403061, 2770549, 4148923, 4368121

Offset: 1

J. M. Bergot and Robert Israel, Feb 13 2022

nonn

A000040(k) for k such that A034961(k) is in A006094.

The Generalized Bunyakovsky Conjecture implies that, for example, there are infinitely many terms of the form 12*s^2+12*s-1 where the next two primes are 12*s^2+12*s+1 and 12*s^2+12*s+5 and the sum of these is (6*s+1)*(6*s+5).

			a(3) = 3671 is a term because 3671, 3673, 3677 are three consecutive primes with 3671+3673+3677 = 11021 = 103*107 and 103 and 107 are two consecutive primes.

Robert Israel, Table of n, a(n) for n = 1..120

Cf. A000040, A006094, A034961.

q:= proc(n) local r,s;
  r:= nextprime(floor(sqrt(n)));
  s:= n/r;
  s::integer and s = prevprime(r)
end proc:
P:= select(isprime,[2,seq(i,i=3..10^7)]):
S:= [0,op(ListTools:-PartialSums(P))]:
map(t -> P[t], select(i -> q(S[i+3]-S[i]), [$1..nops(S)-3]));

prodQ[n_] := Module[{f = FactorInteger[n]}, f[[;; , 2]] == {1, 1} && f[[2, 1]] == NextPrime[f[[1, 1]]]]; q[p_] := PrimeQ[p] && prodQ[p + Plus @@ NextPrime[p, {1, 2}]]; Select[Range[5*10^6], q] (* Amiram Eldar, Feb 14 2022 *)

isok(p) = {if (isprime(p), my(q=nextprime(p+1), f=factor(p+q+nextprime(q+1))); (omega(f) == 2) && (bigomega(f) == 2) && (f[2,1] == nextprime(f[1,1]+1)););} \\ Michel Marcus, Feb 14 2022

A125270 Coefficient of x^2 in polynomial whose zeros are 5 consecutive primes starting with the n-th prime.

Original entry on oeis.org

1358, 3954, 10478, 22210, 43490, 78014, 129530, 206650, 324350, 466270, 621466, 853742, 1132130, 1436690, 1870850, 2388050, 2886370, 3440410, 4133410, 4904906, 5926654, 7195670, 8425430, 9792950, 11040910, 12098990, 13917898, 16097810

Offset: 1

Artur Jasinski, Jan 16 2007

nonn

Sums of all distinct products of 3 out of 5 consecutive primes, starting with the n-th prime; value of 3rd elementary symmetric function on the 5 consecutive primes.

Cf. A001043, A034961, A034963, A034964, A127333, A127334, A127335, A127336, A127337, A127338, A127339, A127340, A127341, A127342, A127343, A127345, A127346, A127347, A127348, A127349, A127351, A037171, A034962, A034965, A082246, A082251, A070934, A006094, A046301, A046302, A046303, A046324, A046325, A046326, A046327, A127489.

a = {}; Do[AppendTo[a, (Prime[x] Prime[x + 1] Prime[x + 2] + Prime[x] Prime[x + 1] Prime[x + 3] + Prime[x] Prime[x + 1] Prime[x + 4] + Prime[x] Prime[x + 2] Prime[x + 3] + Prime[x] Prime[x + 2] Prime[x + 4] + Prime[x] Prime[x + 3] Prime[x + 4] + Prime[x + 1] Prime[x + 2] Prime[x + 3] + Prime[x + 1] Prime[x + 2] Prime[x + 4] + Prime[x + 1] Prime[x + 3] Prime[x + 4] + Prime[x + 2] Prime[x + 3] Prime[x + 4])], {x, 1, 100}]; a
fcp[{p_,q_,r_,s_,t_}]:=p*q(r+s+t)+(p+q)r(s+t)+(p+q+r)s*t; fcp/@Partition[ Prime[ Range[40]],5,1] (* Harvey P. Dale, Sep 05 2014 *)

Let p = Prime(n), q = Prime(n+1), r = Prime(n+2), s = Prime(n+3) and t = Prime(n+4). Then a(n) = p q (r+s+t) + (p + q) r (s + t) + (p + q + r) s t.

Edited and corrected by Franklin T. Adams-Watters, Jan 23 2007

A163488 Primes p such that 5*p is a sum of 3 consecutive primes.

Original entry on oeis.org

2, 3, 47, 79, 113, 197, 227, 257, 263, 317, 347, 383, 431, 443, 491, 499, 541, 557, 617, 757, 811, 887, 929, 977, 1021, 1087, 1093, 1129, 1231, 1237, 1433, 1511, 2111, 2129, 2213, 2347, 2543, 2551, 2609, 2657, 2671, 2803, 2837, 2999, 3011, 3049, 3119, 3187

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 28 2009

nonn

Primes of the form A034961(k)/5, associated with k=1, 2, 21, 31, 42, 66,... - R. J. Mathar, Aug 02 2009

			p=2 is in the sequence because 2*5=10=2+3+5.
p=3 is in the sequence because 3*5=15=3+5+7.

Cf. A006562, A118134, A163487.

lst={};Do[If[PrimeQ[p=(Prime[n]+Prime[n+1]+Prime[n+2])/5],AppendTo[lst, p]],{n,7!}];lst
cp3Q[n_]:=Module[{mid=Floor[PrimePi[(5n)/3]],tst},tst=Total/@ Partition[ Prime[ Range[mid-10,mid+10]],3,1];MemberQ[tst,5n]]; Select[ Prime[ Range[ 500]],cp3Q]//Quiet (* Harvey P. Dale, Jan 02 2018 *)

Entries checked by R. J. Mathar, Aug 02 2009

A288172 a(n) = smallest number that is the sum of 2n - 1, 2n + 1, and 2n + 3 consecutive primes.

Original entry on oeis.org

83, 311, 55813, 42161, 42161, 714295, 113469, 5539053, 20919, 1439643, 7134703, 13432571, 3337639, 6082489, 25241217, 25241217, 2389687, 54309171, 4142423, 63388405, 21570897, 15843991, 62196365, 233917295, 11679841, 96905683, 229821375, 460000131, 125571943

Offset: 1

Zak Seidov, Jun 06 2017

nonn

			n=1: 83 = A000040(23) = A034961(9) = A034964(5) = 23+29+31 = 11+13+17+19+23.
n=20: 63388405 is the sum of 39, 41 and 43 consecutive primes, A000040(122889)+...+A000040(122889+38) = A000040(117360)+...+A000040(122889+40) = A000040(112314)+...+A000040(112314+42).

Cf. A213174.

More terms from Giovanni Resta, Jun 13 2017

cp's OEIS Frontend

A234973 Primes that are 6 plus the sum of three consecutive primes.

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A244186 Primes which are the concatenation of five consecutive primes p, q, r, s, t while the sum (p + q + r + s + t) is another prime.

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A270639 Fermat pseudoprimes (A001567) that are the sum of three consecutive primes.

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A281960 Primes that are the sum of three consecutive odd semiprimes.

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A289361 Least sum s of three consecutive primes such that s is a multiple of the n-th prime.

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A335969 Sphenic numbers that are also the sum of three consecutive primes.

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A351579 Primes p such that the sum of p and the next two primes is the product of two consecutive primes.

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A125270 Coefficient of x^2 in polynomial whose zeros are 5 consecutive primes starting with the n-th prime.

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