A034962 -id:A034962 - cp's OEIS frontend

A179012 Primes that are the sum of three consecutive composite odd numbers.

61, 73, 107, 163, 197, 233, 263, 271, 331, 397, 409, 419, 467, 523, 571, 599, 677, 691, 757, 827, 839, 883, 929, 997, 1039, 1051, 1063, 1097, 1123, 1153, 1163, 1171, 1187, 1223, 1231, 1291, 1301, 1367, 1433, 1493, 1523, 1531, 1571, 1619, 1627, 1637, 1667, 1693, 1783

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 07 2011

nonn

			15+21+25=61, 21+25+27=73, 33+35+39=107.

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

Cf. A034962, A034965, A082246, A179007.

lst={};Do[If[!PrimeQ[n],s=n;k=1,Continue[]];If[!PrimeQ[n+2],s+=n+2;k=2;q=2,If[!PrimeQ[n+4],s+=n+4;k=2;q=4,If[!PrimeQ[n+6],s+=n+6;k=2;q=6]]];If[!PrimeQ[n+q+2],s+=n+q+2;k=3;q+=2,If[!PrimeQ[n+q+4],s+=n+q+4;k=3;q+=4,If[!PrimeQ[n+q+6],s+=n+q+6;k=3;q+=6]]];If[PrimeQ[s],AppendTo[lst,s]],{n,9,6!,2}];lst
nn=1001;With[{compodd=Complement[Range[9,nn,2],Prime[Range[ PrimePi[ nn]]]]}, Select[ Total/@ Partition[compodd,3,1],PrimeQ]] (* Harvey P. Dale, Dec 09 2012 *)

A220569 Smallest prime divisor of prime(n) + prime(n+1) + prime(n+2).

Original entry on oeis.org

2, 3, 23, 31, 41, 7, 59, 71, 83, 97, 109, 11, 131, 11, 3, 173, 11, 199, 211, 223, 5, 251, 269, 7, 7, 311, 11, 7, 349, 7, 5, 11, 5, 439, 457, 3, 487, 503, 3, 13, 19, 5, 7, 19, 607, 3, 661, 7, 13, 701, 23, 17, 7, 3, 3, 11, 19, 829, 29, 857, 883, 911, 7, 941

Offset: 1

Zak Seidov, Dec 16 2012

nonn

			a(6) = 7 because prime(6) + prime(6+1) + prime(6+2) = 13 + 17 + 19 = 49 and the smallest prime factor of 49 is 7.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A034961, A034962, A073681.

{a=2; b=3; c=5; for(n=1, 100, s=a+b+c;
dv=divisors(s); print1(dv[2]", "); a=b; b=c; c=nextprime(c+2))}

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 *)

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

A344868 Primes p that are equal to (prime(k)+2prime(k+1)+3prime(k+2))/2 for some k.

Original entry on oeis.org

17, 101, 191, 227, 293, 431, 461, 557, 571, 757, 821, 863, 1039, 1193, 1213, 1277, 1291, 1307, 1373, 1483, 1499, 1721, 1811, 2239, 2293, 2309, 2447, 2689, 3167, 3181, 3547, 3617, 3701, 3881, 4243, 4441, 4703, 4723, 4871, 5651, 6079, 6101, 6133, 6829, 6907, 6997, 7523, 7853, 7879, 7949

Offset: 1

Zak Seidov, May 31 2021

nonn

Corresponding values of k: 2, 10, 17, 20, 24, 33, 35, 41, 42, 53, 57, 60, 68, 77, 78, 81, 82, 83, 87, 93, 94, 104, 109, 131, 134, 135, 140, 153, 176, 177, 193, 196, 201, 209, 222, 233, 246, 247, 256, 288, 306, 307, 308, 337, 341, 344, 367, 379, 380, 382, 393, 395.

			17 = (3 + 2*5 + 3*7)/2, 101 = (29 + 2*31 + 3*37)/2.

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

Cf. A034962.

s = {}; Do[If[PrimeQ[p = (Prime[k] + 2*Prime[k + 1] + 3*Prime[k + 2])/2], AppendTo[s, p]], {k, 400}]; s
Select[(#[[1]]+2#[[2]]+3#[[3]])/2&/@Partition[Prime[ Range[ 500]],3,1],PrimeQ] (* Harvey P. Dale, Jan 28 2022 *)

{p = 3; q = 5; r = 7; for (k = 1, 400, if (isprime (P = (p + 2*q + 3*r)/2), print1 (P ", ")); p = q; q = r; r = nextprime (r + 2))}

A345472 Emirps p such that both p and its reversal are sums of three consecutive primes.

Original entry on oeis.org

1151, 1249, 1511, 3467, 3697, 7643, 7963, 9421, 11593, 32749, 36467, 39511, 71329, 76463, 92317, 94723, 110119, 111109, 123707, 124309, 124823, 128377, 141371, 146953, 150383, 155153, 160159, 164291, 167779, 173141, 178223, 184609, 190807, 192383, 192461, 199247, 304193, 304879, 306133, 322871

Offset: 1

J. M. Bergot and Robert Israel, Jun 20 2021

Numbers p such that p and its digit reversal are distinct members of A034962.

			a(4) = 1511 because 1511 and its reversal 1151 are distinct primes, and 1511 = 499+503+509 and 1151 = 379+383+389 are sums of three consecutive primes.

Robert Israel, Table of n, a(n) for n = 1..1792

Cf. A006567, A034962.

digrev:= proc(n) local L, i;
  L:= convert(n, base, 10);
  add(L[-i]*10^(i-1), i=1..nops(L))
end proc:P:= select(isprime, [2,seq(i,i=3..nextprime(nextprime(333333)),2)]):
P3:= convert(select(isprime,P[1..-3]+P[2..-2]+P[3..-1]),set):
B:= P3 intersect map(digrev,P3):
sort(convert(remove(t -> digrev(t)=t,B),list);

A050200 Let p = prime(n). Then a(n) = p + (next prime >= p+1) + (next prime >= p+3).

Original entry on oeis.org

10, 15, 23, 29, 41, 47, 59, 65, 81, 97, 105, 119, 131, 137, 153, 171, 187, 195, 209, 223, 231, 245, 261, 283, 299, 311, 317, 329, 335, 367, 389, 405, 425, 437, 457, 465, 483, 497, 513, 531, 551, 563, 581, 587, 607, 621, 657, 677, 689, 695, 711, 731, 743, 765

Offset: 0

Cino Hilliard, May 08 2003

The occurrence of multiples of 3 in the sequence appears to converge to about 0.44.

Cf. A034962.

nextprim[n_] := Block[{k = n}, While[ ! PrimeQ[k], k++ ]; k]; f[n_] := (x = Prime[n]; nextprim[x] + nextprim[x + 1] + nextprim[x + 3]); Table[ f[n], {n, 54}] (* Robert G. Wilson v, Feb 12 2005 *)
np[n_]:=Module[{pr=Prime[n]},pr+NextPrime[pr+1]+NextPrime[pr+3]]; Join[ {10}, Array[ np,60,2]] (* Harvey P. Dale, Mar 04 2015 *)

sumprime3(n) = { c1=0; c2=0; forprime(x=2,n, s = nextprime(x)+nextprime(x+1)+nextprime(x+3); c1++; if(s%3==0,c2++); print1(s" "); ); print(); print(c2/c1+.0) }

Definition corrected by Zak Seidov, Robert G. Wilson v and Ralf Stephan, Feb 10 2005

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

A341338 a(n) is the smallest prime that is simultaneously the sum of 2n-1, 2n+1 and 2n+3 consecutive primes.

Original entry on oeis.org

83, 311, 55813, 437357, 1219789, 8472193, 9496853, 6484103, 2166953, 37296143, 12671599, 13432571, 14968909, 145616561, 732092831, 220872569, 1381099933, 93482633, 4142423, 87030017, 3193060007, 736535783, 6390999871, 280886077, 464341303, 268231657, 686836817, 9000046663

Offset: 1

Zak Seidov, Apr 25 2021

nonn

			For n = 1: 83 = 23 + 29 + 31 = 11 + 13 + 17 + 19 + 23, and 83 is the smallest prime that is the sum of 1, 3 and 5 consecutive primes, so a(1) = 83.

Cf. A034962, A034965, A082246, A082251, A127340, A127341, A161612.

Array[(k=1;
While[(i=Select[Intersection@@((Total/@Subsequences[Prime@Range@Prime[k++],{#}])&/@{2#-1,2#+1,2#+3}),PrimeQ])=={}];First@i)&,4] (* Giorgos Kalogeropoulos, Apr 26 2021 *)

cp's OEIS Frontend

A179012 Primes that are the sum of three consecutive composite odd numbers.

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A220569 Smallest prime divisor of prime(n) + prime(n+1) + prime(n+2).

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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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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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A344868 Primes p that are equal to (prime(k)+2*prime(k+1)+3*prime(k+2))/2 for some k.

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A345472 Emirps p such that both p and its reversal are sums of three consecutive primes.

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Maple

A050200 Let p = prime(n). Then a(n) = p + (next prime >= p+1) + (next prime >= p+3).

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A344868 Primes p that are equal to (prime(k)+2prime(k+1)+3prime(k+2))/2 for some k.