A034962 -id:A034962 - cp's OEIS frontend

A174519 Sum of 3 consecutive primes and of all composite numbers in-between.

14, 25, 56, 70, 98, 112, 140, 264, 243, 297, 396, 280, 308, 528, 689, 513, 567, 726, 490, 675, 858, 924, 1350, 1235, 700, 728, 742, 770, 2242, 2318, 1452, 1215, 1859, 1885, 1377, 2041, 1782, 1848, 2249, 1593, 2405, 2431, 1358, 1372, 3060, 5275, 3723, 1582

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 21 2010

nonn

2+3+4+5=14, 3+4+5+6+7=25, 5+6+7+8+9+10+11=56, ..

Cf. A034962, A054265, A174518

f[n_,x_]:=n*x+x*(x+1)/2;Table[Prime[n]+f[Prime[n],Prime[n+2]-Prime[n]-1]+Prime[n+2],{n,5!}]
sm[{a_,b_,c_}]:=(c-a+1) (a+c)/2; sm/@Partition[Prime[Range[50]],3,1] (* Harvey P. Dale, Mar 11 2012 *)

A188651 Products of two primes (i.e., "semiprimes") that are the sum of three consecutive primes.

Original entry on oeis.org

10, 15, 49, 121, 143, 159, 187, 235, 287, 301, 319, 329, 371, 395, 407, 471, 519, 533, 551, 565, 581, 589, 633, 679, 689, 713, 731, 749, 771, 789, 803, 817, 841, 961, 985, 1079, 1099, 1119, 1135, 1169, 1207, 1271, 1285, 1315, 1349, 1391, 1457, 1477, 1585

Offset: 1

Zak Seidov, Apr 16 2011

nonn

Or, semiprimes in A034961 (Sums of three consecutive primes).
Subsequence of square semiprimes: {49, 121, 841, 961, 1849, 22801, 24649, 36481, 69169, ...} = {7, 11, 29, 31, 43, 151, 157, 191, 263, ...}^2 that is also a subsequence of A080665 (Squares in A034961). Cf. also A034962 (Primes A034961).
Somewhat surprisingly, the sum of two consecutive primes is never a semiprime. This follows from that fact that if p+q = 2r for primes p,q,r, then r must between p and q. So if p and q are consecutive, then r does not exist.

			a(1) = 10 = 2*5 = A034961(1) = prime(1) + prime(2) + prime(3) = 2 + 3 + 5,
a(2) = 15 = 3*5 = A034961(2) = prime(2) + prime(3) + prime(4) = 3 + 5 + 7,
a(3) = 49 = 7*7 = A080665(1) = A034961(6) = prime(6) + prime(7) + prime(8) = 13 + 17 + 19.

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

semiPrimeQ[n_Integer] := Total[FactorInteger[n]][[2]] == 2; Select[Total /@ Partition[Prime[Range[100]], 3, 1], semiPrimeQ] (* T. D. Noe, Apr 20 2011 *)

A244163 Primes which are the concatenation of three consecutive primes p, q, r while the sum (p + q + r) yields another prime.

Original entry on oeis.org

5711, 111317, 171923, 313741, 414347, 229233239, 389397401, 401409419, 409419421, 449457461, 701709719, 773787797, 787797809, 797809811, 140914231427, 157915831597, 163716571663, 202920392053, 212921312137, 252125312539, 259125932609, 263326472657, 268926932699

Offset: 1

K. D. Bajpai, Jun 21 2014

Subsequence of A030469.

The first five terms of this sequence resemble exactly those of A030469.

			5711 is in the sequence because the concatenation of [5, 7, 11] = 5711 which is prime. The sum [5 + 7 + 11] = 23 is another prime.
111317 is in the sequence because the concatenation of [11, 13, 17] = 111317 which is prime. The sum [11 + 13 + 17] = 41 is another prime.

K. D. Bajpai, Table of n, a(n) for n = 1..2250

Cf. A000040, A030469, A034961, A034962, A132903.

A244163:= proc() local a,b,c,k,m; a:=ithprime(n); b:=ithprime(n+1); c:=ithprime(n+2); m:=a+b+c; k:=parse(cat(a,b,c)); if isprime(k) and isprime(m) then RETURN (k); fi; end: seq(A244163 (), n=1..500);

prQ[{a_,b_,c_}]:=Module[{p=FromDigits[Flatten[IntegerDigits/@ {a,b,c}]]}, If[ AllTrue[ {p,a+b+c},PrimeQ],p,Nothing]]; prQ/@Partition[ Prime[ Range[ 500]],3,1] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 05 2021 *)

A258269 Primes of the form p^3 + q^2 + r, where p, q, r are consecutive primes.

Original entry on oeis.org

59, 5297, 7417, 81769, 152419, 714479, 1237037, 3330907, 25248317, 64648901, 84801217, 90728159, 286628773, 530133671, 554065817, 823543381, 1028270917, 1096980919, 1299792317, 1321357391, 1417523659, 1574410169, 1648622903, 1997248987, 2084078057, 2556384373

Offset: 1

K. D. Bajpai, May 25 2015

nonn

			a(1) = 59 is prime of the form 3^3 + 5^2 + 7.
a(2) = 5297 is prime of the form 17^3 + 19^2 + 23.

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

Cf. A000040, A034962, A133529, A133530.

[k: p in PrimesUpTo (3000) | IsPrime(k) where k is (p^3 + NextPrime(p)^2 + NextPrime(NextPrime(p)))];

A258269:= n-> (ithprime(n)^3+ithprime(n+1)^2+ithprime(n+2)): select(isprime, [seq((A258269(n), n=1..5000))]);

Select[Table[p = Prime[n]; q = NextPrime[p]; r = NextPrime[q]; p^3 + q^2 + r, {n, 5000}], PrimeQ]

forprime(p=1, 5000, q=nextprime(p+1); r=nextprime(q+1);  k=(p^3 + q^2 + r); if(isprime(k), print1(k,", ")))

A127493 Indices k such that the coefficient [x^1] of the polynomial Product_{j=0..4} (x-prime(k+j)) is prime.

Original entry on oeis.org

1, 5, 8, 9, 22, 29, 45, 49, 60, 69, 87, 89, 90, 107, 114, 124, 125, 131, 134, 138, 145, 156, 171, 183, 188, 191, 203, 204, 207, 212, 219, 255, 261, 290, 298, 303, 329, 330, 343, 344, 349, 354, 378, 397, 398, 400, 403, 454, 456, 466, 474, 515, 549, 560, 570, 578

Offset: 1

Artur Jasinski, Jan 16 2007

nonn

A fifth-order polynomial with 5 roots which are the five consecutive primes from prime(k) onward is defined by Product_{j=0..4} (x-prime(k+j)). The sequence is a catalog of the cases where the coefficient of its linear term is prime.

Indices k such that e4(prime(k), prime(k+1), ..., prime(k+4)) is prime, where e4 is the elementary symmetric polynomial summing all products of four variables. - Charles R Greathouse IV, Jun 15 2015

			For k=2, the polynomial is (x-3)*(x-5)*(x-7)*(x-11)*(x-13) = x^5-39*x^4+574*x^3-3954*x^2+12673*x-15015, where 12673 is not prime, so k=2 is not in the sequence.
For k=5, the polynomial is x^5-83*x^4+2710*x^3-43490*x^2+342889*x-1062347, where 342889 is prime, so k=5 is in the sequence.

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

Cf. A001043, A034961, A034963, A034964, A127333-A127343, A127345-A127351, A037171, A034962, A034965, A082246, A082251, A070934, A006094, A046301-A046303, A046324-A046327, A127489, A127491, A127492, A024449.

isA127493 := proc(k)
    local x,j ;
    mul( x-ithprime(k+j),j=0..4) ;
    expand(%) ;
    isprime(coeff(%,x,1)) ;
end proc:
A127493 := proc(n)
    option remember ;
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isA127493(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A127493(n),n=1..60) ; # R. J. Mathar, Apr 23 2023

a = {}; Do[If[PrimeQ[(Prime[x] Prime[x + 1]Prime[x + 2]Prime[x + 3] + Prime[x] Prime[x + 2]Prime[x + 3]Prime[x + 4] + Prime[x] Prime[x + 1]Prime[x + 3]Prime[x + 4] + Prime[x] Prime[x + 1]Prime[x + 2]Prime[x + 4] + Prime[x + 1] Prime[x + 2]Prime[x + 3]Prime[x + 4])], AppendTo[a, x]], {x, 1, 1000}]; a

e4(v)=sum(i=1,#v-3, v[i]*sum(j=i+1,#v-2, v[j]*sum(k=j+1,#v-1, v[k]*vecsum(v[k+1..#v]))))
pr(p, n)=my(v=vector(n)); v[1]=p; for(i=2,#v, v[i]=nextprime(v[i-1]+1)); v
is(n,p=prime(n))=isprime(e4(pr(p,5)))
v=List(); n=0; forprime(p=2,1e4, if(is(n++,p), listput(v,n))); Vec(v) \\ Charles R Greathouse IV, Jun 15 2015

Definition and comment rephrased and examples added by R. J. Mathar, Oct 01 2009

A133659 Primes that are the sum of three consecutive primes as well as the sum of three consecutive composite numbers.

Original entry on oeis.org

23, 31, 41, 59, 71, 109, 131, 199, 211, 251, 269, 311, 487, 503, 701, 829, 941, 1049, 1061, 1151, 1229, 1381, 1511, 1931, 2129, 2179, 2251, 2269, 2393, 2579, 2971, 3041, 3271, 3329, 3581, 3851, 3889, 3911, 4289, 4451, 4481, 4679, 4987, 4999

Offset: 1

Randy L. Ekl, Dec 28 2007

			a(3) = 41 because 41 = 11+13+17 and 41 = 12+14+15.

R. J. Mathar, Table of n, a(n) for n=1..287

Cf. A034962, A060328.

a = {}; For[n = 2, n < 10000, n++, If[ ! PrimeQ[n], AppendTo[a, n + Select[Range[n + 1, n + 10], ! PrimeQ[ # ] &][[1]] + Select[Range[n + 1, n + 10], ! PrimeQ[ # ] &][[2]]]]]; b = Table[Prime[i] + Prime[i + 1] + Prime[i + 2], {i, 1, 10000}]; Select[Intersection[a, b], PrimeQ[ # ] &] (* Stefan Steinerberger, Dec 30 2007 *)

Equals A034962 INTERSECT A060328. - R. J. Mathar, Jan 11 2008

More terms from Stefan Steinerberger, Dec 30 2007

A164130 Sums s of squares of three consecutive primes, such that s-+2 are primes.

Original entry on oeis.org

195, 5739, 18459, 32259, 33939, 60291, 74019, 169491, 187131, 244899, 276819, 388179, 783531, 902139, 3588339, 5041491, 5145819, 5193051, 8687091, 9637491, 10227291, 10910019, 11341491, 11757339, 14834379, 15354651, 16115091

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 10 2009

nonn

			5^2 + 7^2 + 11^2 = 195 is a sum of the squared consecutive primes 5, 7 and 11, and 193 and 197 are primes, so 195 is a member of the sequence.

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

Cf. A075893, A133529, A133530, A034962, A164129

q:= 2: r:= 3: R:= NULL: count:= 0:
while count < 100 do
  p:= q; q:= r; r:= nextprime(r);
  s:= p^2+q^2+r^2;
  if isprime(s-2) and isprime(s+2) then
    count:= count+1; R:= R,s;
  fi;
od:
R; # Robert Israel, Apr 21 2023

lst={};Do[p=Prime[n]^2+Prime[n+1]^2+Prime[n+2]^2;If[PrimeQ[p-2]&&PrimeQ[p+2], AppendTo[lst,p]],{n,8!}];lst

A133529 INTERSECT A087679. - R. J. Mathar, Aug 27 2009

Comment turned into example by R. J. Mathar, Aug 27 2009

A174520 Sum of all composite numbers in-between prime numbers p(n) and p(n+2).

Original entry on oeis.org

4, 10, 33, 39, 57, 63, 81, 193, 160, 200, 287, 159, 177, 385, 530, 340, 380, 527, 279, 452, 623, 673, 1081, 948, 399, 417, 423, 441, 1893, 1947, 1057, 808, 1434, 1446, 920, 1570, 1295, 1345, 1730, 1060, 1854, 1866, 777, 783, 2453, 4642, 3062, 903, 921, 1873

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 21 2010

nonn

2_3_4_5 -> 4, 3_4_5_6_7 -> 4+6=10, 5_6_7_8_9_10_11 -> 6+8+9+10=33, ..

Cf. A034962, A054265, A174518, A174519

f[n_,x_]:=n*x+x*(x+1)/2;Table[f[Prime[n],Prime[n+2]-Prime[n]-1]-Prime[n+1],{n,5!}]

A174521 Primes that are the sum of all composite numbers in-between prime numbers p(n) and p(n+2).

Original entry on oeis.org

193, 673, 1873, 2207, 2833, 4391, 3023, 8209, 5903, 8999, 6047, 9643, 7537, 19843, 10273, 29399, 11953, 12433, 20879, 35999, 36241, 23761, 23831, 24907, 20353, 32401, 33403, 22367, 34129, 57367, 49123, 74311, 51197, 40037, 42773, 71399

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 21 2010

nonn

20+21+22+24+25+26+27+28=193,..

Harvey P. Dale, Table of n, a(n) for n = 1..2000

Cf. A034962, A054265, A174518, A174519, A174520

f[n_,x_]:=n*x+x*(x+1)/2;Select[Table[f[Prime[n],Prime[n+2]-Prime[n]-1]-Prime[n+1],{n,6!}],PrimeQ[ # ]&]
Select[Table[Total[Select[Range[Prime[n],Prime[n+2]],CompositeQ]],{n,1000}],PrimeQ] (* Harvey P. Dale, May 13 2017 *)

A179007 Sum of 3 consecutive composite odd numbers.

Original entry on oeis.org

45, 61, 73, 85, 95, 107, 119, 133, 145, 155, 163, 175, 185, 197, 209, 221, 233, 243, 253, 263, 271, 279, 287, 299, 315, 331, 343, 351, 357, 363, 369, 377, 387, 397, 409, 419, 429, 435, 445, 455, 467, 475, 485, 495, 505, 515, 523, 535, 545, 555, 561, 571, 585, 599

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 07 2011

nonn

			9+15+21=45, 15+21+25=61, 21+25+27=73,..

Cf. A034962, A034965, A082246

t=Select[Range[2, 200], OddQ[#] && ! PrimeQ[#] &]; Plus @@@ Partition[t, 3, 1]

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A174519 Sum of 3 consecutive primes and of all composite numbers in-between.

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A188651 Products of two primes (i.e., "semiprimes") that are the sum of three consecutive primes.

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A244163 Primes which are the concatenation of three consecutive primes p, q, r while the sum (p + q + r) yields another prime.

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A258269 Primes of the form p^3 + q^2 + r, where p, q, r are consecutive primes.

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A127493 Indices k such that the coefficient [x^1] of the polynomial Product_{j=0..4} (x-prime(k+j)) is prime.

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A133659 Primes that are the sum of three consecutive primes as well as the sum of three consecutive composite numbers.

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A164130 Sums s of squares of three consecutive primes, such that s-+2 are primes.

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A174520 Sum of all composite numbers in-between prime numbers p(n) and p(n+2).

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