A034961 -id:A034961 - cp's OEIS frontend

A165982 Primes that are 4 plus the sum of three consecutive primes.

19, 53, 101, 113, 163, 191, 227, 239, 353, 443, 461, 491, 523, 569, 593, 683, 821, 887, 1019, 1103, 1123, 1289, 1307, 1319, 1481, 1693, 1721, 1783, 1811, 1871, 1997, 2203, 2237, 2273, 2333, 2459, 2741, 2789, 3001, 3023, 3089, 3251, 3313, 3373, 3407, 3491

Offset: 1

Vincenzo Librandi, Oct 03 2009

nonn

Primes of the form 4+A034961(k).

			A000040(8)  =  19 = 4 + A034961(2).
A000040(16) =  53 = 4 + A034961(6).
A000040(26) = 101 = 4 + A034961(10).

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

Select[Total[#]+4&/@Partition[Prime[Range[200]],3,1],PrimeQ]  (* Harvey P. Dale, Jan 12 2011 *)

1721 inserted by R. J. Mathar, Oct 05 2009

A166039 Sums of three consecutive nonprimes A141468.

Original entry on oeis.org

5, 11, 18, 23, 27, 31, 36, 41, 45, 49, 54, 59, 63, 67, 71, 75, 78, 81, 85, 90, 95, 99, 102, 105, 109, 113, 117, 121, 126, 131, 135, 139, 143, 147, 150, 153, 157, 161, 165, 168, 171, 175, 180, 185, 189, 192, 195, 199, 203, 207, 211, 216, 221, 225, 228, 231, 235

Offset: 1

Juri-Stepan Gerasimov, Oct 05 2009

nonn

			a(1) = 0 + 1 + 4 =  5;
a(2) = 1 + 4 + 6 = 11;
a(3) = 4 + 6 + 8 = 18.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A034961, A141468.

A002808 := proc(n) option remember; if n = 1 then 4; else for a from procname(n-1)+1 do if not isprime(a) then return a; fi; od: fi; end: A141468 := proc(n) if n <= 2 then n-1 ; else A002808(n-2) ; fi; end: A166039 := proc(n) add(A141468(j),j=n..n+2) ; end: seq(A166039(n),n=1..120) ; # R. J. Mathar, Oct 10 2009

With[{nn=100},Total/@Partition[Complement[Range[0,nn],Prime[ Range[ PrimePi[ nn]]]],3,1]] (* Harvey P. Dale, Aug 05 2015 *)

A127492 Indices m of primes such that Sum_{k=0..2, k

Original entry on oeis.org

2, 10, 17, 49, 71, 72, 75, 145, 161, 167, 170, 184, 244, 250, 257, 266, 267, 282, 286, 301, 307, 325, 343, 391, 405, 429, 450, 537, 556, 561, 584, 685, 710, 743, 790, 835, 861, 904, 928, 953

Offset: 1

Artur Jasinski, Jan 16 2007

Let p_0 .. p_4 be five consecutive primes, starting with the m-th prime. The index m is in the sequence if the absolute value [x^0] of the polynomial (x-p_0)*[(x-p_1)*(x-p_2) + (x-p_2)*(x-p_3) + (x-p_3)*(x-p_4)] + (x-p_1)*[(x-p_2)*(x-p_3) + (x-p_3)*(x-p_4)] + (x-p_2)*(x-p_3)*(x-p_4) is two times a prime. The correspondence with A127491: the coefficient [x^2] of the polynomial (x-p_0)*(x-p_1)*..*(x-p_4) is the sum of 10 products of a set of 3 out of the 5 primes. Here the sum is restricted to the 6 products where the two largest of the 3 primes are consecutive. - R. J. Mathar, Apr 23 2023

Harvey P. Dale, Table of n, a(n) for n = 2..1000

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, A127490, A127491.

isA127492 := proc(k)
    local x,j ;
    (x-ithprime(k))* mul( x-ithprime(k+j),j=1..2)
    +(x-ithprime(k))* mul( x-ithprime(k+j),j=2..3)
    +(x-ithprime(k))* mul( x-ithprime(k+j),j=3..4)
    +(x-ithprime(k+1))* mul( x-ithprime(k+j),j=2..3)
    +(x-ithprime(k+1))* mul( x-ithprime(k+j),j=3..4)
    +(x-ithprime(k+2))* mul( x-ithprime(k+j),j=3..4) ;
    p := abs(coeff(expand(%/2),x,0)) ;
    if type(p,'integer') then
        isprime(p) ;
    else
        false ;
    end if ;
end proc:
for k from 1 to 900 do
    if isA127492(k) then
        printf("%a,",k) ;
    end if ;
end do: # R. J. Mathar, Apr 23 2023

a = {}; Do[If[PrimeQ[(Prime[x] Prime[x + 1]Prime[x + 2] + Prime[x] Prime[x + 2]Prime[x + 3] + Prime[x] Prime[x + 3] Prime[x + 4] + Prime[x + 1] Prime[x + 2]Prime[x + 3] + Prime[x + 1] Prime[x + 3]Prime[x + 4] + Prime[x + 2] Prime[x + 3] Prime[x + 4])/2], AppendTo[a, x]], {x, 1, 1000}]; a
prQ[{a_,b_,c_,d_,e_}]:=PrimeQ[(a b c+a c d+a d e+b c d+b d e+c d e)/2]; PrimePi/@Select[ Partition[ Prime[Range[1000]],5,1],prQ][[;;,1]] (* Harvey P. Dale, Apr 21 2023 *)

Definition simplified by R. J. Mathar, Apr 23 2023

Edited by Jon E. Schoenfield, Jul 23 2023

A165985 Primes that are 8 plus the sum of three consecutive primes.

Original entry on oeis.org

23, 31, 67, 79, 139, 151, 167, 181, 277, 337, 379, 433, 479, 541, 641, 709, 739, 757, 797, 811, 919, 1069, 1087, 1237, 1279, 1399, 1619, 1697, 1787, 1801, 1951, 2083, 2137, 2207, 2311, 2503, 2557, 2659, 2713, 2767, 2833, 2939, 3049, 3061, 3079, 3169, 3301, 3547, 3677

Offset: 1

Vincenzo Librandi, Oct 03 2009

nonn

Primes of the form 8+A034961(k).

			A000040(9)= 23 = 8+ A034961(2). A000040(11) = 31 = 8+A034961(3). A000040(19) = 67 = 8+A034961(7).

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

Select[Total[#] + 8&/@Partition[Prime[Range[900]], 3, 1], PrimeQ] (* Vincenzo Librandi, Oct 13 2012 *)

Extended by R. J. Mathar, Oct 05 2009

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 *)

A270267 Carmichael numbers (A002997) that are the sum of three consecutive primes.

Original entry on oeis.org

252601, 410041, 1615681, 2113921, 10606681, 10877581, 11921001, 26932081, 44238481, 54767881, 82929001, 120981601, 128697361, 208969201, 246446929, 255160621, 278152381, 280067761, 311388337, 325546585, 334783585, 416964241, 533860309, 593234929, 672389641

Offset: 1

Altug Alkan, Mar 14 2016

nonn

In other words, Carmichael numbers of the form p + q + r where p, q and r are consecutive primes.
If a Carmichael number is the sum of n consecutive primes, it is so obvious that the minimum value of n is 3.
Intersection of A002997 and A034961.

			84191, 84199 and 84211 are consecutive primes and sum of them is 252601 that is a Carmichael number.
136657, 136691 and 136693 are consecutive primes and sum of them is 410041 that is a Carmichael number.
538553, 538561 and 538567 are consecutive primes and sum of them is 1615681 that is a Carmichael number.

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

Cf. A002997, A034961.

isA002997(n) = {my(f); bittest(n, 0) && !for(i=1, #f=factor(n)~, (f[2, i]==1 && n%(f[1, i]-1)==1)||return) && #f>1}
a034961(n) = my(p=prime(n), q=nextprime(p+1)); p+q+nextprime(q+1);
for(n=1, 1e6, if(isA002997(a034961(n)), print1(a034961(n), ", ")));

More terms from Amiram Eldar, Jun 25 2019

A283873 Smallest number that is the sum of n successive primes and also the sum of n successive semiprimes, n > 1.

Original entry on oeis.org

24, 749, 48, 311, 690, 251, 2706, 2773, 6504, 1081, 2162, 1753, 11356, 6223, 1392, 2303, 9838, 637, 14510, 1995, 3154, 21459, 72960, 5691, 8140, 1475, 2350, 3647, 1593, 7607, 55074, 2719, 9852, 12143, 106562, 12615, 9036, 19883, 15438, 28369, 8560, 8415, 3831

Offset: 2

Zak Seidov, Mar 17 2017

nonn

The sequence is non-monotone.

			a(2) = 24 = A000040(5) + A000040(6) = 11 + 13 = A001358(4) + A001358(5) = 10 + 14,
a(3) = 749 = A000040(53) + A000040(54) + A000040(55) = 241 + 251 + 257 = A001358(79) + A001358(80) + A001358(81) = 247 + 249 + 253.

Alois P. Heinz, Table of n, a(n) for n = 2..1000

Cf. A000040 Primes, A001358 Semiprimes, A118717 Sum of two consecutive semiprimes.

Sum of k consecutive primes: A001043 k=2, A034961 k=3, A034963 k=4, A034964 k=5, A127333 k=6, A127334 k=7, A127335 k=8, A127336 k=9, A127337 k=10, A127338 k=11, A127339 k=12.

issp:= n-> is(not isprime(n) and numtheory[bigomega](n)=2):
ithsp:= proc(n) option remember; local k; for k from 1+
        `if`(n=1, 1, ithsp(n-1)) while not issp(k) do od; k
        end:
ps:= proc(i, j) option remember;
       ithprime(j)+`if`(i=j, 0, ps(i, j-1))
     end:
ss:= proc(i, j) option remember;
       ithsp(j)+`if`(i=j, 0, ss(i, j-1))
     end:
a:= proc(n) option remember; local i, j, k, l, p, s;
      i, j, k, l, p, s:= 1, n, 1, n, ps(1, n), ss(1, n);
      do if p=s then return p
       elif pAlois P. Heinz, Mar 24 2017

Mathematica

sp=Select[Range[4,100000],2==PrimeOmega[#]&];pr=Prime[Range[PrimePi[Max[sp]]]]; Table[Intersection[(Total/@Partition[pr,k,1]),Total/@Partition[sp,k,1]][[1]],{k,2,100}]

Extensions

More terms from Alois P. Heinz, Mar 24 2017

cp's OEIS Frontend

A165982 Primes that are 4 plus the sum of three consecutive primes.

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A166039 Sums of three consecutive nonprimes A141468.

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A127492 Indices m of primes such that Sum_{k=0..2, k

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A165985 Primes that are 8 plus the sum of three consecutive primes.

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Mathematica

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

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Mathematica

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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Maple

Mathematica

A270267 Carmichael numbers (A002997) that are the sum of three consecutive primes.

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PARI

Extensions

A283873 Smallest number that is the sum of n successive primes and also the sum of n successive semiprimes, n > 1.

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