A034961 -id:A034961 - cp's OEIS frontend

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

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

A129109 Sums of three consecutive hexagonal numbers.

Original entry on oeis.org

7, 22, 49, 88, 139, 202, 277, 364, 463, 574, 697, 832, 979, 1138, 1309, 1492, 1687, 1894, 2113, 2344, 2587, 2842, 3109, 3388, 3679, 3982, 4297, 4624, 4963, 5314, 5677, 6052, 6439, 6838, 7249, 7672, 8107, 8554, 9013, 9484, 9967, 10462, 10969, 11488, 12019, 12562

Offset: 0

Jonathan Vos Post, May 24 2007

Arises in hexagonal number analog to A129803 Triangular numbers which are the sum of three consecutive triangular numbers. What are the hexagonal numbers which are the sum of three consecutive hexagonal numbers? Prime for a(0) = 7, a(4) = 139, a(6) = 277, a(8) = 463, a(18) = 2113, a(22) = 3109, a(26) = 4297, a(38) = 9013, a(40) = 9967.

			a(0) = H(0) + H(1) + H(2) = 0 + 1 + 6 = 7 = 6*0^2 + 9*0 + 7.
a(1) = H(1) + H(2) + H(3) = 1 + 6 + 15 = 22 = 6*1^2 + 9*1 + 7.
a(2) = H(2) + H(3) + H(4) = 6 + 15 + 28 = 49 = 6*2^2 + 9*2 + 7.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000384, A007667, A034961, A129803, A129863.

I:=[7,22,49]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 20 2012

LinearRecurrence[{3,-3,1},{7,22,49},50] (* Vincenzo Librandi, Feb 20 2012 *)
Total/@Partition[PolygonalNumber[6,Range[0,50]],3,1] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 14 2020 *)

a(n)=6*n^2+9*n+7 \\ Charles R Greathouse IV, Feb 20 2012

a(n) = H(n) + H(n+1) + H(n+2) where H(n) = A000384(n) = n*(2*n-1).
a(n) = 6*n^2 + 9*n + 7.
From Colin Barker, Feb 20 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: (7 + x + 4*x^2)/(1-x)^3. (End)
E.g.f.: (7 + 15*x + 6*x^2)*exp(x). - Elmo R. Oliveira, Nov 16 2024

A129111 Sums of three consecutive heptagonal numbers.

Original entry on oeis.org

8, 26, 59, 107, 170, 248, 341, 449, 572, 710, 863, 1031, 1214, 1412, 1625, 1853, 2096, 2354, 2627, 2915, 3218, 3536, 3869, 4217, 4580, 4958, 5351, 5759, 6182, 6620, 7073, 7541, 8024, 8522, 9035, 9563, 10106, 10664, 11237, 11825, 12428, 13046, 13679, 14327, 14990

Offset: 0

Jonathan Vos Post, May 24 2007

Arises in heptagonal number analog to A129803 (Triangular numbers which are the sum of three consecutive triangular numbers).
What are the heptagonal numbers which are the sum of three consecutive heptagonal numbers?
Prime for a(2) = 59, a(3) = 107, a(7) = 449, a(10) = 863, a(11) = 1031, a(23) = 4217, a(26) = 5351, a(31) = 7541, a(42) = 13679, a(43) = 14327, a(46) = 16361, a(51) = 20051.

			a(0) = Hep(0) + Hep(1) + Hep(2) = 0 + 1 + 7 = 8 = (15/2)*0^2 + (21/2)*0 + 8.
a(1) = Hep(1) + Hep(2) + Hep(3) = 1 + 7 + 18 = 26 = (15/2)*1^2 + (21/2)*1 + 8.
a(2) = Hep(2) + Hep(3) + Hep(4) = 7 + 18 + 34 = 59 = (15/2)*2^2 + (21/2)*2 + 8.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000566, A007667, A034961, A129803, A129863.

I:=[8,26,59]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 20 2012

LinearRecurrence[{3,-3,1},{8,26,59},50] (* Vincenzo Librandi, Feb 12 2012 *)

a(n)=3*n*(5*n+7)/2+8 \\ Charles R Greathouse IV, Jun 17 2017

def a(n): return 3*n*(5*n+7)//2 + 8
print([a(n) for n in range(44)]) # Michael S. Branicky, Aug 26 2021

a(n) = Hep(n) + Hep(n+1) + Hep(n+2) where Hep(n) = A000566(n) = n*(5*n-3)/2.
a(n) = (15/2)*n^2 + (21/2)*n + 8.
From Colin Barker, Feb 20 2012: (Start)
G.f.: (8 + 2*x + 5*x^2)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
E.g.f.: exp(x)*(16 + 36*x + 15*x^2)/2. - Elmo R. Oliveira, Nov 16 2024

A130704 Palindromic primes whose squares are the sum of three consecutive primes.

Original entry on oeis.org

7, 11, 151, 191, 929, 10301, 14741, 15451, 76667, 98689, 1062601, 1153511, 1175711, 1215121, 1300031, 1317131, 1489841, 1597951, 3075703, 3127213, 3362633, 3441443, 7354537, 7472747, 7662667, 9127219, 9196919, 9451549, 9561659

Offset: 1

Robert G. Wilson v, Jun 19 2007

The number of such palindromic primes less than 10^n: 1, 2, 5, 5, 10, 10, 30, 30, 141, 141, 843, 843, 5856, 5856, 42675, 42675, ....

			7^2 = 49 = 13 + 17 + 19.
11^2 = 121 = 37 + 41 + 43.

Robert G. Wilson v, Table of n, for n = 1..1000.

NextPalindrome[n_] := Block[{l = Floor[Log[10, n] + 1], idn = IntegerDigits[n]}, If[ Union[idn] == {9}, Return[n + 2], If[l < 2, Return[n + 1], If[ FromDigits[ Reverse[ Take[idn, Ceiling[l/2]]]] > FromDigits[ Take[idn, -Ceiling[l/2]]], FromDigits[ Join[ Take[ idn, Ceiling[l/2]], Reverse[Take[idn, Floor[l/2]]]]], idfhn = FromDigits[ Take[idn, Ceiling[l/2]]] + 1; idp = FromDigits[ Join[ IntegerDigits[ idfhn], Drop[ Reverse[ IntegerDigits[ idfhn]], Mod[l, 2]]]]]]]];
PrevPrim[n_] := Block[{k = n - 1}, While[ ! PrimeQ[k], k-- ]; k]; NextPrim[n_] := Block[{k = n + 1}, While[ ! PrimeQ[k], k++ ]; k]; fQ[n_] := Block[{p, q, r, s}, q = PrevPrim[ Ceiling[n^2/3]]; p = PrevPrim@q; r = NextPrim[ Floor[n^2/3]]; s = NextPrim@r; n^2 == p + q + r || n^2 == q + r + s];
pd = 6; lst = {}; Do[pd = NextPalindrome@pd; If[ PrimeQ@pd && fQ@pd, AppendTo[lst, pd]], {n, 10^8}]; lst
Select[Sqrt[#]&/@(Total/@Partition[Prime[Range[10^7]],3,1]),PalindromeQ[#]&&PrimeQ[#]&] (* The program generates the first 8 terms of the sequence. To generate more, increase the Range constant, but the program may take a long time to run. *) (* Harvey P. Dale, Jul 26 2023 *)

Intersection of A002385 and A034961.

A135277 a(n) = prime(2n-1) + prime(2n) + prime(2n+1).

Original entry on oeis.org

10, 23, 41, 59, 83, 109, 131, 159, 187, 211, 235, 269, 301, 319, 349, 395, 425, 457, 487, 519, 551, 581, 607, 661, 689, 713, 749, 789, 817, 841, 883, 931, 961, 1015, 1049, 1079, 1119, 1151, 1187, 1229, 1271, 1303, 1331, 1367, 1391, 1433, 1477, 1511, 1553, 1611

Offset: 1

Cino Hilliard, Dec 02 2007

Original name was: Sum of staircase primes according to the rule: bottom + top + next top.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A034961, A135274.

Table[Prime[n + 1] + Prime[n] + Prime[n + 2], {n, 1, 50}][[;; ;; 2]] (* G. C. Greubel, Oct 08 2016 *)

g(n) = forstep(x=1,n,2,y=prime(x+1) + prime(x) + prime(x+2);print1(y","))

a(n) = prime(2*n-1) + prime(2*n) + prime(2*n+1) \\ Charles R Greathouse IV, Oct 08 2016

from sympy import prime
def a(n): return prime(2*n-1) + prime(2*n) + prime(2*n+1)
print([a(n) for n in range(1, 51)]) # Michael S. Branicky, Oct 23 2021

We list the primes in staircase fashion as in A135274. The right diagonal, RD(n), is the set of top primes and the left diagonal, LD(n), is the set of bottom primes. Then a(n) = LD(n+1) + RD(n) + RD(n+2).

a(n) = A034961(2*n-1). - R. J. Mathar, Sep 10 2016

New name from Charles R Greathouse IV, Oct 08 2016

A165981 Primes p which are equal to 2 plus the sum of three consecutive primes.

Original entry on oeis.org

17, 43, 61, 73, 271, 313, 331, 373, 397, 409, 521, 691, 733, 751, 773, 859, 1051, 1063, 1153, 1171, 1231, 1459, 1613, 1669, 1759, 1823, 1933, 2053, 2131, 2473, 2551, 2707, 2843, 2917, 2953, 2999, 3163, 3221, 3331, 3371, 3469, 3517, 3541, 3583, 3671, 3719

Offset: 1

Vincenzo Librandi, Oct 03 2009

			17 is in the sequence because 2+3+5+7=17; 43=2+11+13+17; 271=2+83+89+97.

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

Cf. A034961. [R. J. Mathar, Oct 14 2009]

Select[Total[#] + 2 & / @Partition[Prime[Range[200]], 3, 1], PrimeQ] (* Vincenzo Librandi, Sep 13 2013 *)

Edited by N. J. A. Sloane, Oct 04 2009

Terms from 313 to 773 inserted by R. J. Mathar, Oct 14 2009

A167807 Square pyramidal numbers which are sums of three consecutive primes.

Original entry on oeis.org

1015, 25585, 1623245, 2127685, 7838831, 8865649, 19849115, 52051769, 73998155, 88409285, 91753025, 161553785, 216862421, 289872105, 347016319, 382029011, 466430159, 835713879, 1077314939, 1223359835, 1509659555, 1584781241

Offset: 1

Zak Seidov, Nov 12 2009

nonn

Intersection of A000330 (Square pyramidal numbers) and A034961 (Sums of three consecutive primes).

			1015=A034961(67)=A000330(14)
25585=A034961(1062)=A000330(42).

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000330, A034961.

from _future_ import division
from sympy import nextprime, prevprime
A167807_list = []
for i in range(3,10**6):
    n = i*(i+1)*(2*i+1)//6
    p2 = prevprime(n//3)
    p1, p3 = prevprime(p2), nextprime(p2)
    q = p1+p2+p3
    while q <= n:
        if q == n:
            A167807_list.append(n)
        p1, p2, p3 = p2, p3, nextprime(p3)
        q = p1+p2+p3 # Chai Wah Wu, Dec 31 2015

a(6)-a(22) from Donovan Johnson, Nov 15 2009

A171444 Sum of three consecutive reversed primes.

Original entry on oeis.org

10, 15, 23, 49, 113, 193, 194, 215, 137, 178, 100, 121, 122, 143, 204, 146, 187, 109, 130, 151, 172, 233, 215, 278, 481, 1103, 1903, 1913, 1933, 1163, 1583, 1793, 2603, 2023, 1843, 1263, 1873, 1493, 2103, 1523, 1343, 763, 1373, 2173, 1894, 1425

Offset: 1

Vincenzo Librandi, Dec 09 2009

			(from primes 11, 13, and 17): 11 + 31 + 71 = 113;
(from primes 13, 17, and 19): 31 + 71 + 91 = 193;
(from primes 173, 179, and 181): 371 + 971 + 181 = 1523.

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

Cf. A034961, A004087.

r[n_] := FromDigits[Reverse[IntegerDigits[n]]]; Table[r[Prime[n]] + r[Prime[n+1]] + r[Prime[n+2]], {n, 50}]

a(n) = r(p(n)) + r(p(n+1)) + r(p(n+2)) where p(n) is the n-th prime number and r(n) is the number obtained by the reversal of the digits of n (e.g., r(1230) = 321).

More terms from Matthew Conroy, Dec 28 2010

A176603 Smallest prime p of three consecutive primes (p,q,r) with p + q + r equal to a lower twin prime.

Original entry on oeis.org

11, 17, 19, 83, 101, 281, 347, 349, 379, 401, 547, 641, 701, 839, 1103, 1151, 1171, 1187, 1279, 1303, 1409, 1439, 1489, 1823, 2089, 2243, 2857, 2861, 2927, 2999, 3083, 3203, 3347, 3359, 3467, 4639, 5087, 5233, 5861, 5879, 5881, 5923, 5953, 6007, 6299, 6491

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 21 2010

nonn

The sequence is constructed by intersecting A034961 and A001359, then printing the smallest of the three primes that sum to A034961.

			11+13+17 = 41 = prime(13), 43 = prime(14), 11 is 1st term.
17+19+23 = 59 = prime(17), 61 = prime(18), 17 is 2nd term.
Detailed list:
11+13+17 = 41, 17+19+23 = 59, 19+23+29 = 71, 83+89+97 = 269,
101+103+107 = 311, 281+283+293 = 857, 347+349+353 = 1049,
349+353+359 = 1061, 379+383+389 = 1151, 401+409+419 = 1229,
547+557+563 = 1667, 641+643+647 = 1931, 701+709+719 = 2129,
839+853+857 = 2549, 1103+1109+1117 = 3329, 1151+1153+1163 = 3467,
1171+1181+1187 = 3539, 1187+1193+1201 = 3581, 1279+1283+1289 = 3851,
1303+1307+1319 = 3929, 1409+1423+1427 = 4259, 1439+1447+1451 = 4337,
1489+1493+1499 = 4481, 1823+1831+1847 = 5501, 2089+2099+2111 = 6299,
2243+2251+2267 = 6761, 2857+2861+2879 = 8597, 2861+2879+2887 = 8627,
2927+2939+2953 = 8819, 2999+3001+3011 = 9011.

Theo Kempermann, Zahlentheoretische Kostproben, Harri Deutsch, 2. aktualisierte Auflage 2005.
Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Band I, B. G. Teubner, Leipzig u. Berlin, 1909.

Amiram Eldar, Table of n, a(n) for n = 1..10000
E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1 and vol. 2, Leipzig, Berlin, B. G. Teubner, 1909.

Cf. A001359, A034961, A174454.

Prime /@ Position[Plus @@@ Partition[ Prime[ Range[1000]], 3, 1] , ?(PrimeQ[#]&&PrimeQ[#+2] &)]//Flatten  (* _Amiram Eldar, Dec 24 2019 *)

my(ppp=2,pp=3); forprime(p=5,6600,my(psum=ppp+pp+p); if(isprime(psum)&&isprime(psum+2), print1(ppp,", ")); ppp=pp; pp=p) \\ Hugo Pfoertner, Dec 24 2019

keyword:base removed, and sequence extended by R. J. Mathar, Apr 23 2010

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))}

cp's OEIS Frontend

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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A129109 Sums of three consecutive hexagonal numbers.

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Magma

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A129111 Sums of three consecutive heptagonal numbers.

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Magma

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A130704 Palindromic primes whose squares are the sum of three consecutive primes.

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Mathematica

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A135277 a(n) = prime(2n-1) + prime(2n) + prime(2n+1).

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Mathematica

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PARI

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A165981 Primes p which are equal to 2 plus the sum of three consecutive primes.

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Mathematica

Extensions

A167807 Square pyramidal numbers which are sums of three consecutive primes.

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