A048058 -id:A048058 - cp's OEIS frontend

A184902 Primes that are not factors of m^2 + m + 11 (A048058).

2, 3, 5, 7, 19, 29, 37, 61, 71, 73, 89, 113, 131, 137, 149, 151, 157, 163, 179, 191, 199, 211, 223, 227, 233, 241, 257, 263, 277, 313, 331, 347, 349, 373, 383, 389, 409, 419, 421, 433, 449, 457, 463, 467, 491, 499, 503, 521, 523, 571, 577

Offset: 1

Zak Seidov, May 18 2011

The discriminant of this polynomial is -43. These are the primes that are not a square (mod 43). These primes are congruent to {2, 3, 5, 7, 8, 12, 18, 19, 20, 22, 26, 27, 28, 29, 30, 32, 33, 34, 37, 39, 42} (mod 43). - T. D. Noe, May 22 2011

Inert rational primes in the field Q(sqrt(-43)). - N. J. A. Sloane, Dec 25 2017

Cf. A048058 (n^2 + n + 11), A048059 (primes of the form n^2 + n + 11), A048097 (n^2 + n + 11 is prime).

Reap[Do[p = Prime[n]; ta = Table[Mod[m(m + 1) + 11, p],{m, 0, p/2 + 1}]; If[FreeQ[ta, 0], Sow[p]], {n, 1000}]][[2, 1]]
Select[Prime[Range[100]], JacobiSymbol[#, 43] == -1 &] (* T. D. Noe, May 22 2011 *)

a(n) ~ 2n log n. - Charles R Greathouse IV, May 22 2011

A176271 The odd numbers as a triangle read by rows.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131

Offset: 1

Reinhard Zumkeller, Apr 13 2010

A108309(n) = number of primes in n-th row.

			From _Philippe Deléham_, Oct 03 2011: (Start)
Triangle begins:
   1;
   3,  5;
   7,  9, 11;
  13, 15, 17, 19;
  21, 23, 25, 27, 29;
  31, 33, 35, 37, 39, 41;
  43, 45, 47, 49, 51, 53, 55;
  57, 59, 61, 63, 65, 67, 69, 71;
  73, 75, 77, 79, 81, 83, 85, 87, 89; (End)

G. C. Greubel, Rows n = 1..100 of the triangle, flattened
Eric Weisstein's World of Mathematics, Nicomachus's Theorem
Wikipedia, Nikomachos von Gerasa

Cf. A000466, A000537, A000578, A001105, A002061, A005408, A014209.
Cf. A016754, A027688, A027690, A027692, A027694, A028387, A048058.
Cf. A053755, A065599, A082111, A108195, A108309, A214604, A214661.

a176271 n k = a176271_tabl !! (n-1) !! (k-1)
a176271_row n = a176271_tabl !! (n-1)
a176271_tabl = f 1 a005408_list where
   f x ws = us : f (x + 1) vs where (us, vs) = splitAt x ws
-- Reinhard Zumkeller, May 24 2012

[n^2-n+2*k-1: k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 10 2024

A176271 := proc(n,k)
    n^2-n+2*k-1 ;
end proc: # R. J. Mathar, Jun 28 2013

Table[n^2-n+2*k-1, {n,15}, {k,n}]//Flatten (* G. C. Greubel, Mar 10 2024 *)

flatten([[n^2-n+2*k-1 for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Mar 10 2024

T(n, k) = n^2 - n + 2*k - 1 for 1 <= k <= n.
T(n, k) = A005408(n*(n-1)/2 + k - 1).
T(2*n-1, n) = A016754(n-1) (main diagonal).
T(2*n, n) = A000466(n).
T(2*n, n+1) = A053755(n).
T(n, k) + T(n, n-k+1) = A001105(n), 1 <= k <= n.
T(n, 1)   = A002061(n), central polygonal numbers.
T(n, 2)   = A027688(n-1) for n > 1.
T(n, 3)   = A027690(n-1) for n > 2.
T(n, 4)   = A027692(n-1) for n > 3.
T(n, 5)   = A027694(n-1) for n > 4.
T(n, 6)   = A048058(n-1) for n > 5.
T(n, n-3) = A108195(n-2) for n > 3.
T(n, n-2) = A082111(n-2) for n > 2.
T(n, n-1) = A014209(n-1) for n > 1.
T(n, n) = A028387(n-1).
Sum_{k=1..n} T(n, k) = A000578(n) (Nicomachus's theorem).
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^(n-1)*A065599(n) (alternating sign row sums).
Sum_{j=1..n} (Sum_{k=1..n} T(j, k)) = A000537(n) (sum of first n rows).

A048059 Primes of the form k^2 + k + 11.

Original entry on oeis.org

11, 13, 17, 23, 31, 41, 53, 67, 83, 101, 167, 193, 251, 283, 317, 353, 431, 563, 661, 823, 881, 941, 1201, 1493, 1571, 1733, 2081, 2267, 2663, 2767, 3203, 3433, 3671, 3793, 3917, 4567, 4703, 5413, 5711, 6173, 6491, 6653, 6983, 7151, 7321, 8753, 8941, 9323, 10111

Offset: 1

N. J. A. Sloane

From Peter Bala, Apr 15 2018: (Start)
The polynomial P(n) := n^2 + n + 11 takes distinct prime values for the 10 consecutive integers n = 0 to 9. It follows that the polynomial P(n-10) = (n - 10)^2 + (n - 10) + 11 takes prime values for the 20 consecutive integers n = 0 to 19, consisting of the 10 primes above each taken twice. We note two consequences of this fact.
1) The polynomial P(2*n-10) = 4*n^2 - 38*n + 101 also takes prime values for the 10 consecutive integers n = 0 to 9.
2)The polynomial P(3*n-10) = 9*n^2 - 57*n + 101 takes prime values for the 7 consecutive integers n = 0 to 6 (= floor(19/3)). In addition, calculation shows that P(3*n-10) also takes prime values for n from -3 to -1. Equivalently put, the polynomial P(3*n-19) = 9*n^2 - 111*n + 353 takes prime values for the 10 consecutive integers n = 0 to 9. Cf. A007635 and A005846. (End)

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.

Cf. A005846, A007635, A048058, A048097, A160548.

[ a: n in [0..200] | IsPrime(a) where a is n^2+n+11 ]; // Vincenzo Librandi, Dec 07 2011

lst={}; Do[p=n^2+n+11; If[PrimeQ[p], AppendTo[lst,p]], {n,0,5*5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 27 2009 *)
Select[Table[n^2+n+11, {n,0,600}], PrimeQ] (* Vincenzo Librandi, Dec 07 2011 *)

a(n) = A048058(A048097(n)). - Elmo R. Oliveira, Apr 20 2025

A048097 Numbers k such that k^2 + k + 11 is prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 15, 16, 17, 18, 20, 23, 25, 28, 29, 30, 34, 38, 39, 41, 45, 47, 51, 52, 56, 58, 60, 61, 62, 67, 68, 73, 75, 78, 80, 81, 83, 84, 85, 93, 94, 96, 100, 101, 103, 106, 108, 113, 114, 122, 123, 124, 125, 127, 130, 135

Offset: 1

N. J. A. Sloane

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

Cf. A048058, A048059.

[n: n in [0..150] |IsPrime(n^2+n+11)]; // Vincenzo Librandi, Aug 06 2010

Select[Range[0,150],PrimeQ[#^2+#+11]&] (* Harvey P. Dale, May 18 2011 *)

is(n)=isprime(n^2+n+11) \\ Charles R Greathouse IV, Feb 17 2017

A117530 Triangle read by rows: T(n,k) = k^2 - k + prime(n), 1<=k<=n.

Original entry on oeis.org

2, 3, 5, 5, 7, 11, 7, 9, 13, 19, 11, 13, 17, 23, 31, 13, 15, 19, 25, 33, 43, 17, 19, 23, 29, 37, 47, 59, 19, 21, 25, 31, 39, 49, 61, 75, 23, 25, 29, 35, 43, 53, 65, 79, 95, 29, 31, 35, 41, 49, 59, 71, 85, 101, 119, 31, 33, 37, 43, 51, 61, 73, 87, 103, 121, 141, 37, 39, 43, 49, 57

Offset: 1

Reinhard Zumkeller, Mar 25 2006

A117531 gives the number of primes in the n-th row;

if T(n,1) is a Lucky Number of Euler then A117531(n)=n, see A014556.

			T(5,k)=A048058(k)=A048059(k), 1<=k<=5: T(5,1)=A014556(4)=11;
T(7,k)=A007635(k), 1<=k<=7: T(7,1)=A014556(5)=17;
T(13,k)=A005846(k), 1<=k<=13: T(13,1)=A014556(6)=41.

Robert Price, Table of n, a(n) for n = 1..10011
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial
Eric Weisstein's World of Mathematics, Lucky Number of Euler

Cf. A005846, A007635, A014556, A048058, A048059, A117531.

Table[k^2 - k + Prime[n], {n, 141}, {k, n}] // Flatten (* Robert Price, Apr 19 2025 *)

T(n,k) = k^2 - k + prime(n) \\ Charles R Greathouse IV, Apr 24 2015

T(n,1) = A000040(k).
T(n,2) = A052147(k) for k>1.
For 1

A173752 a(n) = m-k where (prime(k), prime(m)) is the n-th prime pair (x^2-x+11, x^2+x+11), integer x >= 0.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 3, 4, 3, 5, 7, 5, 5, 9, 8, 10, 17, 15, 15, 16, 15, 17, 18, 20, 23, 27, 25, 30, 27, 26, 30, 32, 39, 43, 49, 48, 55, 54, 48, 64, 66, 62, 61, 62, 68, 63, 65, 77, 65, 73, 79, 85, 73, 86, 93, 98, 84, 100, 107, 113, 110, 105, 107, 121, 119, 120, 119, 121, 125, 114

Offset: 1

Juri-Stepan Gerasimov, Feb 23 2010

nonn

prime(j) is used here to refer to the j-th prime.

			The first prime pair (x^2-x+11, x^2+x+11) is obtained for x = 0: 0^2-0+11 = 11 and 0^2+0+11 = 11; 11 is the fifth prime, hence a(1) = 5-5 = 0.
The second prime pair is obtained for x = 1: 1^2-1+11 = 11 and 1^2+1+11 = 13; 11 is the fifth prime and 13 is the sixth prime, hence a(2) = 6-5 = 1.
The third prime pair is obtained for x = 2: 2^2-2+11 = 13 and 2^2+2+11 = 17; 13 is the sixth prime and 17 is the seventh prime, hence a(3) = 7-6 = 1.
The eleventh prime pair is obtained for x = 13: 13^2-13+11 = 167 and 13^2+13+11 = 193; 167 is prime(39) and 193 is prime(44), hence a(11) = 44-39 = 5.

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

Cf. A048058 (n^2+n+11), A048059 (primes of the form n^2+n+11), A048097 (n such that n^2+n+11 is prime), A175154 (prime index of A048059(n)).

PrimePi:=func< n | #PrimesUpTo(n) >; [ PrimePi(p)-PrimePi(q): x in [0..850] | IsPrime(p) and IsPrime(q) where p is x^2+x+11 where q is x^2-x+11 ]; // Klaus Brockhaus, Feb 27 2010

for x from 0 to 1000 do mp := x^2+x+11 ; kp := x^2-x+11 ; if isprime(mp) and isprime(kp) then m := numtheory[pi](mp) ; k := numtheory[pi](kp) ; printf("%d,",m-k) ; end if; end do : # R. J. Mathar, Mar 01 2010

pp[n_]:=Module[{c=n^2+11},If[AllTrue[c+{n,-n},PrimeQ],PrimePi[c+n]- PrimePi[ c-n],0]]; Join[{0},Array[pp,1000]/.(0->Nothing)] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 15 2017 *)

Edited and extended by Klaus Brockhaus, Feb 27 2010

a(15) corrected and sequence extended by R. J. Mathar, Mar 01 2010

A297786 Decimal expansion of 10980011/999^3.

Original entry on oeis.org

0, 1, 1, 0, 1, 3, 0, 1, 7, 0, 2, 3, 0, 3, 1, 0, 4, 1, 0, 5, 3, 0, 6, 7, 0, 8, 3, 1, 0, 1, 1, 2, 1, 1, 4, 3, 1, 6, 7, 1, 9, 3, 2, 2, 1, 2, 5, 1, 2, 8, 3, 3, 1, 7, 3, 5, 3, 3, 9, 1, 4, 3, 1, 4, 7, 3, 5, 1, 7, 5, 6, 3, 6, 1, 1, 6, 6, 1, 7, 1, 3, 7, 6, 7, 8, 2, 3

Offset: 0

Seiichi Manyama, Jan 06 2018

			0.011013017023031041053067083101121...

John D. Cook, Prime generating fractions

Cf. A048058, A048059, A272066, A297785, A297787.

Join[{0},RealDigits[10980011/999^3,10,120][[1]]] (* Harvey P. Dale, Aug 20 2021 *)

Sum_{k>=0} 10^(-3*k-3)*A048058(k) = 10980011/999^3.

A175154 a(n) = prime index of A048059(n).

Original entry on oeis.org

5, 6, 7, 9, 11, 13, 16, 19, 23, 26, 39, 44, 54, 61, 66, 71, 83, 103, 121, 143, 152, 160, 197, 238, 248, 270, 313, 336, 386, 403, 453, 481, 512, 527, 542, 619, 635, 714, 752, 804, 842, 857, 898, 915, 933, 1092, 1112, 1154, 1242, 1265, 1307, 1372, 1412, 1534, 1561

Offset: 1

Klaus Brockhaus, Feb 27 2010

nonn

Indices of primes of the form n^2+n+11.
PrimePi(A048059(n)), where PrimePi(k) is the number of primes <= k (A000720).
prime(a(n)) = A048059(n).

			A048059(7) = 53 is the seventh prime of the form n^2+n+11; 53 has prime index 16 (PrimePi(53) = 16, prime(16) = 53). Hence a(7) = 16.

Cf. A048058 (n^2+n+11), A048097 (n such that n^2+n+11 is prime), A048059 (primes of the form n^2+n+11), A000720 (PrimePi(n), number of primes <= n), A173752.

PrimePi:=func< n | #PrimesUpTo(n) >; [ PrimePi(p): x in [0..120] | IsPrime(p) where p is x^2+x+11 ];

a(n) = A000720(A048059(n)).

A214870 Natural numbers placed in table T(n,k) layer by layer. The order of placement: at the beginning filled odd places of layer clockwise, next - even places counterclockwise. T(n,k) read by antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 10, 9, 8, 13, 17, 16, 6, 14, 21, 26, 25, 11, 12, 22, 31, 37, 36, 18, 15, 20, 32, 43, 50, 49, 27, 24, 23, 30, 44, 57, 65, 64, 38, 35, 19, 33, 42, 58, 73, 82, 81, 51, 48, 28, 29, 45, 56, 74, 91, 101, 100, 66, 63, 39, 34, 41, 59, 72, 92, 111

Offset: 1

Boris Putievskiy, Mar 11 2013

Permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1).
Enumeration table T(n,k) layer by layer. The order of the list:
  T(1,1)=1;
  T(1,2), T(2,1), T(2,2);
  . . .
  T(1,n), T(3,n), ... T(n,3), T(n,1); T(n,2), T(n,4), ... T(4,n), T(2,n);
  . . .

			The start of the sequence as table:
   1   2   5  10  17  26 ...
   3   4   9  16  25  36 ...
   7   8   6  11  18  27 ...
  13  14  12  15  24  35 ...
  21  22  20  23  19  28 ...
  31  32  30  33  29  34 ...
  ...
The start of the sequence as triangle array read by rows:
   1;
   2,  3;
   5,  4,  7;
  10,  9,  8, 13;
  17, 16,  6, 14, 21;
  26, 25, 11, 12, 22, 31;
  ...

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A060734, A060736, A185725, A213921, A213922; table T(n,k) contains: in rows A002522, A000290, A059100, A005563, A117950, A008865, A087475, A028872, A117951, A028347, A114949, A028875, A117619, A028878, A189833, A028881, A189834, A028884, A114948, A028560, A189836; in columns A002061, A014206, A002378, A027688, A028387, A027689, A028552, A027690, A014209, A027691, A027692, A082111, A027693, A028557, A027694, A108195, A187710, A048058, A048840.

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
if i > j:
   result=i*i-i+(j%2)*(2-(j+1)/2)+((j+1)%2)*(j/2+1)
else:
   result=j*j-2*(i%2)*j + (i%2)*((i+1)/2+1) + ((i+1)%2)*(-i/2+1)

As table
T(n,k) = k*k-2*(n mod 2)*k+(n mod 2)*((n+1)/2+1)+((n+1) mod 2)*(-n/2+1), if n<=k;
T(n,k) = n*n-n+(k mod 2)*(2-(k+1)/2)+((k+1) mod 2)*(k/2+1), if n>k.
As linear sequence
a(n) = j*j-2*(i mod 2)*j+(i mod 2)*((i+1)/2+1)+((i+1) mod 2)*(-i/2+1), if i<=j;
a(n) = i*i-i+(j mod 2)*(2-(j+1)/2)+((j+1) mod 2)*(j/2+1), if i>j; where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).

A268101 Smallest prime p such that some polynomial of the form ax^2 - bx + p generates distinct primes in absolute value for x = 1 to n, where 0 < a < p and 0 <= b < p.

Original entry on oeis.org

2, 3, 5, 5, 7, 7, 11, 11, 11, 11, 13, 13, 17, 17, 17, 17, 19, 19, 23, 23, 23, 23, 29, 29, 29, 29, 29, 29, 31, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 41, 647, 1277, 1979, 2753

Offset: 1

Arkadiusz Wesolowski, Jan 26 2016

			a(1) = 2 (a prime), x^2 + 2 gives a prime for x = 1.
a(2) = 3 (a prime), 2*x^2 + 3 gives distinct primes for x = 1 to 2.
a(4) = 5 (a prime), 2*x^2 + 5 gives distinct primes for x = 1 to 4.
a(6) = 7 (a prime), 4*x^2 + 7 gives distinct primes for x = 1 to 6.
a(10) = 11 (a prime), 2*x^2 + 11 gives distinct primes for x = 1 to 10.
a(12) = 13 (a prime), 6*x^2 + 13 gives distinct primes for x = 1 to 12.
a(16) = 17 (a prime), 6*x^2 + 17 gives distinct primes for x = 1 to 16.
a(18) = 19 (a prime), 10*x^2 + 19 gives distinct primes for x = 1 to 18.
a(22) = 23 (a prime), 3*x^2 - 3*x + 23 gives distinct primes for x = 1 to 22.
a(28) = 29 (a prime), 2*x^2 + 29 gives distinct primes for x = 1 to 28.
a(29) = 31 (a prime), 2*x^2 - 4*x + 31 gives distinct primes for x = 1 to 29.
a(40) = 41 (a prime), x^2 - x + 41 gives distinct primes for x = 1 to 40.
a(41) = 647 (a prime), abs(36*x^2 - 594*x + 647) gives distinct primes for x = 1 to 41.
a(42) = 1277 (a prime), abs(36*x^2 - 666*x + 1277) gives distinct primes for x = 1 to 42.
a(43) = 1979 (a prime), abs(36*x^2 - 738*x + 1979) gives distinct primes for x = 1 to 43.
a(44) = 2753 (a prime), abs(36*x^2 - 810*x + 2753) gives distinct primes for x = 1 to 44.

Carlos Rivera, Problem 12
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial

Cf. A027688, A027753, A027690, A027755, A048058, A048059, A007635, A007639, A007637, A007641, A202018, A005846, A117081, A050268, A268109.

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cp's OEIS Frontend

A184902 Primes that are not factors of m^2 + m + 11 (A048058).

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A176271 The odd numbers as a triangle read by rows.

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A048059 Primes of the form k^2 + k + 11.

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A048097 Numbers k such that k^2 + k + 11 is prime.

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A117530 Triangle read by rows: T(n,k) = k^2 - k + prime(n), 1<=k<=n.

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A173752 a(n) = m-k where (prime(k), prime(m)) is the n-th prime pair (x^2-x+11, x^2+x+11), integer x >= 0.

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A297786 Decimal expansion of 10980011/999^3.

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A268101 Smallest prime p such that some polynomial of the form ax^2 - bx + p generates distinct primes in absolute value for x = 1 to n, where 0 < a < p and 0 <= b < p.