A090108 -id:A090108 - cp's OEIS frontend

A090109 Values of k such that {P(k), P(k+1), ..., P(k+10)} are all prime numbers, where P(k) = k^2 - 79*k + 1601.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 259, 260, 261

Offset: 1

Labos Elemer, Dec 29 2003

nonn

a(n) is the first argument providing 11 "polynomially consecutive" primes with respect to the polynomial x^2 - 79*x + 1601 described by Escott in 1899.

			k = 1 provides the following non-monotonic (!) chain of 11 "polynomially consecutive" primes as follows: {1523, 1447, 1373, 1301, 1231, 1163, 1097, 1033, 971, 911, 853}.

Amiram Eldar, Table of n, a(n) for n = 1..83
E. B. Escott, Réponse 1133, Formule d'Euler x^2 + x + 41 et formules analogues, L'Intermédiaire des mathématiciens, Vol. 6 (1899), pp. 10-11.

Cf. A055561, A090101, A090102, A090107, A090108, A090562, A090563.

Position[Times @@@ Partition[Table[Boole@PrimeQ[k^2 - 79*k + 1601], {k, 1, 1000}], 11, 1], 1] // Flatten (* Amiram Eldar, Sep 27 2024 *)

isp(x) = isprime(x^2 - 79*x + 1601);
lista(kmax) = {my(v = vector(11, k, isp(k))); for(k = 12, kmax, if(vecprod(v) == 1, print1(k - 11, ", ")); v = concat(vecextract(v, "^1"), isp(k)));} \\ Amiram Eldar, Sep 27 2024

Data corrected by Amiram Eldar, Sep 27 2024

A090110 Values of k such that {P(k), P(k+1), ..., P(k+7)} are all prime numbers, where P(k) = 4k^2 - 154k + 1523.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 66, 129, 130, 328, 1619, 7509, 29714, 45905, 447588, 509862, 1022565, 1102373, 1388125, 1665379, 1762387, 1786292, 2111602, 2962834, 3391838

Offset: 1

Labos Elemer, Dec 30 2003

nonn

The terms are arguments introducing a sequence of 8 polynomially consecutive primes with respect to 4*x^2 - 154*x + 1523, a polynomial communicated by Rivera (2003).

			k = 1 provides {1373, 1231, 1097, 971, 853, 743, 641, 547}, an 8-chain of primes.

Amiram Eldar, Table of n, a(n) for n = 1..1000
Carlos Rivera, Puzzle 232. Primes and Cubic polynomials, The Prime Puzzles and Problems Connection.

Cf. A090101, A090102, A090107, A090108, A090109, A055561, A090563.

okQ[x_] := And@@PrimeQ[Table[4n^2-154n+1523, {n,x,x+7}]];
Select[Range[ 510000], okQ] (* Harvey P. Dale, May 25 2011 *)

isp(x) = isprime(4*x^2 - 154*x + 1523);
lista(kmax) = {my(v = vector(8, k, isp(k))); for(k = 9, kmax, if(vecprod(v) == 1, print1(k - 8, ", ")); v = concat(vecextract(v, "^1"), isp(k)));} \\ Amiram Eldar, Sep 27 2024

a(43)-a(51) from Amiram Eldar, Sep 27 2024

A090111 Values of k such that {P(k), P(k+1), ..., P(k+6)} are all prime numbers, where P(k) = 4k^2 - 154k + 1523.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 45, 53, 66, 67, 84, 129, 130, 131, 266, 328, 329, 1619, 1620, 2655, 2937, 7509, 7510, 18030, 29283, 29714, 29715, 37630, 42037, 44473, 45905

Offset: 1

Labos Elemer, Dec 30 2003

nonn

The terms are arguments providing a sequence of 7 polynomially consecutive primes with respect to 4*x^2 - 154*x + 1523, a polynomial communicated by Rivera (2003).

			k = 1 provides {1373, 1231, 1097, 971, 853, 743, 641}, a 7-chain of primes.

Amiram Eldar, Table of n, a(n) for n = 1..6400 (terms 1..200 from Harvey P. Dale)
Carlos Rivera, Puzzle 232. Primes and Cubic polynomials, The Prime Puzzles and Problems Connection.

Cf. A090101, A090102, A090107, A090108, A090109, A090110, A055561, A090563.

Flatten[Position[Partition[Table[If[PrimeQ[4n^2-154n+1523],1,0],{n,46000}],7,1],{1,1,1,1,1,1,1}]] (* Harvey P. Dale, Mar 06 2015 *)

isp(x) = isprime(4*x^2 - 154*x + 1523);
lista(kmax) = {my(v = vector(7, k, isp(k))); for(k = 8, kmax, if(vecprod(v) == 1, print1(k - 7, ", ")); v = concat(vecextract(v, "^1"), isp(k)));} \\ Amiram Eldar, Sep 27 2024

A373300 Sum of successive integers in a row of length p(n) where p counts integer partitions.

Original entry on oeis.org

1, 5, 15, 45, 105, 264, 555, 1221, 2445, 4935, 9324, 17941, 32522, 59400, 104808, 184569, 315711, 540540, 902335, 1504800, 2462724, 4014513, 6444425, 10316250, 16283707, 25610886, 39841865, 61720659, 94687230, 144731706, 219282679, 330996105, 495901413, 740046425

Offset: 1

Olivier Gérard, May 31 2024

nonn

The length of each row is given by A000041.
As many sequences start like the positive integers, their row sums when disposed in this shape start with the same values.
Here is a sample list by A-number order of the sequences which are sufficiently close to A000027 to have the same row sums for at least 8 terms.
A069782, A088480, A090107, A090108, A090109, A115510, A115511, A130446, A131717, A132086, A153671, A167904, A296879, A296882, A296885, A296888, A296891, A296894, A296897, A296900, A296903, A296906, A303502, A317945, A335280, A361374.

			Let's put the list of integers in a triangle whose rows have length p(n), number of integer partitions of n.
.
    1 |  1
    5 |  2  3
   15 |  4  5  6
   45 |  7  8  9 10 11
  105 | 12 13 14 15 16 17 18
  264 | 19 20 21 22 23 24 25 26 27 28 29
  555 | 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
.
The sequence gives the row sums of this triangle.

Cf. A000027, seen as a triangle with shape A000041.
Cf. A373301, the same principle, but starting from integer zero instead of 1.
Cf. A006003, row sums of the integers but for the linear triangle.

Module[{s = 0},
 Table[s +=
   PartitionsP[n - 1]; (s + PartitionsP[n])*(s + PartitionsP[n] - 1)/2 -
   s*(s - 1)/2, {n, 1, 30}]]

cp's OEIS Frontend

A090109 Values of k such that {P(k), P(k+1), ..., P(k+10)} are all prime numbers, where P(k) = k^2 - 79*k + 1601.

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A090110 Values of k such that {P(k), P(k+1), ..., P(k+7)} are all prime numbers, where P(k) = 4*k^2 - 154*k + 1523.

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A090111 Values of k such that {P(k), P(k+1), ..., P(k+6)} are all prime numbers, where P(k) = 4*k^2 - 154*k + 1523.

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A373300 Sum of successive integers in a row of length p(n) where p counts integer partitions.

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Author

Keywords

Comments

Examples

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Programs

Mathematica

A090110 Values of k such that {P(k), P(k+1), ..., P(k+7)} are all prime numbers, where P(k) = 4k^2 - 154k + 1523.

A090111 Values of k such that {P(k), P(k+1), ..., P(k+6)} are all prime numbers, where P(k) = 4k^2 - 154k + 1523.