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A086153 Special prime numbers arranged in a triangle: n-th row contains m primes p (where m = pi(2n + A020483(n)) - pi(A020483(n))) with following properties. 2
3, 7, 3, 23, 5, 89, 23, 3, 139, 19, 7, 3, 199, 47, 17, 5, 113, 83, 23, 17, 3, 1831, 211, 43, 13, 7, 3, 523, 109, 79, 19, 11, 5, 887, 317, 107, 47, 17, 11, 3, 1129, 619, 109, 79, 19, 7, 1669, 199, 113, 73, 43, 13, 5, 2477, 1373, 197, 113, 71, 41, 11, 3, 2971, 1123, 199, 109 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

1: q = p + 2n is also a prime, although not necessarily the next after p;

2: the k-th position of the n-th row gives is a prime p such that the number of further primes between p and q = p + 2n (not counting p and q) is k-1;

3: the primes p are the smallest with these properties.

Thus each row only contains primes. The first term in the n-th row is A000230(n). The last one in the same row is A020483(n). The length of the n-th row is pi(2n + A020483(n)) - pi(A020483(n)).

From Martin Raab, Aug 29 2021: (Start)

T(n,k) is zero if there is no admissible pattern with k+1 primes for the interval of length 2n under the given properties.

T(38,16) > 2^48. It requires a pattern of 17 primes with a difference of 76 between the first and the last prime. Admissible patterns of this kind exist, but solutions with 17 primes are rather hard to find. (End)

LINKS

Martin Raab, Rows n = 1..37, flattened

Martin Raab, The first 370 rows of the table arranged as a triangle, including unknown terms.

EXAMPLE

The table begins as follows:

    3;

    7,  3;

   23,  5;

   89, 23,  3;

  139, 19,  7,  3;

  199, 47, 17,  5;

  113, 83, 23, 17,  3;

  ...

For example, suppose n = 50: d = 2n = 100; the 50th row consists of 25 terms as follows: {396733, 58789, 142993, 38461, 37699, 7351, 5881, 1327, 2557, 1879, 1621, 1117, 463, 457, 283, 331, 211, 127, 73, 67, 31, ?, ?, 7, 3};

A000230(50)=396733, A020483(50)=3; between 143093 and 142993 two primes {143053,143063} occur because 142993 is the 3rd (from 2+1) entry in the 50th row.

The length of 50th row is pi(100+3) - pi(3) = pi(103) - pi(3) = 27 - 2 = 25, number of primes between 103 and 3 is 24 (not counting 103 and 3).

MATHEMATICA

(* Program to generate the 19th row *) cp[x_, y_] := Count[Table[PrimeQ[i], {i, x, y}], True] {d=38, k=0, mxc=Ceiling[d/3]; vg=PrimePi[30593]} t=Table[0, {mxc}]; t1=Table[0, {mxc}]; Do[s=cp[1+Prime[n], Prime[n]+d-1]; np=d+Prime[n]; If[PrimeQ[np]&&s<(1+mxc)&&t[[s+1]]==0, t[[s+1]]=n; t1[[s+1]]=Prime[n]], {n, 1, 5000}]; {t, t1}

PROG

(PARI) {z=concat(vector(13), binary(8683781)); for(n=1, 37, p1=3; while(!isprime(p1+2*n), p1=nextprime(p1+2)); p2=p1+2*n; k=primepi(p2)-primepi(p1); r=vector(k); r[k]=p1; i1=1; i2=0; s=vecsort(r); while(s[1+z[n]]==0, while(i1*i2==0, p1+=2; p2+=2; i2=isprime(p2); k=k-i1+i2; i1=isprime(p1)); if(!r[k], r[k]=p1; s=vecsort(r)); i2=0); print("row "n": "r))} \\ Martin Raab, Oct 21 2021

CROSSREFS

Cf. A000720, A000230, A020483, A023190, A086155, A086138-A086149, A086155.

Sequence in context: A282160 A338266 A019158 * A049479 A125314 A213244

Adjacent sequences:  A086150 A086151 A086152 * A086154 A086155 A086156

KEYWORD

nonn,tabf

AUTHOR

Labos Elemer, Aug 08 2003

STATUS

approved

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Last modified January 17 17:47 EST 2022. Contains 350402 sequences. (Running on oeis4.)