

A086153


Special prime numbers arranged in a triangle: nth 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
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OFFSET

1,1


COMMENTS

1: q = p + 2n is also a prime, although not necessarily the next after p;
2: the kth position of the nth 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 k1;
3: the primes p are the smallest with these properties.
Thus each row only contains primes. The first term in the nth row is A000230(n). The last one in the same row is A020483(n). The length of the nth 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]+d1]; 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=ki1+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, A086138A086149, 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



