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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
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
Sequence in context: A338266 A019158 A367865 * A366141 A049479 A125314
KEYWORD
nonn,tabf
AUTHOR
Labos Elemer, Aug 08 2003
STATUS
approved

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Last modified March 28 12:59 EDT 2024. Contains 371254 sequences. (Running on oeis4.)