A090116 a(n)=x is the least number such that x^2 is "surrounded" by two closest primes, prevprime(x^2) and nextprime(x^2), whose difference nextprime - prevprime = 2n.

2, 3, 5, 19, 12, 25, 11, 44, 23, 30, 57, 41, 50, 102, 76, 104, 100, 149, 175, 159, 348, 276, 305, 397, 461, 189, 345, 1059, 437, 820, 833, 1002, 509, 1283, 822, 1099, 729, 1090, 693, 2710, 1110, 1284, 3563, 1823, 1370, 4332, 3771, 1380, 4394, 2160, 2011, 1498

Offset: 1

Author

Labos Elemer, Jan 09 2004

Keywords

• nonn

Comments

a(14) > 16*10^6. - David A. Corneth, Jun 12 2017

Examples

			n=5: a(5)=12 because the primes closest to 12^2 = 144 are {139,149} whose difference 149 - 139 = 10 = 2n and 144 is the smallest square with this property;
n=1: a(1)=2 because 2^2=4 is surrounded by primes {3,5} with difference 5 - 3 = 2 = 2n.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..213
• David A. Corneth, Table of n, a(n) for n = 1..302 where a(n) <= (about) 16*10^6

Crossrefs

Cf. A090117, A090118, A090119.

Programs

Mathematica

de[x_] := Prime[PrimePi[x]+1]-Prime[PrimePi[x]]; de[1] = 0; t=Table[de[w^2], {w, 1, 50000}]; mt=Table[Min[Flatten[Position[t, 2*j]]], {j, 1, 100}]

PARI

first(n) = my(todo = n, res = vector(n), p, x = 2); while(todo > 0, m = nextprime(x^2) - precprime(x^2); if(m <= 2*n, if(res[m/2]==0, res[m/2] = x; todo--)); x++); res \\ David A. Corneth, Jun 12 2017