A282687
a(n) = strictly increasing number m, such that m+n is the next prime and m-n is the previous prime.
Original entry on oeis.org
4, 5, 26, 93, 144, 157, 300, 1839, 1922, 3099, 3240, 4189, 5544, 5967, 6506, 10815, 11760, 12871, 30612, 33267, 35002, 36411, 81486, 86653, 95676, 103263, 106060, 153219, 181332, 189097, 190440, 288615, 294596, 326403, 399318, 507253, 515004, 570291, 642320
Offset: 1
For n = 5, a(5) = 144, because the next prime after 144 is 149 and the previous prime before 144 is 139, where both have an equal distance of 5 from 144.
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a = {}; Do[If[n == 1, k = 1, k = Max@ a + 1]; While[Nand[k - n == NextPrime[k, -1], k + n == NextPrime@ k], k++]; AppendTo[a, k], {n, 41}]; a (* Michael De Vlieger, Feb 20 2017 *)
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use ntheory qw(:all);
for (my ($n, $k) = (1, 1) ; ; ++$n) {
my $p = prev_prime($n) || next;
my $q = next_prime($n);
if ($n-$p == $k and $q-$n == $k) {
printf("%s %s\n", $k++, $n);
}
}
A306475
Smallest nonprime number <= 10^n (n>=1) with maximum distance from a prime.
Original entry on oeis.org
9, 93, 897, 9569, 31433, 492170, 4652430, 47326803, 436273150, 4302407536, 42652618575, 738832928197, 7177162612050, 90874329411895, 218209405436996, 1693182318746937, 80873624627235459, 804212830686678390
Offset: 1
For n=1: first prime numbers are 2, 3, 5, 7 and 11. Maximum difference between two consecutive primes is 4 between 7 and 11 thus a(1)=9.
For n=4: maximum difference between two primes less than 10^4 is 36, which occurs once: between 9551 and 9587. a(4)=(9551 + 9587)/2 = 9569.
A353089
Least number which differs from both of its prime neighbors by n^2, and -1 if no such number exists.
Original entry on oeis.org
4, 93, 532, 5607, 31932, 31433, 604122, 3851523, 39175298, 378044079, 367876650, 383204683, 22076314482
Offset: 1
a(1) = 4, because 3 and 5 are the prime neighbors of 4, and 5 - 4 = 4 - 3 = 1 = 1^2 and no number less than 4 differs from both of its prime neighbors by 1^2.
a(2) = 93, because 97 and 89 are the prime neighbors of 93, and 97 - 93 = 93 - 89 = 4 = 2^2 and no number less than 93 differs from both of its prime neighbors by 2^2.
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a[n_] := a[n] = Module[{diff, diff2, p, q, r},
{diff, diff2, p} = {n*n, 2*n*n, NextPrime[1 + n^2]};
q = NextPrime[p];
r = NextPrime[q];
While[!(q - p == diff2 || (q - p == diff && r - q == diff)),
{p, q, r} = {q, r, NextPrime[r]}];
Return[If[q - p == diff2, Floor[(q + p)/2], q]]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 10}] (* Jean-François Alcover, Jun 07 2022, after Michael S. Branicky's code *)
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a(n) = my(k=2); while (((nextprime(k+1)-k) != n^2) || ((k-precprime(k-1)) != n^2), k++); k; \\ Michel Marcus, Jul 10 2022
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from sympy import nextprime
def a(n):
diff, diff2, p = n*n, 2*n*n, nextprime(1+n**2)
q = nextprime(p)
r = nextprime(q)
while not (q-p == diff2 or (q-p == diff and r-q == diff)):
p, q, r = q, r, nextprime(r)
return (q+p)//2 if q-p == diff2 else q
print([a(n) for n in range(1, 9)]) # Michael S. Branicky, Apr 22 2022
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