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A339188
Highly insulated primes (see Comments for definition).
2
23, 53, 89, 211, 293, 409, 479, 631, 797, 839, 919, 1039, 1259, 1409, 1471, 1511, 1637, 1709, 1847, 1889, 2039, 2099, 2179, 2503, 2579, 2633, 2777, 2819, 2939, 3011, 3049, 3137, 3229, 3271, 3433, 3499, 3593, 3659, 3709, 3779, 3967, 4111, 4177, 4253, 4327, 4409, 4493, 4621, 4703, 4831
OFFSET
1,1
COMMENTS
Let degree of insulation D(p) for a prime p be defined as the largest m such that the prime between p-m and p+m is p only. Then the n-th insulated prime is said to be highly insulated if and only if D(A339148(n)) > D(A339148(n+1)) and D(A339148(n)) > D(A339148(n-1)).
LINKS
Abhimanyu Kumar and Anuraag Saxena, Insulated primes, arXiv:2011.14210 [math.NT], 2020. See also Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 602-612. See p. 610.
EXAMPLE
For the triplet (13,23,37) of insulated primes, the values of degree of insulation are D(13)=2, D(23)=4, and D(37)=3. Hence, 23 is the highly insulated prime.
MATHEMATICA
Block[{s = {0}~Join~Array[Min[NextPrime[# + 1] - # - 1, # - NextPrime[# - 1, -1]] &@ Prime@ # &, 660, 2], t}, t = Array[If[#1 < #2 > #3, #4, Nothing] & @@ Append[s[[# - 1 ;; # + 1]], #] &, Length@ s - 2, 2]; Array[If[s[[#1]] < s[[#2]] > s[[#3]], #4, Nothing] & @@ Append[t[[# - 1 ;; # + 1]], Prime@ t[[#]]] &, Length@ t - 2, 2] ] (* Michael De Vlieger, Dec 11 2020 *)
PROG
(PARI)
A339188(n) = { \\ Return the list of the first n highly insulated primes
my( HighInsulated=List([]), D(p)=min(nextprime(p+1)-p-1, p-precprime(p-1)); );
my( Dpred_ins=D(7), Pcur_ins=13, Dcur_ins=D(Pcur_ins) );
local( Dpred=D(Pcur_ins), p=nextprime(Pcur_ins+1), Dp=D(p), Pnext=nextprime(p+1), Dnext=D(Pnext) );
my(SearchNextInsulated() =
until(Dp > max(Dpred, Dnext),
Dpred = Dp; p = Pnext; Dp = Dnext;
Pnext = nextprime(p+1); Dnext = D(Pnext);
);
\\ At this point p is the first insulated prime > Dcur_ins
);
while(#HighInsulated<n,
until(Dcur_ins > max(Dpred_ins, Dp),
Dpred_ins = Dcur_ins; Pcur_ins = p; Dcur_ins = Dp;
SearchNextInsulated();
);
listput(HighInsulated, Pcur_ins);
);
return(HighInsulated);
} \\ François Marques, Dec 01 2020
CROSSREFS
Cf. A000040, A339148 (insulated primes).
Sequence in context: A128473 A132235 A277993 * A051650 A049438 A078854
KEYWORD
nonn
AUTHOR
Abhimanyu Kumar, Nov 27 2020
STATUS
approved