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A186995 Smallest weak prime in base n.

%I A186995 #50 Feb 16 2025 08:33:14
%S A186995 127,2,373,83,28151,223,6211,2789,294001,3347,20837899,4751,6588721,
%T A186995 484439,862789,10513,2078920243,10909,169402249,2823167,267895961,
%U A186995 68543,1016960933671,181141,121660507,6139219,11646280537,488651
%N A186995 Smallest weak prime in base n.
%C A186995 In base b, a prime is said to be weakly prime if changing any digit produces only composite numbers. Tao proves that in any fixed base there are an infinite number of weakly primes.
%C A186995 In particular, changing the leading digit to 0 must produce a composite number. These are also called weak primes, weakly primes, and isolated primes. - _N. J. A. Sloane_, May 06 2019
%C A186995 a(24) > 10^11. - _Jon E. Schoenfield_, May 06 2019
%C A186995 a(30) > 2*10^12. - _Giovanni Resta_, Jun 17 2019
%C A186995 a(30) > 10^13. - _Dana Jacobsen_, Mar 25 2023
%C A186995 a(n) appears to be relatively smaller for n odd than for n even. For instance, a(31) = 356479, a(33) = 399946711, a(35) = 22549349, a(37) = 8371249. a(n) for n of the form 6k+3 appear to be relatively larger than a(n) for other odd n. a(n) for n of the form 6k appear to be relatively larger than a(n) for other even n. - _Chai Wah Wu_, Mar 24 2024
%H A186995 Terence Tao, <a href="https://arxiv.org/abs/0802.3361">A remark on primality testing and decimal expansions</a>, arXiv:0802.3361 [math.NT], 2008.
%H A186995 Terence Tao, <a href="https://doi.org/10.1017/S1446788712000043">A remark on primality testing and decimal expansions</a>, Journal of the Australian Mathematical Society 91:3 (2011), pp. 405-413.
%H A186995 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WeaklyPrime.html">Weakly Prime</a>
%t A186995 isWeak[n_, base_] := Module[{d, e, weak, num}, d = IntegerDigits[n, base]; weak = True; Do[e = d; e[[i]] = j; num = FromDigits[e, base]; If[num != n && PrimeQ[num], weak = False; Break[]], {i, Length[d]}, {j, 0, base - 1}]; weak]; Table[p = 2; While[! isWeak[p, n], p = NextPrime[p]]; p, {n, 2, 16}]
%o A186995 (Python)
%o A186995 from itertools import count
%o A186995 from sympy import isprime
%o A186995 from sympy.ntheory.digits import digits
%o A186995 def fromdigits(d, b):
%o A186995     n = 0
%o A186995     for di in d: n *= b; n += di
%o A186995     return n
%o A186995 def h1(n, b): # hamming distance 1 neighbors of n in base b
%o A186995     d = digits(n, b)[1:]; L = len(d)
%o A186995     yield from (fromdigits(d[:i]+[c]+d[i+1:], b) for c in range(b) for i in range(L) if c!=d[i])
%o A186995 def ok(n, b): return isprime(n) and all(not isprime(k) for k in h1(n, b))
%o A186995 def a(n): return next(k for k in count(2) if ok(k, n))
%o A186995 print([a(n) for n in range(2, 12)]) # _Michael S. Branicky_, Jul 31 2022
%o A186995 (Python)
%o A186995 from sympy import isprime, nextprime
%o A186995 from sympy.ntheory import digits
%o A186995 def A186995(n):
%o A186995     p = 2
%o A186995     while True:
%o A186995         s = digits(p,n)[1:]
%o A186995         l = len(s)
%o A186995         for i,j in enumerate(s[::-1]):
%o A186995             m = n**i
%o A186995             for k in range(n):
%o A186995                 if k!=j and isprime(p+(k-j)*m):
%o A186995                     break
%o A186995             else:
%o A186995                 continue
%o A186995             break
%o A186995         else:
%o A186995             return p
%o A186995         p = nextprime(p) # _Chai Wah Wu_, Mar 24 2024
%Y A186995 Cf. A050249 (base 10), A137985 (base 2).
%K A186995 nonn,base,more
%O A186995 2,1
%A A186995 _T. D. Noe_, Mar 01 2011
%E A186995 a(17)-a(23) from _Terentyev Oleg_, Sep 04 2011
%E A186995 a(24)-a(29) from _Giovanni Resta_, Jun 17 2019