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A065585 Smallest prime beginning with exactly n 2's.

%I A065585 #23 Mar 03 2021 02:55:51
%S A065585 3,2,223,2221,22229,2222203,22222253,22222223,222222227,22222222273,
%T A065585 22222222223,2222222222243,22222222222201,22222222222229,
%U A065585 222222222222227,222222222222222043,222222222222222281,222222222222222221,22222222222222222253,222222222222222222277
%N A065585 Smallest prime beginning with exactly n 2's.
%H A065585 M. F. Hasler, <a href="/A065585/b065585.txt">Table of n, a(n) for n = 0..200</a>
%t A065585 Do[a = Table[2, {n}]; k = 0; While[b = FromDigits[ Join[a, IntegerDigits[k] ]]; First[ IntegerDigits[k]] == 2 || !PrimeQ[b], k++ ]; Print[b], {n, 1, 17} ]
%o A065585 (PARI) A065585(n)={n=10^n\9*2; n>2&for(d=1, 9e9, n*=10; for(t=1, 10^d-1, t\10^(d-1)==2 & t+= 10^(d-1)+(t>2); ispseudoprime(n+t) & return(n+t))); 2+!n} \\ _M. F. Hasler_, Oct 17 2012
%o A065585 (Python)
%o A065585 from sympy import isprime
%o A065585 def a(n):
%o A065585   if n < 2: return list([3, 2])[n]
%o A065585   n2s, i, pow10, end_digits = int('2'*n), 1, 1, 0
%o A065585   while True:
%o A065585     i = 1
%o A065585     while i < pow10:
%o A065585       istr = str(i)
%o A065585       if istr[0] == '2' and len(istr) == end_digits:
%o A065585         i += pow10 // 10
%o A065585       else:
%o A065585         t = n2s * pow10 + i
%o A065585         if isprime(t): return t
%o A065585         i += 2
%o A065585     pow10 *= 10; end_digits += 1
%o A065585 print([a(n) for n in range(20)]) # _Michael S. Branicky_, Mar 02 2021
%Y A065585 Cf. A037057, A065584 - A065592.
%Y A065585 A068103 is a lower bound, but most often equality holds. - _M. F. Hasler_, Oct 17 2012
%K A065585 nonn,base
%O A065585 0,1
%A A065585 _Robert G. Wilson v_, Nov 28 2001
%E A065585 Corrected by _Don Reble_, Jan 17 2007