cp's OEIS Frontend

A371194 a(n) = smallest penholodigital prime in base n.

%I A371194 #22 Apr 04 2024 10:14:17
%S A371194 3,5,103,823,10061,157427,2439991,49100173,1123465789,31148488997,
%T A371194 816695154683,25401384476191,859466293047623,33373273595699879,
%U A371194 1234907033823334111,51892599148660469993,2322058300483667372689,115713970660820468376569,5533344265927977839343539
%N A371194 a(n) = smallest penholodigital prime in base n.
%C A371194 a(n) is the smallest prime whose base-n representation is zeroless and contains all nonzero digits (i.e., 1,...,n-1) at least once.
%H A371194 Chai Wah Wu, <a href="/A371194/b371194.txt">Table of n, a(n) for n = 2..387</a>
%H A371194 Chai Wah Wu, <a href="https://arxiv.org/abs/2403.20304">Pandigital and penholodigital numbers</a>, arXiv:2403.20304 [math.GM], 2024. See p. 3.
%F A371194 a(n) >= A023811(n).
%e A371194 The corresponding base-n representations are:
%e A371194 n  a(n) in base n
%e A371194 ------------------------
%e A371194 2  11
%e A371194 3  12
%e A371194 4  1213
%e A371194 5  11243
%e A371194 6  114325
%e A371194 7  1223654
%e A371194 8  11235467
%e A371194 9  112345687
%e A371194 10 1123465789
%e A371194 11 1223456789a
%e A371194 12 11234567a98b
%e A371194 13 112345678abc9
%e A371194 14 112345678cadb9
%e A371194 15 1223456789adcbe
%e A371194 16 1123456789abcedf
%e A371194 17 1123456789abdgfec
%e A371194 18 1123456789abcehfgd
%e A371194 19 1223456789abcdefghi
%e A371194 20 1123456789abcdefhigj
%e A371194 21 1123456789abcdefgihjk
%e A371194 22 1123456789abcdefgjhikl
%e A371194 23 1223456789abcdefghjimlk
%e A371194 24 1123456789abcdefghkmijln
%e A371194 25 1123456789abcdefghijklnom
%e A371194 26 1123456789abcdefghijkmnpol
%e A371194 27 1223456789abcdefghijklmqnop
%e A371194 28 1123456789abcdefghijklmnqorp
%e A371194 29 1123456789abcdefghijklmnrqspo
%e A371194 30 1123456789abcdefghijklmnosqprt
%e A371194 31 1223456789abcdefghijklmnoptusrq
%e A371194 32 1123456789abcdefghijklmnopqrvust
%e A371194 33 1123456789abcdefghijklmnopqsrtuvw
%e A371194 34 1123456789abcdefghijklmnopqrstuxwv
%e A371194 35 1223456789abcdefghijklmnopqrstuxwvy
%e A371194 36 1123456789abcdefghijklmnopqrstuwzyxv
%o A371194 (Python)
%o A371194 from math import gcd
%o A371194 from sympy import nextprime
%o A371194 from sympy.ntheory import digits
%o A371194 def A371194(n):
%o A371194     m, j = 1, 0
%o A371194     if n > 3:
%o A371194         for j in range(1,n):
%o A371194             if gcd((n*(n-1)>>1)+j,n-1) == 1:
%o A371194                  break
%o A371194     if j == 0:
%o A371194         for i in range(2,n):
%o A371194             m = n*m+i
%o A371194     elif j == 1:
%o A371194         for i in range(1,n):
%o A371194             m = n*m+i
%o A371194     else:
%o A371194         for i in range(2,1+j):
%o A371194             m = n*m+i
%o A371194         for i in range(j,n):
%o A371194             m = n*m+i
%o A371194     m -= 1
%o A371194     while True:
%o A371194         s = digits(m:=nextprime(m),n)[1:]
%o A371194         if 0 not in s and len(set(s))==n-1:
%o A371194             return m
%Y A371194 Cf. A023811, A185122.
%K A371194 nonn
%O A371194 2,1
%A A371194 _Chai Wah Wu_, Mar 14 2024