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