A109938 - OEIS

%I #18 Jul 08 2021 00:46:40

%S 7,7,0,0,89,79,0,0,47,59,0,0,83,0,0,0,0,0,0,0,0,0,0,0,0,809,619,857,0,

%T 227,509,787,823,557,0,907,0,653,0,347,359,0,383,773,0,797,0,0,0,0,

%U 467,479,0,503,0,0,0,0,0,563,0,587,0,0,0,0,0,0,0,0,0,719,0,0,0,0,0,0,0,0

%N Largest k-digit prime == 1 (mod prime(n)) where k is the number of digits in prime(n), or 0 if no such prime exists.

%H Michael S. Branicky, <a href="/A109938/b109938.txt">Table of n, a(n) for n = 1..10000</a>

%e a(13) = 83 as prime(13) = 41 and 83 == 1 (mod 41). 83 is the largest such two-digit prime.

%p A055642 := proc(n) max(1, ilog10(n)+1) ; end: A109938 := proc(n) local p,k,a; p := ithprime(n) ; k := A055642(p) ; a := 0; q := numtheory[pi](10^(k-1)) ; q := ithprime(q+1) ; while A055642(q) < k+1 do if q mod p = 1 and q > a then a := q ; fi ; q := nextprime(q) ; od ; RETURN(a) ; end: seq(A109938(n),n=1..90) ; # _R. J. Mathar_, Aug 17 2007

%o (Python)

%o from sympy import prime, prevprime

%o def a(n):

%o     pn = prime(n); k = len(str(pn))

%o     p = prevprime(10**k); lb = max(10**(k-1), 2)

%o     while p > lb and p%pn != 1: p = prevprime(p)

%o     return p if p > lb else 0

%o print([a(n) for n in range(1, 81)]) # _Michael S. Branicky_, Jul 07 2021

%o (Python) # faster version for initial segment of sequence

%o from sympy import prime, primerange

%o def aupto(limit):

%o     alst, primeswithkdigs, plimit = [], dict(), prime(limit)

%o     for k in range(1, len(str(plimit))+1):

%o         primeswithkdigs[k] = list(primerange(10**(k-1), 10**k))[::-1]

%o     for pn in primerange(1, plimit+1):

%o         k, found = len(str(pn)), False

%o         for pk in primeswithkdigs[k]:

%o             if pk%pn == 1: alst.append(pk); found = True; break

%o         if not found: alst.append(0)

%o     return alst

%o print(aupto(80)) # _Michael S. Branicky_, Jul 07 2021

%Y Cf. A109939.

%K base,nonn

%O 1,1

%A _Amarnath Murthy_, Jul 19 2005

%E More terms from _R. J. Mathar_, Aug 17 2007