A109938 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.

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

Offset

1,1

Links

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Example

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

Maple

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

Prog

(Python)
from sympy import prime, prevprime
def a(n):
    pn = prime(n); k = len(str(pn))
    p = prevprime(10**k); lb = max(10**(k-1), 2)
    while p > lb and p%pn != 1: p = prevprime(p)
    return p if p > lb else 0
print([a(n) for n in range(1, 81)]) # Michael S. Branicky, Jul 07 2021
(Python) # faster version for initial segment of sequence
from sympy import prime, primerange
def aupto(limit):
    alst, primeswithkdigs, plimit = [], dict(), prime(limit)
    for k in range(1, len(str(plimit))+1):
        primeswithkdigs[k] = list(primerange(10**(k-1), 10**k))[::-1]
    for pn in primerange(1, plimit+1):
        k, found = len(str(pn)), False
        for pk in primeswithkdigs[k]:
            if pk%pn == 1: alst.append(pk); found = True; break
        if not found: alst.append(0)
    return alst
print(aupto(80)) # Michael S. Branicky, Jul 07 2021

Crossrefs

Cf. A109939.

Sequence in context: A105167 A217227 A217346 * A372609 A215735 A344382

Adjacent sequences: A109935 A109936 A109937 * A109939 A109940 A109941

Keyword

base,nonn

Author

Amarnath Murthy, Jul 19 2005

Extensions

More terms from R. J. Mathar, Aug 17 2007

Status

approved