A076623 Total number of left truncatable primes (without zeros) in base n.

0, 3, 16, 15, 454, 22, 446, 108, 4260, 75, 170053, 100, 34393, 9357, 27982, 362, 14979714, 685, 3062899, 59131, 1599447, 1372, 1052029701, 10484, 7028048, 98336, 69058060, 3926

Offset: 2

Author

Martin Renner, Oct 22 2002, Nov 03 2002, Sep 24 2007, Feb 20 2008, Apr 20 2008

Keywords

• nonn
• base
• more

Comments

Approximation of a(b) by (PARI) code: l(b)=c=b*(b-1)/log(b)/eulerphi(b);\ return(floor((primepi(b)-omega(b))*exp(c)/c)); - Robert Gerbicz, Nov 02 2008

a(24) = 1052029701 based on strong BPSW pseudoprimes. Other terms up to a(29) use proved primes. - Martin Fuller, Nov 24 2008

Links

• I. O. Angell and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977.
• Michael S. Branicky, String-based Python Program
• Martin Fuller, Table of n, a(n) for n= 2..53, with question marks where unknown
• Hans Havermann, A076623 Decomposed
• Index entries for sequences related to truncatable primes

Crossrefs

Cf. A024779, A024780, A024781, A024782, A024783, A024784, A024785, A076586, A103443, A103463.

Programs

Maple

Lton := proc(L,b) add( op(i,L)*b^(i-1),i=1..nops(L)) ; end proc:
A076623rec := proc(L,b) local a,d,Lext,p ; a := 0 ; for d from 1 to b-1 do Lext := [op(L),d] ; p := Lton(Lext,b) ; if isprime(p) then a := a+1 ;  a := a+procname(Lext,b) ; end if; end do: a ;end proc:
A076623 := proc(b) A076623rec([],b) ; end proc:
for b from 2 do print(b,A076623(b)) ; end do: # R. J. Mathar, Jun 01 2011

PARI

f(b)=ct=0;A=[0];n=-1;L=1;while(L,n++;B=vector(L*b);M=0;\
for(i=1,L,for(j=1,b-1,x=A[i]+j*b^n;if(isprime[x],M++;B[M]=x;ct++)));\
L=M;A=vector(L,i,B[i]));return(ct) \\ Robert Gerbicz, Oct 31 2008

Python

# works for all n; link has faster string-based version for n < 37
from sympy import isprime, primerange
from sympy.ntheory.digits import digits
def fromdigits(digs, base):
    return sum(d*base**i for i, d in enumerate(digs))
def a(n):
    prime_lists, an = [(p,) for p in primerange(1, n)], 0
    while len(prime_lists) > 0:
        an += len(prime_lists)
        candidates = set(p+(d,) for p in prime_lists for d in range(1, n))
        prime_lists = [c for c in candidates if isprime(fromdigits(c, n))]
    return an
print([a(n) for n in range(2, 12)]) # Michael S. Branicky, Apr 27 2022

Extensions

a(12) corrected from 170051 to 170053 by Martin Fuller, Oct 31 2008

a(18) corrected by Robert Gerbicz, Nov 02 2008

a(24)-a(29) from Martin Fuller, Nov 24 2008

Entries in a-file corrected by N. J. A. Sloane, Jun 02 2011