A076623 -id:A076623 - cp's OEIS frontend

A076586 Total number of right truncatable primes in base n.

0, 4, 7, 14, 36, 19, 68, 68, 83, 89, 179, 176, 439, 373, 414, 473, 839, 1010, 1577, 2271, 2848, 1762, 3376, 5913, 6795, 6352, 10319, 5866, 14639, 13303, 19439, 29982, 38956, 39323, 58857, 41646, 68371, 80754, 128859, 81453, 175734, 161438, 228543, 396274, 538797

Offset: 2

Martin Renner, Oct 20 2002, Sep 24 2007

Seth A. Troisi, Table of n, a(n) for n = 2..100 (terms n=2..53 from Martin Renner)
I. O. Angell and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977.
Index entries for sequences related to truncatable primes

Cf. A024763, A024764, A024765, A024766, A024767, A024768, A024769, A024770, A076623.

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
    digits = 1
    while len(prime_lists) > 0:
        an += len(prime_lists)
        candidates = set((d,)+p for p in prime_lists for d in range(1, n))
        prime_lists = [c for c in candidates if isprime(fromdigits(c, n))]
        digits += 1
    return an
print([a(n) for n in range(2, 27)]) # Michael S. Branicky, May 03 2022

A103443 Largest left-truncatable prime in base n (decimal expansion).

Original entry on oeis.org

23, 4091, 7817, 4836525320399, 817337, 14005650767869, 1676456897, 357686312646216567629137, 2276005673, 13092430647736190817303130065827539, 812751503, 615419590422100474355767356763

Offset: 3

Martin Renner, Mar 21 2005, Sep 24 2007, Apr 20 2008

Martin Fuller, Table of n, a(n) for n = 3..29
I. O. Angell, and H. J. Godwin, On Truncatable Primes Math. Comput. 31, 265-267, 1977. (See also Sloane link below)
James Grime and Brady Haran, 357686312646216567629137, Numberphile video (2018).
Martin Renner, Table of n, a(n) for n = 3..53 (with some question marks). Corrected and expanded by Hans Havermann, Jan 25 2014.
N. J. A. Sloane, The Angell-Godwin table of initial terms of A023107 and A103443
Eric Weisstein's World of Mathematics, Truncatable Prime.
Index entries for sequences related to truncatable primes

Cf. A076623, A023107, A103463, A254755.

a(n)=my(v=primes(primepi(n-1)),u,t,b=1,best); while(#v, best=vecmax(v); b*=n; u=List(); for(i=1,#v,for(k=1,n-1,if(isprime(t=v[i]+k*b), listput(u,t)))); v=Vec(u)); best \\ Charles R Greathouse IV, Feb 05 2013

Base-14 entry corrected by Hans Havermann, May 30 2011
Corresponding entry in a-file corrected by N. J. A. Sloane, Jun 02 2011
a-file corrected and expanded by Hans Havermann, Jan 25 2014

A323137 Largest prime that is both left-truncatable and right-truncatable in base n.

Original entry on oeis.org

23, 11, 67, 839, 37, 1867, 173, 739397, 79, 105691, 379, 37573, 647, 3389, 631, 202715129, 211, 155863, 1283, 787817, 439, 109893629, 577, 4195880189, 1811, 14474071, 379, 21335388527, 2203, 1043557, 2939, 42741029, 2767, 50764713107, 853, 65467229, 4409, 8524002457

Offset: 3

Felix Fröhlich, Jan 05 2019

			For n = 12: 105691 is 511B7 in base 12. Successively removing the leftmost digit yields the base-12 numbers 11B7, 1B7, B7 and 7. When converted to base 10, these are 2011, 283, 139 and 7, respectively, all primes. Successively removing the rightmost digit yields the base-12 numbers 511B, 511, 51 and 5. When converted to base 10, these are 8807, 733, 61 and 5, respectively, all primes. Since no larger prime with this property in base 12 exists (as proven by Daniel Suteu), a(12) = 105691.

Daniel Suteu, Table of n, a(n) for n = 3..64
Wikipedia, Truncatable prime

Cf. A020994, A023107, A076586, A076623, A103443, A323390, A323396.

digitsToNum(d, base) = sum(k=1, #d, base^(k-1) * d[k]);
isLeftTruncatable(d, base) = my(ok=1); for(k=1, #d, if(!isprime(digitsToNum(d[1..k], base)), ok=0; break)); ok;
generateFromPrefix(p, base) = my(seq = [p]); for(n=1, base-1, my(t=concat(n, p)); if(isprime(digitsToNum(t, base)), seq=concat(seq, select(v -> isLeftTruncatable(v, base), generateFromPrefix(t, base))))); seq;
bothTruncatablePrimesInBase(base) = my(t=[]); my(P=primes(primepi(base-1))); for(k=1, #P, t=concat(t, generateFromPrefix([P[k]], base))); vector(#t, k, digitsToNum(t[k], base));
a(n) = vecmax(bothTruncatablePrimesInBase(n)); \\ for n>=3; Daniel Suteu, Jan 22 2019

a(n) <= min(A023107(n), A103443(n)). - Daniel Suteu, Feb 24 2019

a(17)-a(40) from Daniel Suteu, Jan 11 2019

A103463 Length of the largest left-truncatable prime (in base n).

Original entry on oeis.org

0, 3, 6, 6, 17, 7, 15, 10, 24, 9, 32, 8, 26, 22, 25, 11, 43, 14, 37, 27, 37, 17, 53, 20, 39, 28, 46, 19

Offset: 2

Martin Renner, Mar 21 2005, Feb 20 2008, Apr 20 2008

The next term (base 30) will be difficult to calculate because there are over a trillion left-truncatable primes in that base for each of digit-lengths 29-34. Nevertheless, the largest left-truncatable prime in this base can be estimated by theory to have a length of about 82. [Hans Havermann, Aug 16 2011]

I. O. Angell and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977.
Roman Maeder, A Prime Pencil
Eric Weisstein's World of Mathematics, Truncatable Prime.
Index entries for sequences related to truncatable primes

Cf. A076623, A103443.

a(24)-a(29) from Hans Havermann, Aug 16 2011

A323396 Irregular array read by rows, where T(n, k) is the k-th prime that is both left-truncatable and right-truncatable in base n.

Original entry on oeis.org

2, 23, 2, 3, 11, 2, 3, 13, 17, 67, 2, 3, 5, 17, 23, 83, 191, 479, 839, 2, 3, 5, 17, 19, 23, 37, 2, 3, 5, 7, 19, 23, 29, 31, 43, 47, 59, 61, 139, 157, 239, 251, 331, 349, 379, 479, 491, 1867, 2, 3, 5, 7, 23, 29, 47, 173, 2, 3, 5, 7, 23, 37, 53, 73, 313, 317, 373, 797, 3137, 3797, 739397

Offset: 3

Daniel Suteu and Felix Fröhlich, Jan 13 2019

The n-th row contains A323390(n) terms.

The largest term in the n-th row is given by A323137(n).

			Rows for n = 3..7:
  [2, 23]
  [2,  3, 11]
  [2,  3, 13, 17, 67]
  [2,  3,  5, 17, 23, 83, 191, 479, 839]
  [2,  3,  5, 17, 19, 23,  37]

Daniel Suteu, Table of n, a(n) for n = 3..6587
Wikipedia, Truncatable prime

Cf. A020994, A076586, A076623, A323137, A323390.

digitsToNum(d, base) = sum(k=1, #d, base^(k-1) * d[k]);
isLeftTruncatable(d, base) = my(ok=1); for(k=1, #d, if(!isprime(digitsToNum(d[1..k], base)), ok=0; break)); ok;
generateFromPrefix(p, base) = my(seq = [p]); for(n=1, base-1, my(t=concat(n, p)); if(isprime(digitsToNum(t, base)), seq=concat(seq, select(v -> isLeftTruncatable(v, base), generateFromPrefix(t, base))))); seq;
bothTruncatablePrimesInBase(base) = my(t=[]); my(P=primes(primepi(base-1))); for(k=1, #P, t=concat(t, generateFromPrefix([P[k]], base))); vector(#t, k, digitsToNum(t[k], base));
row(n) = vecsort(bothTruncatablePrimesInBase(n));
T(n,k) = row(n)[k];

A323390 Total number of primes that are both left-truncatable and right-truncatable in base n.

Original entry on oeis.org

0, 2, 3, 5, 9, 7, 22, 8, 15, 6, 35, 11, 37, 17, 22, 12, 69, 12, 68, 18, 44, 13, 145, 16, 47, 20, 77, 13, 291, 15, 89, 27, 74, 20, 241, 18, 106, 25, 134, 15, 450, 23, 144, 33, 131, 24, 491, 27, 235, 29, 187, 23, 575, 30, 218, 31, 183, 25, 1377, 26, 247, 37, 231

Offset: 2

Daniel Suteu, Jan 13 2019

			For n = 2, there are no both-truncatable primes, therefore a(2) = 0.
For n = 3, there are 2 both-truncatable primes: 2, 23.
For n = 4, there are 3 both-truncatable primes: 2, 3, 11.
For n = 5, there are 5 both-truncatable primes: 2, 3, 13, 17, 67.
For n = 6, there are 9 both-truncatable primes: 2, 3, 5, 17, 23, 83, 191, 479, 839.

Chris Caldwell, right-truncatable prime, The Prime Glossary.
Eric Weisstein's World of Mathematics, Truncatable Prime

Cf. A020994, A076623, A076586, A323137, A323396.

digitsToNum(d, base) = sum(k=1, #d, base^(k-1) * d[k]);
isLeftTruncatable(d, base) = my(ok=1); for(k=1, #d, if(!isprime(digitsToNum(d[1..k], base)), ok=0; break)); ok;
generateFromPrefix(p, base) = my(seq = [p]); for(n=1, base-1, my(t=concat(n, p)); if(isprime(digitsToNum(t, base)), seq=concat(seq, select(v -> isLeftTruncatable(v, base), generateFromPrefix(t, base))))); seq;
bothTruncatablePrimesInBase(base) = my(t=[]); my(P=primes(primepi(base-1))); for(k=1, #P, t=concat(t, generateFromPrefix([P[k]], base))); vector(#t, k, digitsToNum(t[k], base));
a(n) = #(bothTruncatablePrimesInBase(n));

A129940 Left truncatable primes in base 4 (written in decimal form).

Original entry on oeis.org

2, 3, 7, 11, 23, 43, 59, 107, 151, 251, 619, 919, 1019, 3067, 3691, 4091

Offset: 1

Martin Renner, Jun 09 2007

There are 16 left truncatable primes in base 4.

I. O. Angell and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977.
Eric Weisstein's World of Mathematics, Truncatable Prime.
Index entries for sequences related to truncatable primes

Cf. A024779, A076623.

Edited by Franklin T. Adams-Watters, Jan 25 2010

A129941 Left truncatable primes in base 5 (written in decimal form).

Original entry on oeis.org

2, 3, 7, 13, 17, 23, 67, 73, 107, 113, 317, 607, 613, 1567, 7817

Offset: 1

Martin Renner, Jun 09 2007

There are a total number of 15 left truncatable primes in base 5.

I. O. Angell and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977.
Eric Weisstein's World of Mathematics, Truncatable Prime.
Index entries for sequences related to truncatable primes

Cf. A024780, A076623.

A129942 Left truncatable primes in base 6 (written in decimal form).

Original entry on oeis.org

2, 3, 5, 11, 17, 23, 29, 47, 53, 59, 83, 89, 101, 131, 137, 167, 173, 191, 197, 263

Offset: 1

Martin Renner, Jun 09 2007

There is a total number of 454 left truncatable primes in base 6. Last term is a(454) = 4836525320399.

I. O. Angell and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977.
Eric Weisstein's World of Mathematics, Truncatable Prime
Index entries for sequences related to truncatable primes

Cf. A024781, A076623.

A129943 Left truncatable primes in base 7 (written in decimal form).

Original entry on oeis.org

2, 3, 5, 17, 19, 23, 31, 37, 47, 227, 233, 311, 313, 317, 331, 919, 997, 2371, 2389, 10601, 111443, 817337

Offset: 1

Martin Renner, Jun 09 2007

There are a total number of 22 left truncatable primes in base 7.

I. O. Angell and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977.
Eric Weisstein's World of Mathematics, Truncatable Prime
Index entries for sequences related to truncatable primes

Cf. A024782, A076623.

cp's OEIS Frontend

A076586 Total number of right truncatable primes in base n.

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A103443 Largest left-truncatable prime in base n (decimal expansion).

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A323137 Largest prime that is both left-truncatable and right-truncatable in base n.

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A103463 Length of the largest left-truncatable prime (in base n).

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A323396 Irregular array read by rows, where T(n, k) is the k-th prime that is both left-truncatable and right-truncatable in base n.

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PARI

A323390 Total number of primes that are both left-truncatable and right-truncatable in base n.

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Programs

PARI

A129940 Left truncatable primes in base 4 (written in decimal form).

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Extensions

A129941 Left truncatable primes in base 5 (written in decimal form).

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A129942 Left truncatable primes in base 6 (written in decimal form).

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A129943 Left truncatable primes in base 7 (written in decimal form).

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