A323137 -id:A323137 - cp's OEIS frontend

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.

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));

A326609 Largest minimal prime in base n (written in base 10).

Original entry on oeis.org

3, 13, 5, 3121, 5209, 2801, 76695841, 811, 66600049, 29156193474041220857161146715104735751776055777, 388177921

Offset: 2

Richard N. Smith, Jul 13 2019

a(13) is (probably) 13^32020*8+183, it has 35670 digits, a(14) = 14^85*4+65, it has 99 digits, a(15) = (15^106*66-619)/7, it has 126 digits, a(16) = 16^3544*9+145, it has 4269 digits.
a(17) is the smallest prime of the form (4105*17^k-9)/16 if it exists, otherwise (probably) (73*17^111333-9)/16 (136991 digits), a(18) = 18^31*304+1 (42 digits).
Other known terms: a(20) = (20^449*16-2809)/19 (585 digits), a(22) = 22^763*20+7041 (1026 digits), a(23) is (probably) (23^800873*106-7)/11 (1090573 digits), a(24) = (24^99*512-121)/23 (138 digits), a(30) = 30^1023*12+1 (1513 digits), a(42) = (42^487*27-1093)/41 (791 digits).
a(19) is the smallest prime of the form (15964*19^k-1)/3 if it exists, otherwise (probably) (904*19^110984-1)/3 (141924 digits), a(21) is the smallest prime of the form 16*21^k+335 if it exists, otherwise (probably) (51*21^479149-1243)/4 (633542 digits).

Richard N. Smith, Table of n, a(n) for n = 2..16
Curtis Bright, Raymond Devillers, and Jeffrey Shallit, Minimal Elements for the Prime Numbers, preprint, 2014.
Curtis Bright, Raymond Devillers, and Jeffrey Shallit, Minimal Elements for the Prime Numbers, Experimental Mathematics, Volume 25, 2016 - Issue 3.
C. K. Caldwell, The Prime Glossary, minimal prime
Richard N. Smith, List of all known terms
Top PRPs, Search for 8*13^n+183
Top PRPs, Search for (73*17^n-9)/16
Top PRPs, Search for (106*23^n-7)/11
Wikipedia, Minimal prime (recreational mathematics)

Cf. A071062 (base 10 minimal primes), A110600 (base 12 minimal primes).

Cf. A293142 (largest non-repunit permutable prime), A317689 (largest non-repunit circular prime), A103443 (largest left-truncatable prime), A023107 (largest right-truncatable prime), A323137 (largest two-sided prime), A084738 (smallest repunit prime), A186995 (smallest weakly prime).

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

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A326609 Largest minimal prime in base n (written in base 10).

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