A048054 -id:A048054 - cp's OEIS frontend

A375171 Square array T(n,k), n>0 and k>0, read by antidiagonals in ascending order, giving the smallest n*k-digit number that, if arranged in an n X k matrix, form k-digit reversible prime in each row and n-digit reversible prime in each column, or -1 if no such number exists.

2, 37, 37, 337, 1111, 337, 3257, 111331, 113131, 3257, 32233, 13139731, 113101311, 11933371, 32233, 322573, 1111179779, 113101929311, 119310213191, 1119711779, 322573, 3222223, 111111131397, 113101167919739, 1193100990013911, 111971042937997, 111119111337, 3222223

Offset: 1

Jean-Marc Rebert, Aug 06 2024

			T(3,2) = 111331 is the smallest 3*2-digit number that if arranged in a 3 X 2 matrix yields in each row and column an reversible prime, i.e.,
  11
  13
  31
-> 11 (1 time), 13 (1 time), 31 (1 time), 113 (1 time), 131 (1 time) are all reversible primes.
Table begins (upper left corner = T(1,1)):
     2       37          337             3257 ...
    37     1111       113131         11933371 ...
   337   111331    113101311     119310213191 ...
  3257 13139731 113101929311 1193100990013911 ...
   ...      ...          ...              ... ...

Cf. A007500, A048054, A177513, A224164, A375234.

isp(x) = ispseudoprime(x) && ispseudoprime(fromdigits(Vecrev(digits((x)))));
ispd(x) = ispseudoprime(fromdigits(x)) && ispseudoprime(fromdigits(Vecrev(x)));
vp(n) = select(isp, [10^(n-1)..10^n-1]);
isok(val, n, k) = my(d=digits(val), v=vector(k, i, []), j=1); for (i=1, #d, v[j] = concat(v[j], d[i]); j++; if (j>k, j=1);); for (i=1, k, if (!ispd(v[i]), return(0));); return(1);
T(n,k) = my(v = vp(k), nbp = #v, nb = nbp^n); for (i=0, nb-1, my(d=digits(i, nbp)); if (d==[], d=vector(n)); while(#d x+1, d); my(s=""); for (i=1, #d, s = concat(s, Str(v[d[i]]))); my(val = eval(s)); if (isok(val, n, k), return(val));); \\ Michel Marcus, Aug 08 2024

T(1,n) = T(n,1) <= A177513(n) for n >1.

T(1,n) = T(n,1) = A177513(n) for n = 2..6.

A172384 Partial sums of A048895.

Original entry on oeis.org

1061, 2152, 3753, 5654, 15715, 25806, 41807, 60808, 167669, 277560, 446161, 645062, 1751943, 2861824, 4467905, 6273966, 8083057, 9969068, 11858079, 13767160, 24574041, 35383922, 46445733, 57537544, 69147225, 80845916

Offset: 1

Jonathan Vos Post, Feb 01 2010

None of these partial sums of "bemirps: primes that yield a different prime when turned upside down with reversals of both being two more different primes" is itself prime. So what is the first (nontrivial) prime partial sum of bemirps? Of emirps? Of "norep emirps": primes with distinct digits which remain prime when reversed? Of emirpimes? I suspect that G. L. Honaker, Jr. would be delighted to have any of these.

			a(26) = 1061 + 1091 + 1601 + 1901 + 10061 + 10091 + 16001 + 19001 + 106861 + 109891 + 168601 + 198901 + 1106881 + 1109881 + 1606081 + 1806061 + 1809091 + 1886011 + 1889011 + 1909081 + 10806881 + 10809881 + 11061811 + 11091811 + 11609681 + 11698691.

Cf. A000040, A003684, A006567, A007628, A046732, A048051, A048052, A048053, A048054, A048895.

A256623 Number of distinct n-digit patterns in base 10 such that the pattern and its reverse are prime.

Original entry on oeis.org

4, 5, 29, 102, 796, 4769, 35905, 267789, 2101184, 16809690, 137487157, 1147385627, 9745119882

Offset: 1

Russell Y. Webb, Jul 11 2015

Here, distinct numbers means under reversal. 13 and 31 are the same pattern under reversal and only count as one. The sequence can be calculated from the number of palindrome primes (A016115), p_i, and number of reversal primes (A048054), r_i. X_i = (r_i - p_i)/2 + p_i. The (r_i - p_i) term is always even, by construction (it is the count of reversible primes that are not their own reverse).
This sequence is the set cardinality of the prime numbers under a base-10 digit reversal identity operator.
Since there are no palindrome primes with even digits > 11 we know that the even entries are the same as half the number of reversible primes.

Cf. A016115, A048054.

a(n) = (A048054(n) + A016115(n))/2.

a(11)-a(13) from Giovanni Resta, Jul 19 2015

A346021 Primes that are the first in a run of exactly 1 emirp.

Original entry on oeis.org

97, 107, 113, 149, 157, 167, 179, 199, 311, 359, 389, 907, 1009, 1061, 1069, 1091, 1181, 1301, 1321, 1429, 1439, 1453, 1471, 1487, 1559, 1619, 1657, 1669, 1753, 1789, 1811, 1867, 1879, 1901, 1913, 1979, 3049, 3067, 3121, 3163, 3169, 3221, 3251, 3257, 3319

Offset: 1

Lars Blomberg, Jul 14 2021

There are large gaps in this sequence because all terms need to begin with 1, 3, 7, or 9 otherwise the reversal is composite.

			a(1) = 97 because of the three consecutive primes 89, 97, 101 only 97 is an emirp and this is the first such occurrence.

Subsequence of A006567 (emirps).

Cf. A003684, A048052, A048054, A071612, A346022, A346023, A346024, A346025, A346026, A346027, A346028, A346029.

emirpQ[p_] := (r = IntegerReverse[p]) != p && PrimeQ[r]; p = Select[Range[3400], PrimeQ]; p[[1 + Position[Partition[emirpQ /@ p, 3, 1], {False, True, False}] // Flatten]] (* Amiram Eldar, Jul 14 2021 *)

from sympy import isprime, nextprime
def isemirp(p): s = str(p); return s != s[::-1] and isprime(int(s[::-1]))
def aupto(limit):
  alst, pvec, evec, p = [], [2, 3, 5], [0, 0, 0], 7
  while pvec[1] <= limit:
    if evec == [0, 1, 0]: alst.append(pvec[1])
    pvec = pvec[1:] + [p]; evec = evec[1:] + [isemirp(p)]; p = nextprime(p)
  return alst
print(aupto(3319)) # Michael S. Branicky, Jul 14 2021

cp's OEIS Frontend

A375171 Square array T(n,k), n>0 and k>0, read by antidiagonals in ascending order, giving the smallest n*k-digit number that, if arranged in an n X k matrix, form k-digit reversible prime in each row and n-digit reversible prime in each column, or -1 if no such number exists.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

A172384 Partial sums of A048895.

Views

Author

Keywords

Comments

Examples

Crossrefs

A256623 Number of distinct n-digit patterns in base 10 such that the pattern and its reverse are prime.

Views

Author

Keywords

Comments

Crossrefs

Formula

Extensions

A346021 Primes that are the first in a run of exactly 1 emirp.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Python