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A317248 Semiprimes which when truncated arbitrarily on either side in base 10 yield semiprimes.

%I A317248 #27 Aug 23 2018 20:59:19
%S A317248 4,6,9,46,49,69,94,469,694,949,4694
%N A317248 Semiprimes which when truncated arbitrarily on either side in base 10 yield semiprimes.
%C A317248 There are exactly 3 1-digit terms, 4 2-digit terms, 3 3-digit terms, and 1 4-digit term.
%C A317248 After the 4-digit term, there are no more terms in this sequence. This is provable by induction: There are no 5-digit terms. If there are no k-digit terms, there are no (k+1)-digit terms. (If there were, then said term, when truncated on either side, would produce a k-digit number that is in the sequence.) Therefore, there are no terms that have at least 5 digits.
%C A317248 The sequence 3, 4, 3, 1, 0, 0, ... does not appear to be significant.
%C A317248 This sequence depends on base 10 and is nonnegative.
%C A317248 Any truncation of a number in this sequence yields another number in this sequence. If one did not, then truncating the number more would yield a non-semiprime, which is impossible.
%C A317248 Base 10 is the first base in which this sequence contains 3-digit terms.
%H A317248 Keith J. Bauer, <a href="http://tpcg.io/cSo9J1">"A317248 (Python v2.7.13)"</a>
%e A317248 4694 is a semiprime (2 * 2347), and its truncations are, too: 469 (7 * 67), 694 (2 * 347), 46 (2 * 23), etc.
%t A317248 ok[w_, n_] := AllTrue[Flatten@ Table[ FromDigits@ Take[w, {i, j}], {i, n}, {j, i, n}], PrimeOmega[#] == 2 &]; Union @@ Reap[ Do[Sow[ FromDigits /@ Select[Tuples[{4, 6, 9}, n], ok[#, n] &]], {n, 5}]][[2, 1]] (* _Giovanni Resta_, Jul 26 2018 *)
%o A317248 (Python) #v2.7.13, see LINKS to run it online.
%o A317248 #semitest(number, 0) returns True iff number is a semiprime
%o A317248 def semitest(number, factors):
%o A317248     if number != 2:
%o A317248         for p in [2] + range(3, int(number ** 0.5) + 1, 2):
%o A317248             if number % p == 0:
%o A317248                 if factors < 2:
%o A317248                     return semitest(number / p, factors + 1)
%o A317248                 else:
%o A317248                     return False
%o A317248     if factors == 1:
%o A317248         return True
%o A317248     else:
%o A317248         return False
%o A317248 #main function
%o A317248 def doIt(base):
%o A317248     #initialization
%o A317248     numbers = [[]]
%o A317248     indices_list = [[]]
%o A317248     i = 0
%o A317248     for number in range(1, base):
%o A317248         if semitest(number, 0):
%o A317248             numbers[0].append(number)
%o A317248             indices_list[0].append([i])
%o A317248             i += 1
%o A317248     #numbers[0] is the digit pool
%o A317248     #numbers[-1] is to be appended to
%o A317248     #numbers[-2] is for reference to past numbers
%o A317248     #indices_list records the indices of numbers
%o A317248     numbers.append([])
%o A317248     indices_list.append([])
%o A317248     #main while loop, go until there are no numbers left in the sequence
%o A317248     indices = [0, 0]
%o A317248     while len(numbers[-2]) > 0:
%o A317248         #test number
%o A317248         if indices[:-1] in indices_list[-2]:
%o A317248             if indices[1:] in indices_list[-2]:
%o A317248                 #little-endian
%o A317248                 number = 0
%o A317248                 power = 0
%o A317248                 for index in indices:
%o A317248                     number += numbers[0][index] * base ** power
%o A317248                     power += 1
%o A317248                 if semitest(number, 0):
%o A317248                     numbers[-1].append(number)
%o A317248                     indices_list[-1].append(indices[:])
%o A317248         #increment indices
%o A317248         for i in range(len(indices)):
%o A317248             indices[i] += 1
%o A317248             if indices[i] == len(numbers[0]):
%o A317248                 indices[i] = 0
%o A317248                 if i == len(indices) - 1:
%o A317248                     indices = [0] * len(indices) + [0]
%o A317248                     numbers.append([])
%o A317248                     indices_list.append([])
%o A317248             else:
%o A317248                 break
%o A317248     #print results after while loop has run
%o A317248     print base, sum(numbers, [])
%o A317248     print numbers
%o A317248 #call main function
%o A317248 doIt(10)
%o A317248 (Python)
%o A317248 from sympy import factorint
%o A317248 A317248_list = xlist = [4,6,9]
%o A317248 for n in range(1,10):
%o A317248     ylist = []
%o A317248     for i in (4,6,9):
%o A317248         for x in xlist:
%o A317248             if sum(factorint(10*x+i).values()) == 2 and (10*x+i) % 10**n in xlist:
%o A317248                 ylist.append(10*x+i)
%o A317248             elif sum(factorint(x+i*10**n).values()) == 2 and (x//10+i*10**(n-1)) in xlist:
%o A317248                 ylist.append(x+i*10**n)
%o A317248     xlist = set(ylist)
%o A317248     if not len(xlist):
%o A317248         break
%o A317248     A317248_list.extend(xlist)
%o A317248 A317248_list.sort() # _Chai Wah Wu_, Aug 23 2018
%Y A317248 Cf. A001358.
%Y A317248 Subset of A107342 and A086698.
%K A317248 base,fini,full,nonn
%O A317248 1,1
%A A317248 _Keith J. Bauer_, Jul 24 2018