A107342 -id:A107342 - cp's OEIS frontend

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

4, 6, 9, 46, 49, 69, 94, 469, 694, 949, 4694

Offset: 1

Keith J. Bauer, Jul 24 2018

There are exactly 3 1-digit terms, 4 2-digit terms, 3 3-digit terms, and 1 4-digit term.
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.
The sequence 3, 4, 3, 1, 0, 0, ... does not appear to be significant.
This sequence depends on base 10 and is nonnegative.
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.
Base 10 is the first base in which this sequence contains 3-digit terms.

			4694 is a semiprime (2 * 2347), and its truncations are, too: 469 (7 * 67), 694 (2 * 347), 46 (2 * 23), etc.

Keith J. Bauer, "A317248 (Python v2.7.13)"

Cf. A001358.

Subset of A107342 and A086698.

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 *)

#v2.7.13, see LINKS to run it online.
#semitest(number, 0) returns True iff number is a semiprime
def semitest(number, factors):
    if number != 2:
        for p in [2] + range(3, int(number ** 0.5) + 1, 2):
            if number % p == 0:
                if factors < 2:
                    return semitest(number / p, factors + 1)
                else:
                    return False
    if factors == 1:
        return True
    else:
        return False
#main function
def doIt(base):
    #initialization
    numbers = [[]]
    indices_list = [[]]
    i = 0
    for number in range(1, base):
        if semitest(number, 0):
            numbers[0].append(number)
            indices_list[0].append([i])
            i += 1
    #numbers[0] is the digit pool
    #numbers[-1] is to be appended to
    #numbers[-2] is for reference to past numbers
    #indices_list records the indices of numbers
    numbers.append([])
    indices_list.append([])
    #main while loop, go until there are no numbers left in the sequence
    indices = [0, 0]
    while len(numbers[-2]) > 0:
        #test number
        if indices[:-1] in indices_list[-2]:
            if indices[1:] in indices_list[-2]:
                #little-endian
                number = 0
                power = 0
                for index in indices:
                    number += numbers[0][index] * base ** power
                    power += 1
                if semitest(number, 0):
                    numbers[-1].append(number)
                    indices_list[-1].append(indices[:])
        #increment indices
        for i in range(len(indices)):
            indices[i] += 1
            if indices[i] == len(numbers[0]):
                indices[i] = 0
                if i == len(indices) - 1:
                    indices = [0] * len(indices) + [0]
                    numbers.append([])
                    indices_list.append([])
            else:
                break
    #print results after while loop has run
    print base, sum(numbers, [])
    print numbers
#call main function
doIt(10)

from sympy import factorint
A317248_list = xlist = [4,6,9]
for n in range(1,10):
    ylist = []
    for i in (4,6,9):
        for x in xlist:
            if sum(factorint(10*x+i).values()) == 2 and (10*x+i) % 10**n in xlist:
                ylist.append(10*x+i)
            elif sum(factorint(x+i*10**n).values()) == 2 and (x//10+i*10**(n-1)) in xlist:
                ylist.append(x+i*10**n)
    xlist = set(ylist)
    if not len(xlist):
        break
    A317248_list.extend(xlist)
A317248_list.sort() # Chai Wah Wu, Aug 23 2018

A321046 Semiprimes for which the concatenation of the digits in the even positions and the concatenation of the digits in the odd positions are semiprimes.

Original entry on oeis.org

46, 49, 69, 94, 145, 194, 262, 265, 291, 295, 365, 393, 394, 395, 398, 446, 466, 469, 545, 565, 591, 597, 649, 662, 669, 695, 699, 767, 794, 842, 862, 865, 866, 895, 943, 961, 965, 993, 995, 1006, 1046, 1059, 1145, 1154, 1202, 1205, 1241, 1255, 1343, 1345, 1349, 1354, 1355, 1501, 1507, 1541, 1555, 1642, 1649, 1655

Offset: 1

Marius A. Burtea, Oct 26 2018

			46 is a term because 46 = 2*23, 4 = 2*2 and 6 = 2*3 are semiprimes.
469 is a term because 469 = 7*67, 49 = 7*7 and 6 = 2*3 are semiprimes.
1145 is a term because 1145 = 5*229, 14 = 2*7 and 15 = 3*5 are semiprimes.
Also 38159 belongs to the sequence. In fact: 38159 = 11*3469, 319 = 11*29 and 85 = 5*17 are semiprimes.

Marius A. Burtea, Table of n, a(n) for n = 1..22530

Cf. A001358, A100484, A046315, A107342.

spQ[n_] := Plus @@ Last /@ FactorInteger[n] == 2; ok[n_] := spQ[n] && Block[{d = IntegerDigits[n]},If[OddQ@ Length@ d, PrependTo[d, 0]]; AllTrue[ FromDigits /@ Transpose[ Partition[d, 2]], spQ]]; Select[ Range@ 1655, ok] (* Giovanni Resta, Oct 29 2018 *)

A365009 Semiprimes that are the concatenation of two or more semiprimes.

Original entry on oeis.org

46, 49, 69, 94, 106, 146, 159, 214, 219, 226, 254, 259, 334, 339, 346, 386, 394, 415, 422, 446, 451, 458, 466, 469, 482, 485, 493, 514, 519, 554, 559, 579, 586, 589, 614, 622, 626, 629, 633, 634, 635, 649, 655, 662, 669, 674, 685, 687, 694, 695, 699, 746, 749, 779, 866, 869, 879, 914, 921, 922

Offset: 1

Zak Seidov and Robert Israel, Aug 15 2023

Conjecture: The fraction of semiprimes <= N that are in this sequence goes to 1 as N -> infinity. What is the first N for which that fraction >= 1/2?

			a(3) = 69 is a term because 69 = 3 * 23 is a semiprime and is the concatenation of the semiprimes 6 = 2 * 3 and 9 = 3 * 3.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001358, A001238, A019549. Contains A107342.

filter:= proc(n) local d,v;
  if numtheory:-bigomega(n) <> 2 then return false fi;
  for d from 1 to length(n)-1 do
     v:= n  mod 10^d;
     if v >= 10^(d-1) and numtheory:-bigomega(v)=2 and g((n-v)/10^d) then return true fi
  od;
  false
end proc:
g:= proc(n) local d,v; option remember;
  if numtheory:-bigomega(n) = 2 then return true fi;
  for d from 1 to length(n)-1 do
    v:= n mod 10^d;
    if v >= 10^(d-1) and numtheory:-bigomega(v)=2 and procname((n-v)/10^d) then return true fi
  od;
  false
end proc:
select(filter, [$10..1000]);

A111730 6-almost primes with semiprime digits (digits 4, 6, 9 only).

Original entry on oeis.org

64, 96, 4644, 4944, 6664, 6966, 9464, 9996, 44464, 44944, 46496, 46644, 49644, 49696, 64449, 64496, 66444, 66696, 69444, 69496, 69966, 94496, 94644, 94696, 96496, 96944, 99666, 99944, 444496, 444664, 444696, 444996, 446664, 446944, 446964, 449469, 449694, 449964, 464496, 464646, 464664, 464994, 469464, 494494, 494944, 494949, 494964, 496464, 499446, 499944, 644464, 644944

Offset: 1

Jonathan Vos Post, Nov 18 2005

			64 = 2^6
96 = 2^5 * 3
4644 = 2^2 * 3^3 * 43
4944 = 2^4 * 3 * 103
6664 = 2^3 * 7^2 * 17
6966 = 2 * 3^4 * 43
9464 = 2^3 * 7 * 13^2
9996 = 2^2 * 3 * 7^2 * 17
44464 = 2^4 * 7 * 397
44944 = 2^4 * 53^2 = 212^2
46496 = 2^5 * 1453

Intersection of A046306 and A107665.

Cf. A001358, A107342, A111494, A111496, A111697.

Select[Range[645000],ContainsOnly[IntegerDigits[#],{4,6,9}]&&PrimeOmega[#]==6&] (* James C. McMahon, Jun 05 2024 *)

isok(k) = (bigomega(k) == 6) && (#setminus(Set(digits(k)), Set([4,6,9])) == 0); \\ Michel Marcus, Apr 13 2022

Missing a(1)=64 prepended and several terms corrected by Georg Fischer and Michel Marcus, Apr 13 2022

A349275 Semiprimes with only semiprime digits, each appearing at least once.

Original entry on oeis.org

469, 649, 694, 4469, 4694, 4699, 4946, 6499, 6649, 6694, 9446, 9466, 9469, 9946, 44669, 44966, 44969, 46469, 46946, 46969, 46994, 46999, 49466, 49649, 49694, 49699, 49969, 64469, 64649, 64669, 64949, 64994, 66469, 66494, 66694, 69449, 69469, 69494, 69694, 69949, 94469, 94669, 94699, 94969, 96449, 96494, 96649, 96946, 96949, 96994, 99646, 99649, 444694

Offset: 1

Zak Seidov, Nov 12 2021

Each digits 4, 6, 9 occur at least once. Minimal difference is 3.

Cf. A001358.

Subsequence of A107342.

is(n)=bigomega(n)==2 && Set(digits(n))==[4, 6, 9] \\ Charles R Greathouse IV, Nov 12 2021

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Python

A321046 Semiprimes for which the concatenation of the digits in the even positions and the concatenation of the digits in the odd positions are semiprimes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A365009 Semiprimes that are the concatenation of two or more semiprimes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A111730 6-almost primes with semiprime digits (digits 4, 6, 9 only).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A349275 Semiprimes with only semiprime digits, each appearing at least once.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI