A046328 -id:A046328 - cp's OEIS frontend

A144766 Terms in A144719 that are themselves decimal palindromes.

121, 10001, 10201, 36763, 1226221, 7673767, 12467976421, 1030507050301, 1120237320211, 1225559555221, 1234469644321, 1334459544331, 100330272033001, 100827848728001, 101222252222101, 103023070320301, 121363494363121, 134312696213431, 10022212521222001

Offset: 1

Reikku Kulon, Sep 20 2008

Presumed infinite, but it is difficult to find more terms.
The earlier claim that this sequence is a subsequence of A046450 was incorrect, as the counterexample of 7673767 =97*79111 shows. The reason is that A046450 checks only concatenations in the natural order of the prime factors, but this sequence here allows for both orders, 97//79111 as well as 79111//97, to be palindromic. - R. J. Mathar, Jan 22 2009
3*10^14 < a(19) <= 10022212521222001. - Donovan Johnson, Dec 08 2010

			10001 = 73 * 137; 73137 is a palindrome.

Giovanni Resta, Table of n, a(n) for n = 1..41

Cf. A002113, A046450, A144719.

Subsequence of (A144719 INTERSECT A046328). - R. J. Mathar, Jan 22 2009

a(7)-a(18) from Donovan Johnson, Dec 08 2010

a(19) from Giovanni Resta, Aug 31 2018

A174924 Semiprimes sp(k) = q * r such that sum of digits of sp(k) equals sum of digits of the semiprime index k.

Original entry on oeis.org

14, 15, 55, 121, 122, 123, 214, 215, 265, 287, 407, 481, 482, 535, 667, 813, 851, 901, 951, 1119, 1149, 1174, 1537, 1538, 1639, 1681, 1961, 2059, 2117, 2165, 2209, 2245, 2246, 2386, 2419, 2458, 2501, 2513, 2537, 2603, 2629, 2641, 2642, 2643, 2807, 2845

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Apr 02 2010

Numbers of the form q * r where q and r are primes, not necessarily distinct.
These numbers are also called semiprimes or 2-almost primes.
For primes with such a property see A033548

			sp(5) = 14 = 2 * 7 is the 5th semiprime, sum of digits sod(14) = 1+4 = 5, 1st term
sp(6) = 15 = 3 * 5 is the 6th semiprime, sum of digits sod(15) = 1+5 = 6. 2nd term
sp(40) = 121 = 11^2 is the 40th semiprime, sum of digits sod(121) = 1+2+1 = 4, 4th term
Additionally for the prime based (q=r=11) square 121: sod(q) + sod(r) = 2 * sod(11) = 4
The first 110 such semiprimes:
14, 15, 55, 121, 122, 123, 214, 215, 265, 287, 407, 481, 482, 535, 667, 813, 851, 901, 951, 1119,
1149, 1174, 1537, 1538, 1639, 1681, 1961, 2059, 2117, 2165, 2209, 2245, 2246, 2386, 2419,
2458, 2501, 2513, 2537, 2603, 2629, 2641, 2642, 2643, 2807, 2845, 2846, 2858, 2859, 2921,
3158, 3205, 3218, 3427, 3439, 4322, 4333, 4367, 4661, 4713, 4714, 4735, 4811, 5221, 5317,
5318, 5615, 5707, 5753, 6009, 6022, 6023, 6046, 6081, 6082, 6117, 6193, 6283, 6371, 6411,
6423, 6514, 6515, 6527, 6541, 6542, 6593, 6635, 6649, 6683, 6694, 6905, 7251, 7291, 7363,
7387, 8023, 8102, 8153, 8203, 8401, 8402, 8403, 8503, 8531, 9019, 9201, 9223, 9271, 9902

Cf. A033548, A046328

A075813 Palindromic even numbers with exactly 2 prime factors (counted with multiplicity). Equivalently, palindromic numbers of the form 2*p with p prime.

Original entry on oeis.org

4, 6, 22, 202, 262, 454, 626, 818, 838, 878, 898, 20302, 20602, 22322, 22522, 22622, 22822, 24142, 24842, 26662, 26762, 28682, 41014, 41414, 41614, 41714, 43034, 43234, 43534, 43634, 45454, 45554, 45754, 47074, 47374, 47774, 49094, 49394

Offset: 1

Jani Melik, Oct 13 2002

			4=2^2, 6=2*3 and 22=2*11 are palindromic, even and have exactly 2 prime factors.

Cf. A001747.

Even subsequence of A046328.

test := proc(n) local d; d := convert(n,base,10); return ListTools[Reverse](d)=d and numtheory[bigomega](n)=2; end; a := []; for n from 2 to 50000 by 2 do if test(n) then a := [op(a),n]; end; od; a;

Select[Range[50000],EvenQ[#]&&PalindromeQ[#]&&PrimeOmega[#]==2&] (* Harvey P. Dale, Oct 16 2024 *)

Edited by Dean Hickerson, Oct 21 2002

A189484 Numbers that can be factored into semiprimes which, when concatenated in increasing order, produce a palindrome.

Original entry on oeis.org

1, 4, 6, 9, 16, 22, 33, 36, 55, 56, 64, 77, 81, 111, 121, 136, 141, 156, 161, 202, 216, 256, 262, 276, 296, 303, 323, 351, 376, 393, 441, 454, 484, 505, 515, 516, 535, 545, 560, 565, 621, 626, 707, 717, 729, 737, 765, 767, 784, 818, 838, 878, 898, 939, 949

Offset: 1

Jonathan Vos Post, Apr 22 2011

This is to semiprimes A001358 as A046447 is to primes A000040.

The initial 1 represents the empty product.

			The first value not itself a semiprime palindrome (A046328) or power of semiprimes (i.e., 16 = 4 * 4 which concatenate to the palindrome 44, 484 = 22^2) is 56 = 4 * 14. The first where additionally the first factor is not a single digit is 765 = 15 * 51 = 3^2 * 5 * 17 since (15, 51) are a pair of emirpimes A097393, and 765 = A158126(1).

Cf. A001358, A002113, A046328, A046447, A097393, A158126 Products of emirpimes pairs, sorted.

ok[n_] := n == 1 || Block[{d, p = Join @@ mu /@ FactorInteger[n]}, EvenQ@ Length[p] && AnyTrue[ Union[ Sort /@ ((Times @@@ #) & /@ Union[ (Sort /@ Partition[#, 2]) & /@ Permutations[p]])], (d = Join @@ IntegerDigits[#]; d == Reverse[d]) &]]; Select[ Range[1000], ok] (* Giovanni Resta, Sep 15 2018 *)

Additional terms from Franklin T. Adams-Watters, Apr 28 2011

More terms from Giovanni Resta, Sep 15 2018

A348050 Palindromes setting a new record of their number of prime divisors A001222.

Original entry on oeis.org

1, 2, 4, 8, 88, 252, 2112, 4224, 8448, 44544, 48384, 405504, 4091904, 405909504, 677707776, 4285005824, 21128282112, 29142024192, 4815463645184, 445488555884544, 27874867776847872, 40539458585493504, 63556806860865536, 840261068860162048, 4870324782874230784

Offset: 1

Hugo Pfoertner, Oct 25 2021

Cf. A001222, A002113, A046399, A335645.

Cf. A046328, A046329, A046330, A046331, A046332, A046333, A046334, A046335, A046336.

m=0;lst=Union@Flatten[Table[{FromDigits@Join[s=IntegerDigits@n,Reverse@s],FromDigits@Join[w=IntegerDigits@n,Rest@Reverse@w]},{n,10^5}]];Do[t=PrimeOmega@lst[[n]];If[t>m,Print@lst[[n]];m=t],{n,Length@lst}] (* Giorgos Kalogeropoulos, Oct 25 2021 *)

from sympy import factorint
from itertools import product
def palsthru(maxdigits):
    midrange = [[""], [str(i) for i in range(10)]]
    for digits in range(1, maxdigits+1):
        for p in product("0123456789", repeat=digits//2):
            left = "".join(p)
            if len(left) and left[0] == '0': continue
            for middle in midrange[digits%2]:
                yield int(left+middle+left[::-1])
def afind(maxdigits):
    record = -1
    for p in palsthru(maxdigits):
        f = factorint(p, multiple=True)
        if p > 0 and len(f) > record:
            record = len(f)
            print(p, end=", ")
afind(10) # Michael S. Branicky, Oct 25 2021

a(1) = 1 from David A. Corneth, Oct 25 2021
a(16)-a(19) from Giorgos Kalogeropoulos, Oct 25 2021
a(20) from Michael S. Branicky, Oct 25 2021
a(21)-a(25) from Chai Wah Wu, Oct 28 2021

cp's OEIS Frontend

A144766 Terms in A144719 that are themselves decimal palindromes.

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A174924 Semiprimes sp(k) = q * r such that sum of digits of sp(k) equals sum of digits of the semiprime index k.

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A075813 Palindromic even numbers with exactly 2 prime factors (counted with multiplicity). Equivalently, palindromic numbers of the form 2*p with p prime.

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A189484 Numbers that can be factored into semiprimes which, when concatenated in increasing order, produce a palindrome.

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A348050 Palindromes setting a new record of their number of prime divisors A001222.

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