A046328 -id:A046328 - cp's OEIS frontend

A084994 Duplicate of A046328.

4, 6, 9, 22, 33, 55, 77, 111, 121, 141, 161, 202, 262, 303, 323, 393, 454, 505, 515, 535

Offset: 1

dead

A097393 Emirpimes: numbers n such that n and its reversal are distinct semiprimes.

15, 26, 39, 49, 51, 58, 62, 85, 93, 94, 115, 122, 123, 129, 143, 155, 158, 159, 169, 177, 178, 183, 185, 187, 203, 205, 221, 226, 265, 289, 302, 314, 319, 321, 326, 327, 329, 335, 339, 341, 355, 381, 394, 398, 413, 415, 437, 493, 497, 502, 511, 514, 533

Offset: 1

Jonathan Vos Post

Computed by Eric W. Weisstein, Aug 13 2004.

			26 is a semiprime, as it is 2 * 13, and so is 62 = 2 * 31. 26 and 62 are therefore both in the sequence.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Emirpimes

Cf. A001358, A097394.

Equals A085751 \ A046328.

isA097393 := proc(n)
    local R ;
    R := digrev(n) ;
    if R <> n then
        if numtheory[bigomega](R) = 2 and numtheory[bigomega](n) = 2 then
            return true;
        else
            false;
        end if;
      else
        false;
    end if;
end proc:
for n from 1 to 500 do
    if isA097393(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Apr 05 2012

Cases[{#, IntegerReverse@#} & /@ DeleteCases[Range@5000, ?PalindromeQ], {?(PrimeOmega@# == 2 &) ..}][[All,1]] (* Hans Rudolf Widmer, Jan 07 2024 *)

rev(n)=subst(Polrev(digits(n)),'x,10)
issemi(n)=bigomega(n)==2
list(lim)=my(v=List(),r);forprime(p=2,lim\2,forprime(q=2,min(lim\p,p),r=rev(p*q);if(issemi(r)&&r!=p*q,listput(v,p*q))));Set(v) \\ Charles R Greathouse IV, Jan 27 2015

from sympy import factorint
from itertools import islice
def sp(n): f = factorint(n); return sum(f[p] for p in f) == 2
def ok(n): r = int(str(n)[::-1]); return r != n and sp(n) and sp(r)
print([k for k in range(534) if ok(k)]) # Michael S. Branicky, Jul 03 2022

A046376 Palindromes with exactly 2 palindromic prime factors (counted with multiplicity), and no other prime factors.

Original entry on oeis.org

4, 6, 9, 22, 33, 55, 77, 121, 202, 262, 303, 393, 505, 626, 707, 939, 1111, 1441, 1661, 1991, 3443, 3883, 7997, 10201, 13231, 15251, 18281, 19291, 20602, 22622, 22822, 24842, 26662, 28682, 30903, 31613, 33933, 35653, 37673, 38683, 39993, 60206, 60406, 60806

Offset: 1

Patrick De Geest, Jun 15 1998

Equivalently, semiprime palindromes where both prime factors are palindromes. - Franklin T. Adams-Watters, Apr 11 2011
The sequence "trivially" includes products of palindromic primes p*q where
  a) p = 2 or 3 and q has only digits < 4, as q = 11, 101, 131, 10301, 30103, ...
  b) p <= 11 and q has only digits 0 and 1, as q = 101 and repunit primes A004022
  c) p = 11 and q has only digits spaced out by zeros, as q = 101, 10301, 10501, 10601, 30103, 30203, 30403, 30703, 30803, ... - M. F. Hasler, Jan 04 2022

			The palindrome 35653 is a term since it has 2 factors, 101 and 353, both palindromic.

Lars Blomberg, Table of n, a(n) for n = 1..1000

Cf. A001358 (semiprimes), A002113 (palindromes), A002385 (palindromic primes).
Cf. A046328, A046408.
Subsequence of A188650.

Take[Select[Times@@@Tuples[Select[Prime[Range[5000]],PalindromeQ],2], PalindromeQ]// Union,50] (* Harvey P. Dale, Aug 25 2019 *)

{first(N=50, p=1) = vector(N, i, until( bigomega( p=nxt_A002113(p))==2 && vecmin( apply( is_A002113, factor(p)[,1])),); p)} \\ M. F. Hasler, Jan 04 2022

from sympy import factorint
from itertools import product
def ispal(n): s = str(n); return s == s[::-1]
def pals(d, base=10): # all d-digit palindromes
    digits = "".join(str(i) for i in range(base))
    for p in product(digits, repeat=d//2):
        if d > 1 and p[0] == "0": continue
        left = "".join(p); right = left[::-1]
        for mid in [[""], digits][d%2]: yield int(left + mid + right)
def ok(pal):
    f = factorint(pal)
    return sum(f.values()) == 2 and all(ispal(p) for p in f)
print(list(filter(ok, (p for d in range(1, 6) for p in pals(d) if ok(p))))) # Michael S. Branicky, Aug 14 2022

Intersection of A002113 and A046368; A188649(a(n)) = a(n). - Reinhard Zumkeller, Apr 11 2011

Definition clarified by Franklin T. Adams-Watters, Apr 11 2011

More terms from Lars Blomberg, Nov 06 2015

A046408 Palindromes with exactly 2 distinct palindromic prime factors.

Original entry on oeis.org

6, 22, 33, 55, 77, 202, 262, 303, 393, 505, 626, 707, 939, 1111, 1441, 1661, 1991, 3443, 3883, 7997, 13231, 15251, 18281, 19291, 20602, 22622, 22822, 24842, 26662, 28682, 30903, 31613, 33933, 35653, 37673, 38683, 39993, 60206, 60406, 60806, 62026, 64646, 64846

Offset: 1

Patrick De Geest, Jun 15 1998

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A046328, A046376.

Select[Range[65000],PalindromeQ[#]&&Total[Boole[PalindromeQ/@ FactorInteger[ #][[All,1]]]]==2&&PrimeOmega[#]==2&] (* Harvey P. Dale, Aug 07 2021 *)

ispal(n) = my(d=digits(n)); d == Vecrev(d) \\ A002113
for(k=1, 1e5, if(ispal(k)&&bigomega(k)==2,a=divisors(k); if(#a==4&&ispal(a[2])&&ispal(a[3]), print1(k,", ")))) \\ Alexandru Petrescu, Aug 14 2022

from sympy import factorint
from itertools import product
def ispal(n): s = str(n); return s == s[::-1]
def pals(d, base=10): # all d-digit palindromes
    digits = "".join(str(i) for i in range(base))
    for p in product(digits, repeat=d//2):
        if d > 1 and p[0] == "0": continue
        left = "".join(p); right = left[::-1]
        for mid in [[""], digits][d%2]: yield int(left + mid + right)
def ok(pal):
    f = factorint(pal)
    return len(f) == 2 and sum(f.values()) == 2 and all(ispal(p) for p in f)
print(list(filter(ok, (p for d in range(1, 6) for p in pals(d) if ok(p))))) # Michael S. Branicky, Jun 22 2021

a(41) and beyond from Michael S. Branicky, Jun 22 2021

A196104 Repdigit semiprimes (semiprimes composed of identical digits).

Original entry on oeis.org

4, 6, 9, 22, 33, 55, 77, 111, 1111, 11111, 1111111, 11111111111, 11111111111111111, 2222222222222222222, 3333333333333333333, 5555555555555555555, 7777777777777777777, 22222222222222222222222, 33333333333333333333333, 55555555555555555555555

Offset: 1

Michel Lagneau, Oct 27 2011

A semiprime can be repdigit (base 10) in only three ways. It can be a single-digit semiprime, a repunit semiprime (A102782), or a repunit prime times a prime digit {2, 3, 5, 7}. Occurs in proof that the sequence is infinite in which a(n) is the least semiprime > a(n-1) such that a(n) has no digit in common with a(n-1). - Jonathan Vos Post; corrected by Max Alekseyev, Sep 14 2022

			a(12) = 11111111111 = 21649 * 513239 is semiprime.

Max Alekseyev, Table of n, a(n) for n = 1..35

Subsequence of A046328.
Except for initial terms, subsequence of A116063.
Cf. A000042, A001358, A004023, A046413, A102782.

with(numtheory):for n from 1 to 23 do:for b from 1 to 9 do:x:=(((10^n)- 1)/9)*b:if bigomega(x)=2 then printf(`%d, `,x):else fi:od:od:

Select[FromDigits/@Flatten[Table[PadRight[{},i,n],{i,25},{n,9}],1], PrimeOmega[ #] ==2&] (* Harvey P. Dale, Mar 11 2019 *)

print1("4, 6, 9");for(n=1,20,t=10^n\9;if(bigomega(t)==2,print1(", "t)); if(isprime(t),forprime(p=2,7,print1(", "p*t)))) \\ Charles R Greathouse IV, Oct 27 2011

Union of {4, 6, 9}, A102782, 2*A004022, 3*A004022, 5*A004022, and 7*A004022. - Jonathan Vos Post and R. J. Mathar, Oct 27 2011

Edited by Max Alekseyev, Sep 14 2022

A046392 Palindromes with exactly 2 distinct prime factors.

Original entry on oeis.org

6, 22, 33, 55, 77, 111, 141, 161, 202, 262, 303, 323, 393, 454, 505, 515, 535, 545, 565, 626, 707, 717, 737, 767, 818, 838, 878, 898, 939, 949, 959, 979, 989, 1111, 1441, 1661, 1991, 3113, 3223, 3443, 3883, 7117, 7447, 7997, 9119, 9229, 9449, 10001

Offset: 1

Patrick De Geest, Jun 15 1998

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

Cf. A046328, A046408.

Intersection of A002113 and A006881.

revdigs:= proc(n) local L,i;
  L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
f:= proc(n) local F;
  F:= ifactors(n)[2];
  if nops(F) = 2 and F[1,2]=1 and F[2,2]=1 then n fi
end proc:
N:=5: # for terms of up to N digits.
Res:= 6:
for d from 2 to N do
  if d::even then
    m:= d/2;
    Res:= Res, seq(f(n*10^m + revdigs(n)), n=10^(m-1)..10^m-1);
  else
    m:= (d-1)/2;
    Res:= Res, seq(seq(f(n*10^(m+1)+y*10^m+revdigs(n)), y=0..9), n=10^(m-1)..10^m-1);
  fi
od:
Res; # Robert Israel, Mar 24 2020

pdpfQ[n_]:=Module[{idn=IntegerDigits[n]},idn==Reverse[idn] && PrimeNu[n] == PrimeOmega[n] == 2]; Select[Range[11000],pdpfQ] (* Harvey P. Dale, Dec 16 2012 *)

from sympy import factorint
from itertools import product
def pals(d, base=10): # all d-digit palindromes
    digits = "".join(str(i) for i in range(base))
    for p in product(digits, repeat=d//2):
        if d > 1 and p[0] == "0": continue
        left = "".join(p); right = left[::-1]
        for mid in [[""], digits][d%2]: yield int(left + mid + right)
def ok(pal): f = factorint(pal); return len(f) == 2 and sum(f.values()) == 2
print(list(filter(ok, (p for d in range(1, 5) for p in pals(d) if ok(p))))) # Michael S. Branicky, Jun 22 2021

A210616 Digit reversal of n-th semiprime.

Original entry on oeis.org

4, 6, 9, 1, 41, 51, 12, 22, 52, 62, 33, 43, 53, 83, 93, 64, 94, 15, 55, 75, 85, 26, 56, 96, 47, 77, 28, 58, 68, 78, 19, 39, 49, 59, 601, 111, 511, 811, 911, 121, 221, 321, 921, 331, 431, 141, 241, 341, 541, 641, 551, 851, 951, 161, 661, 961, 771, 871, 381

Offset: 1

Jonathan Vos Post, Mar 23 2012

The fixed points where a(n) = n are A046328.

Alois P. Heinz, Table of n, a(n) for n = 1..2000

Cf. A001358, A004086, A046328.

t = Select[Range[200], PrimeOmega[#] == 2 &]; FromDigits[Reverse[IntegerDigits[#]]] & /@ t (* T. D. Noe, Mar 26 2012 *)
IntegerReverse/@Select[Range[500],PrimeOmega[#]==2&] (* Harvey P. Dale, Dec 06 2021 *)

a(n) = R(A001358(n)) = A004086(A001358(n)).

A324988 Palindromes whose number of divisors is palindromic.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, 222, 232, 242, 262, 282, 292, 303, 313, 323, 333, 343, 353, 363, 373, 383, 393, 404, 424, 434, 454, 474, 484, 494, 505, 515, 535, 545

Offset: 1

Jaroslav Krizek, Mar 23 2019

Numbers m such that m and A000005(m) = tau(m) are both in A002113.

			Number of divisors of palindrome number 22 with divisors 1, 2, 11 and 22 is 4 (palindrome number).

Cf. A000005, A002113, A069747.
Similar sequences for functions sigma(m) and pod(m): A028986, A324989.
Includes A002385, A046328 and A046329.

[n: n in [1..1000] | Intseq(n, 10) eq Reverse(Intseq(n, 10)) and Intseq(NumberOfDivisors(n), 10) eq Reverse(Intseq(NumberOfDivisors(n), 10))]

ispali:= proc(n) local L; L:= convert(n,base,10); L = ListTools:-Reverse(L) end proc:
select(t -> ispali(t) and ispali(numtheory:-tau(t)), [$1..10000]); # Robert Israel, Mar 26 2019

Select[Range@ 600, And[PalindromeQ@ #, PalindromeQ@ DivisorSigma[0, #]] &] (* Michael De Vlieger, Mar 24 2019 *)

ispal(n) = my(d=digits(n)); Vecrev(d) == d;
isok(n) = ispal(n) && ispal(numdiv(n)); \\ Michel Marcus, Mar 23 2019

A075812 Palindromic odd numbers with exactly 2 prime factors (counted with multiplicity).

Original entry on oeis.org

9, 33, 55, 77, 111, 121, 141, 161, 303, 323, 393, 505, 515, 535, 545, 565, 707, 717, 737, 767, 939, 949, 959, 979, 989, 1111, 1441, 1661, 1991, 3113, 3223, 3443, 3883, 7117, 7447, 7997, 9119, 9229, 9449, 10001, 10201, 10401, 10801, 10901, 11111

Offset: 1

Jani Melik, Oct 13 2002

			9=3^2, 33=3*11 and 55=5*11 are palindromic, odd and are products of exactly 2 prime factors.

Zak Seidov, Table of n, a(n) for n = 1..2000

Cf. A046315.

Odd 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 1 to 14000 by 2 do if test(n) then a := [op(a),n]; end; od; a;

Edited by Dean Hickerson, Oct 21 2002

A108505 Number of palindromic semiprimes less than 10^n.

Original entry on oeis.org

0, 3, 7, 36, 50, 269, 367, 2181, 2816, 18391, 23617, 160773, 203733, 1429749, 1788486, 12808711, 15889727

Offset: 0

Robert G. Wilson v, Jun 06 2005

Does the limit n-> inf. a(n+2)/a(n) =~ 8*Pi^2/9?

Cf. A046328.

NextPalindrome[n_] := Block[ {l = Floor[ Log[10, n] + 1], idn = IntegerDigits[n]}, If[ Union[ idn] == {9}, Return[n + 2], If[l < 2, Return[n + 1], If[ FromDigits[ Reverse[ Take[ idn, Ceiling[l/2]]]] > FromDigits[ Take[ idn, -Ceiling[l/2]]], FromDigits[ Join[ Take[ idn, Ceiling[l/2]], Reverse[ Take[ idn, Floor[l/2]]] ]], idfhn = FromDigits[ Take[ idn, Ceiling[l/2]]] + 1; idp = FromDigits[ Join[ IntegerDigits[ idfhn], Drop[ Reverse[ IntegerDigits[ idfhn]], Mod[l, 2]]]] ]]]]; fQ[n_] := Plus @@ Last /@ FactorInteger[n] == 2; c = np = 0; Do[ While[np < 10^n, If[ fQ[np], c++ ]; np = NextPalindrome[np]]; Print[c], {n, 0, 12}]

a(15)-a(16) from Donovan Johnson, Mar 14 2010

cp's OEIS Frontend

A084994 Duplicate of A046328.

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A097393 Emirpimes: numbers n such that n and its reversal are distinct semiprimes.

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A046376 Palindromes with exactly 2 palindromic prime factors (counted with multiplicity), and no other prime factors.

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A046408 Palindromes with exactly 2 distinct palindromic prime factors.

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A196104 Repdigit semiprimes (semiprimes composed of identical digits).

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A046392 Palindromes with exactly 2 distinct prime factors.

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A210616 Digit reversal of n-th semiprime.

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A324988 Palindromes whose number of divisors is palindromic.

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