A046376 -id:A046376 - cp's OEIS frontend

A084189 Minimum digit out of both prime factors of the numbers in A046376.

2, 2, 3, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 0, 0, 0, 2, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1

Offset: 0

Jason Earls, Jun 20 2003

Conjecture: a(n) is never greater than 3.

Suggested by this old JRM problem: "Find a composite palindrome larger than 343 whose prime factors are palindromes consisting of digits greater than 3."

			a(9)=1 because the factors of A046376(9)= 262 are 2*131 and the minimum digit is 1.

A046368 Products of two palindromic primes.

Original entry on oeis.org

4, 6, 9, 10, 14, 15, 21, 22, 25, 33, 35, 49, 55, 77, 121, 202, 262, 302, 303, 362, 382, 393, 453, 505, 543, 573, 626, 655, 706, 707, 746, 755, 766, 905, 917, 939, 955, 1057, 1059, 1111, 1119, 1149, 1267, 1337, 1441, 1454, 1514, 1565, 1574, 1594, 1661, 1765, 1838, 1858

Offset: 1

Patrick De Geest, Jun 15 1998

Intersection of A001358 and A033620; A046368 is a subsequence. - Reinhard Zumkeller, Apr 10 2011
Equivalently, semiprimes where both prime factors are palindromes. - Franklin T. Adams-Watters, Apr 11 2011
See A046376 for the subsequence of palindromic terms. - M. F. Hasler, Jan 04 2022

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A001358 (semiprimes), A002113 (palindromes), A002385 (palindromic primes), A046376 (subsequence of palindromes), A046400.

palQ[n_] := Reverse[x = IntegerDigits[n]] == x; Select[Range[1800], PrimeOmega[#] == 2 && And @@ palQ /@ First /@ FactorInteger[#] &] (* Jayanta Basu, Jun 23 2013 *)

select( {is_A046368(n)=bigomega(n)==2 && vecmin( apply( is_A002113, factor(n)[, 1]))}, [1..9999]) \\ M. F. Hasler, Jan 04 2022

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

Definition simplified by M. F. Hasler, Jan 04 2022

A188650 Fixed points of A188649: numbers divisible by the reverse of all their divisors.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 22, 33, 40, 44, 55, 66, 77, 88, 99, 101, 110, 121, 131, 151, 181, 191, 202, 220, 242, 262, 303, 313, 353, 363, 373, 383, 393, 404, 440, 484, 505, 606, 626, 707, 727, 757, 787, 797, 808, 909, 919, 929, 939, 1010, 1111, 1331, 1441, 1661, 1991, 2020, 2222, 2662, 2882, 3333, 3443, 3883, 3993, 4040, 4444, 5555, 6666, 6886, 7777, 7997, 8888

Offset: 1

Reinhard Zumkeller, Apr 11 2011

A188649(a(n)) = a(n);

A002385 and A046376 are subsequences; subsequence of A002113.

Michael S. Branicky, Table of n, a(n) for n = 1..10700

import Data.List (elemIndices)
a188650 n = a188650_list !! (n-1)
a188650_list =
   map succ $ elemIndices 0 $ zipWith (-) [1..] $ map a188649 [1..]

rev(n:int,B=10)=my(m=n%B);n\=B;while(n>0,m=m*B+n%B;n\=B);m
is(n)=fordiv(n,d,if(n%rev(d),return(0)));1 \\ Charles R Greathouse IV, Jul 14 2011

from math import lcm
from sympy import divisors
def ok(n): return n == lcm(*(int(str(d)[::-1]) for d in divisors(n)))
print([k for k in range(1, 10000) if ok(k)]) # Michael S. Branicky, Sep 30 2022

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

A046385 Smallest palindrome with exactly n palindromic prime factors (counted with multiplicity), and no other prime factors.

Original entry on oeis.org

1, 2, 4, 8, 88, 252, 2772, 82728, 2112, 4224, 8448, 236989632, 48384, 2977792

Offset: 0

Patrick De Geest, Jun 15 1998

Initial terms of sequences A046376-A046384.
Note that 48384 (k=12) is a 'Droll' number: see A019507.
There are 3 more known terms: a(15)=405504, a(16)=40955904, a(20)=677707776. Any other terms would have at least 18 decimal digits. Conjecture: The sequence is finite and has no other terms than those shown here. - Hugo Pfoertner, Aug 13 2019

			a(7) = 82728 because it is the smallest palindrome with 7 palindromic and no other prime factors: 82728 = 2^3 * 3^3 * 383. If other prime factors are not excluded, A309565(7) = 29792 =  2^5 * 7^2 * 19 also has exactly 7 palindromic factors and the additional factor 19.

Cf. A309565 (additional non-palindromic prime factors allowed).

is_A002113(n)={Vecrev(n=digits(n))==n}; \\ M. F. Hasler in A002113
arepalf(nf,x)={forstep(j=nf,1,-1,if(is_A002113(x[j,1]),,return(0)));return(1)};
md=[0,1,2,3,4,5,6,7,8,9]; \\ Middle digits in odd length palindromes
a=vector(64);a[1]=2;a[2]=4;a[3]=8;
for(d=2,11,print("Digits: ",d);if(d%2==0,for(k=10^((d-2)/2),10*10^((d-2)/2)-1,my(dv=digits(k));P=fromdigits(concat(dv,Vecrev(dv)));x=factor(P);bigom=vecsum(x[,2]);nf=#x[,2];for(j=1,#a,if(a[j],,if(j==bigom,if(arepalf(nf,x),print("a(",j,")=",a[j]=P)))))),for(k=10^((d-3)/2),10*10^((d-3)/2)-1,my(dv=digits(k));for(m=1,10,P=fromdigits(concat(concat(dv,md[m]),Vecrev(dv)));x=factor(P);bigom=vecsum(x[,2]);nf=#x[,2];for(j=1,#a,if(a[j],,if(j==bigom,if(arepalf(nf,x),print("a(",j,")=",a[j]=P)))))))));a \\ Hugo Pfoertner, Aug 13 2019

Definition clarified by Hugo Pfoertner, Aug 08 2019

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A084189 Minimum digit out of both prime factors of the numbers in A046376.

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A046368 Products of two palindromic primes.

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A188650 Fixed points of A188649: numbers divisible by the reverse of all their divisors.

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A046408 Palindromes with exactly 2 distinct palindromic prime factors.

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A046385 Smallest palindrome with exactly n palindromic prime factors (counted with multiplicity), and no other prime factors.

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PARI

Extensions