cp's OEIS Frontend

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

%I A046385 #21 Jan 24 2021 13:28:31
%S A046385 1,2,4,8,88,252,2772,82728,2112,4224,8448,236989632,48384,2977792
%N A046385 Smallest palindrome with exactly n palindromic prime factors (counted with multiplicity), and no other prime factors.
%C A046385 Initial terms of sequences A046376-A046384.
%C A046385 Note that 48384 (k=12) is a 'Droll' number: see A019507.
%C A046385 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
%e A046385 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.
%o A046385 (PARI) is_A002113(n)={Vecrev(n=digits(n))==n}; \\ _M. F. Hasler_ in A002113
%o A046385 arepalf(nf,x)={forstep(j=nf,1,-1,if(is_A002113(x[j,1]),,return(0)));return(1)};
%o A046385 md=[0,1,2,3,4,5,6,7,8,9]; \\ Middle digits in odd length palindromes
%o A046385 a=vector(64);a[1]=2;a[2]=4;a[3]=8;
%o A046385 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
%Y A046385 Cf. A309565 (additional non-palindromic prime factors allowed).
%K A046385 nonn,base,more,hard
%O A046385 0,2
%A A046385 _Patrick De Geest_, Jun 15 1998
%E A046385 Definition clarified by _Hugo Pfoertner_, Aug 08 2019