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

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

Offset: 0

Author

Patrick De Geest, Jun 15 1998

Keywords

• nonn
• base
• more
• hard

Comments

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

Examples

			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.
		

Crossrefs

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

Programs

PARI

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

Extensions

Definition clarified by Hugo Pfoertner, Aug 08 2019