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

0,2

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

Links

Table of n, a(n) for n=0..13.

Example

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.

Prog

(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

Crossrefs

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

Sequence in context: A237913 A076886 A309565 * A068664 A199166 A018605

Adjacent sequences: A046382 A046383 A046384 * A046386 A046387 A046388

Keyword

nonn,base,more,hard

Author

Patrick De Geest, Jun 15 1998

Extensions

Definition clarified by Hugo Pfoertner, Aug 08 2019

Status

approved