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A046385 Smallest palindrome with exactly n palindromic prime factors (counted with multiplicity), and no other prime factors. 2
1, 2, 4, 8, 88, 252, 2772, 82728, 2112, 4224, 8448, 236989632, 48384, 2977792 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 10 04:15 EST 2019. Contains 329885 sequences. (Running on oeis4.)