A260705 Least integer k such that the set of the divisors of k contains exactly n pairs of numbers having the following property: for each pair of two distinct divisors, the reversal of one is equal to the other.

84, 168, 336, 1008, 3024, 5544, 11088, 16632, 33264, 49896, 99792, 182952, 365904, 249480, 498960, 1097712, 2162160, 3359664, 1846152, 3027024, 5538456, 6054048, 9081072, 9230760, 14270256, 19891872, 20307672, 25197480, 33297264, 45405360, 55135080, 71351280

Offset: 1

Author

Michel Lagneau, Nov 17 2015

Keywords

• nonn
• base

Comments

It seems that a(n)==0 (mod 84).

Additional terms with n > 75: a(77) = 15455984544, a(80) = 27719972280, a(83) = 22439977560, a(84) = 18479981520, a(86) = 28559971440. - Lars Blomberg, Jan 04 2016

Examples

			a(4)=1008 because the set of the divisors {1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 18, 21, 24, 28, 36, 42, 48, 56, 63, 72, 84, 112, 126, 144, 168, 252, 336, 504, 1008} contains 4 pairs (12, 21), (24, 42), (36, 63) and (48, 84) with the property 21 = reversal(12), 42 = reversal(24), 63 = reversal(36) and 84 = reversal(48).
		

Links

• Lars Blomberg, Table of n, a(n) for n = 1..75.

Crossrefs

Cf. A000005, A027750, A083815, A206703, A260704.

Programs

Maple

with(numtheory):nn:=10^8:
for n from 1 to 16 do:
ii:=0:
for m from 1 to nn while(ii=0) do:
it:=0:d:=divisors(m):d0:=nops(d):
  for i from 1 to d0 do:
   dd:=d[i]:y:=convert(dd,base,10):n1:=length(dd):
   s:=sum('y[j]*10^(n1-j)', 'j'=1..n1):
    for k from i+1 to d0 do:
     if s=d[k]
     then
     it:=it+1:
     else fi:
    od:
    od:
    if it=n
    then
    ii:=1:printf("%d %d \n",n,m):
    else fi:
od:
od:

PARI

nbr(vd) = {nb = 0; for (j=1, #vd, da = vd[j]; rda = eval(concat(Vecrev(Str(da)))); rrda = eval(concat(Vecrev(Str(rda)))); if ((da != rda) && vecsearch(vd,rda) && (da == rrda), nb++);); nb/2;}
a(n) = {k=1; while (nbrp(divisors(k)) != n, k++); k;} \\ Michel Marcus, Dec 27 2015

Extensions

a(14)-a(15) corrected by Lars Blomberg, Dec 27 2015

a(7), a(19), a(20) corrected and a(21)-a(32) added by Lars Blomberg, Jan 04 2016