A257973 - OEIS

%I #16 Jun 07 2015 17:52:54

%S 323,767,949,989,10001,11111,11611,11911,12121,12521,13031,13631,

%T 14041,14141,14941,15151,15751,16861,17071,17371,17671,17771,18581,

%U 18881,19091,31813,32023,32723,33133,33233,33433,33833,34043,34343,34943,35953,36163,36763,37973

%N Palindromes with no palindromic aliquot parts except 1.

%C The definition implies that the prime factors are not palindromes. The converse is not true. For instance, prime factors of 1681861 are 17, 19, 41, 127, which are not palindromes, but its aliquot parts are 1, 17, 19, 41, 127, 323, 697, 779, 2159, 2413, 5207, 13243, 41021, 88519, 98933, and 323 is a palindrome.

%C Mainly squarefree numbers. Not squarefree are 36163 = 29^2*43, 39593 = 17^2*137, 71117 = 19^2*197, 94249 = 307^2, 95659 = 17^2*331, 1069601 = 13^2*6329, etc.

%C There are large jumps: from a(4) = 989 to a(5) = 10001; from a(97) = 99499 to a(98) = 1005001; from a(380) = 1996991 to a(381) = 3004003; from a(677) = 3995993 to a(678) = 7002007; from a(985) = 7999997 to a(986) = 9001009; etc.

%C The first two squares in the sequence are 94249 = 307^2 and 1230127210321 = 509^2 * 2179^2. - _Giovanni Resta_, May 15 2015

%H Paolo P. Lava, <a href="/A257973/b257973.txt">Table of n, a(n) for n = 1..1000</a>

%e Aliquot parts of 323 are 1, 17, 19 that are not palindromes, except 1.

%e Aliquot parts of 767 are 1, 13, 59 that are not palindromes, except 1.

%e Aliquot parts of 12121 are 1, 17, 23, 31, 391, 527, 713 that are not palindromes, except 1.

%p with(numtheory): T:=proc(w) local x,y,z; x:=w; y:=0;

%p for z from 1 to ilog10(x)+1 do y:=10*y+(x mod 10);

%p x:=trunc(x/10); od; y; end:

%p P:=proc(q) local a,j,k,ok,n;

%p for n from 2 to q do if n=T(n) then a:=ifactors(n)[2]; ok:=1;

%p for k from 1 to nops(a) do if a[k][1]=T(a[k][1]) then ok:=0; break; fi;

%p od; if ok=1 then print(n); fi; fi; od; end: P(10^9);

%Y Cf. A002113.

%K nonn,easy,base

%O 1,1

%A _Paolo P. Lava_, May 15 2015