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%I #18 Oct 28 2014 05:10:33
%S 16331,98639,161051,179641,272802,1206611,1226221,1649431,1794971,
%T 6061206,6177253,8792914
%N Numbers n with at least one nonpalindromic divisor such that the sum of sigma(x) = the sum of sigma(reverse(x)), where x runs over the divisors of n.
%e Divisors of 16331 are 1, 7, 2333, 16331;
%e sigma(1) = 1, sigma(7) = 8, sigma(2333) = 2334, sigma(16331) = 18672 and 1 + 8 + 2334 + 18672 = 21015.
%e sigma(1) = 1, sigma(7) = 8, sigma(3332) = 7182, sigma(13361) = 13824 and 1 + 8 + 7182 + 13824 = 21015.
%e Divisors of 98639 are 1, 98639;
%e sigma(1) = 1, sigma(98639) = 98640, and 1 + 98640 = 98641.
%e sigma(1) = 1, sigma(93689) = 98640, and 1 + 98640 = 98641.
%p with(numtheory); T:=proc(h) local x,y,w; x:=h; y:=0;
%p for w from 1 to ilog10(h)+1 do y:=10*y+(x mod 10); x:=trunc(x/10); od; y; end:
%p P:=proc(q) local a,b,c,k,n,ok;
%p for n from 1 to q do a:=divisors(n); b:=0; c:=0; ok:=0;
%p for k from 1 to nops(a) do b:=b+sigma(T(a[k])); c:=c+sigma(a[k]);
%p if a[k]<>T(a[k]) then ok:=1; fi; od;
%p if ok=1 and c=b then print(n); fi; od; end: P(10^9);
%o (PARI) rev(n) = subst(Polrev(digits(n)), x, 10);
%o isok(n) = {nbpal = sumdiv(n, d, rev(d)==d); if (nbpal == numdiv(n), return(0)); sumdiv(n, d, sigma(d)) == sumdiv(n, d, sigma(rev(d)));} \\ _Michel Marcus_, Oct 04 2014
%o (PARI) rev(n)=r="";d=digits(n);for(i=1,#d,r=concat(Str(d[i]),r));eval(r)
%o for(n=1,10^6,D=divisors(n);c=0;for(k=1,#D,if(D[k]==rev(D[k]),c++));if(c!=#D,if(sumdiv(n,i,sigma(i))==sumdiv(n,j,sigma(rev(j))),print1(n,", ")))) \\ _Derek Orr_, Oct 26 2014
%Y Cf. A000203, A196677, A246545.
%K nonn,more,base,hard
%O 1,1
%A _Paolo P. Lava_, Sep 30 2014
%E a(6)-a(12) from _Michel Marcus_, Oct 04 2014
%E Definition edited by _Derek Orr_, Oct 26 2014
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