A247826 Numbers k with at least one nonpalindromic divisor such that the sum of sigma(d) = the sum of sigma(reverse(d)), where d runs over the divisors of k.
16331, 98639, 161051, 179641, 272802, 1206611, 1226221, 1649431, 1794971, 6061206, 6177253, 8792914, 13488431, 16266151, 29498851, 61887064, 66673266, 69536743, 123848321, 440664044
Offset: 1
Examples
Divisors of 16331 are 1, 7, 2333, 16331; sigma(1) = 1, sigma(7) = 8, sigma(2333) = 2334, sigma(16331) = 18672 and 1 + 8 + 2334 + 18672 = 21015. sigma(1) = 1, sigma(7) = 8, sigma(3332) = 7182, sigma(13361) = 13824 and 1 + 8 + 7182 + 13824 = 21015. Divisors of 98639 are 1, 98639; sigma(1) = 1, sigma(98639) = 98640, and 1 + 98640 = 98641. sigma(1) = 1, sigma(93689) = 98640, and 1 + 98640 = 98641.
Programs
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Maple
with(numtheory); T:=proc(h) local x,y,w; x:=h; y:=0; for w from 1 to ilog10(h)+1 do y:=10*y+(x mod 10); x:=trunc(x/10); od; y; end: P:=proc(q) local a,b,c,k,n,ok; for n from 1 to q do a:=divisors(n); b:=0; c:=0; ok:=0; for k from 1 to nops(a) do b:=b+sigma(T(a[k])); c:=c+sigma(a[k]); if a[k]<>T(a[k]) then ok:=1; fi; od; if ok=1 and c=b then print(n); fi; od; end: P(10^9);
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PARI
rev(n) = subst(Polrev(digits(n)), x, 10); 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 -
PARI
rev(n)=r="";d=digits(n);for(i=1,#d,r=concat(Str(d[i]),r));eval(r) 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
Extensions
a(6)-a(12) from Michel Marcus, Oct 04 2014
Definition edited by Derek Orr, Oct 26 2014
a(13)-a(20) from Jinyuan Wang, Apr 08 2025