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Numbers that divide the reverse of the sum of their aliquot parts.
2

%I #30 Sep 08 2022 08:46:11

%S 1,6,2274,44304,229974,498906,4177662,20671542,22999974,41673714,

%T 73687923,403999652,479444901,4158499614,27378395352,209659386726,

%U 216276435966,229999999974,406406685462,922964834547

%N Numbers that divide the reverse of the sum of their aliquot parts.

%C Noting 2274, 229974, and 22999974, 23*10^n-26 is a term herein for any n in A253968. - _Hans Havermann_, Jan 24 2015

%C Additionally, 404*10^(6*n)-348 is a term herein if this is 28 times a prime. Three such numbers are known: n = 1, 10, and 1608. - _Hans Havermann_, Jan 28 2015

%C a(21) > 10^12. - _Giovanni Resta_, May 09 2015

%e sigma(1) - 1 = 0, Rev(0) = 0 and 0 / 1 = 0.

%e sigma(6) - 6 = 6, Rev(6) = 6 and 6 / 6 = 1.

%e sigma(2274) - 2274 = 2286, Rev(2286) = 6822 and 6822 / 2274 = 3.

%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); x:=trunc(x/10);

%p od; y; end:

%p P:=proc(q) local n; for n from 1 to q do

%p if type(T(sigma(n)-n)/n,integer) then print(n);

%p fi; od; end: P(10^9);

%t fQ[n_] := Mod[ FromDigits@ Reverse@ IntegerDigits[ DivisorSigma[1, n] - n], n] == 0; k = 1; lst = {}; While[k < 1000000001, If[fQ@ k, AppendTo[lst, k]]; k++]; lst (* _Robert G. Wilson v_, Jan 28 2015 *)

%o (PARI) rev(n) = subst(Polrev(digits(n)), x, 10);

%o isok(n) = rev(sigma(n)-n) % n == 0; \\ _Michel Marcus_, Jan 25 2015

%o (Magma) [n: n in [1..10^7] | Seqint(Reverse(Intseq(SumOfDivisors(n)-n))) mod n eq 0]; // _Vincenzo Librandi_, May 09 2015

%Y Cf. A000203, A001065, A253968, A254004.

%K nonn,base,more

%O 1,2

%A _Paolo P. Lava_, Jan 22 2015

%E More terms from _Hans Havermann_, Jan 24 2015

%E a(13) from _Robert G. Wilson v_, Jan 29 2015

%E a(14)-a(20) from _Giovanni Resta_, May 09 2015