login
Integers k for which there exists a nonnegative integer j such that (s(k) + j) * reversal(s(k) + j) = k where s(k) is the sum of digits of k.
1

%I #54 Dec 10 2023 18:08:32

%S 1,10,40,81,90,100,121,160,250,252,360,400,403,484,490,574,640,736,

%T 765,810,900,976,1000,1008,1089,1207,1210,1300,1458,1462,1600,1612,

%U 1729,1855,1936,1944,2268,2296,2430,2500,2520,2668,2701,2944,3025,3154,3478,3600,3627,3640,4000,4030,4032,4275

%N Integers k for which there exists a nonnegative integer j such that (s(k) + j) * reversal(s(k) + j) = k where s(k) is the sum of digits of k.

%C Subsequence of A305231. This sequence excludes for example 4 = (s(4) + (-2)) * (s(4) + (-2)) from that sequence. - _David A. Corneth_, Apr 15 2019

%H David A. Corneth, <a href="/A306830/b306830.txt">Table of n, a(n) for n = 1..17624</a>

%H Viorel Nitica, Andrei Török, <a href="https://arxiv.org/abs/1908.00713">About Some Relatives of Palindromes</a>, arXiv:1908.00713 [math.NT], 2019.

%H Viorel Niţică, Jeroz Makhania, <a href="https://doi.org/10.3390/sym11111374">About the Orbit Structure of Sequences of Maps of Integers</a>, Symmetry (2019), Vol. 11, No. 11, 1374.

%e The sum of the digits of 90 is 9 and (9+21)*reversal(9+21) = 30*3 = 90, so 90 is in the sequence.

%e The sum of the digits of 2268 is 18 and (18 + 18)*reversal(18 + 18) = 36*63 = 2268, so 2268 is in the sequence.

%t okQ[k_] := Module[{s, j}, s = Total[IntegerDigits[k]]; For[j = 0, j<k, j++, If[(s+j)IntegerReverse[s+j] == k, Print["k = ", k , ", j = ", j]; Return[ True]]]; False]; Reap[Do[If[okQ[k], Sow[k]], {k, 1, 4275}]][[2, 1]] (* _Jean-François Alcover_, Mar 17 2019 *)

%o (PARI) isok(k) = {my(s = sumdigits(k)); fordiv(k, d, if ((d>=s) && (k/d == fromdigits(Vecrev(digits(d)))), return (1));); return (0);} \\ _Michel Marcus_, Mar 13 2019

%o (PARI) upto(n) = {my(res = List([1, 10, 40, 81, 90]), m = 0); for(i = 10, 10*sqrtint(n), revi = fromdigits(Vecrev(digits(i))); if(revi <= i && i * revi <= n, m = i; listput(res, i * revi); ) ); q = #res; for(i = 1, #q, for(j = 1, logint(n \ res[i], 10), listput(res, res[i]*10^j); ) ); listsort(res, 1); res } \\ _David A. Corneth_, Apr 15 2019

%Y Cf. A004086 (reversal), A007953 (sum of digits), A027750 (divisors), A305231.

%K nonn,base

%O 1,2

%A _Viorel Nitica_, Mar 12 2019

%E Name clarified by _David A. Corneth_, Apr 15 2019