A305130 -id:A305130 - cp's OEIS frontend

A305131 Numbers k with the property that there exists a positive integer multiplier M such that M times the sum of the digits of k, multiplied further by the reversal of this product, gives k.

1, 10, 40, 81, 100, 400, 640, 736, 810, 1000, 1300, 1458, 1729, 1944, 2268, 2430, 3640, 4000, 6400, 7360, 7744, 8100, 10000, 12070, 12100, 13000, 14580, 16120, 17290, 19440, 22680, 23632, 24300, 27010, 30250, 31003, 36400, 38152, 40000, 42282, 51142, 63504

Offset: 1

Viorel Nitica, May 26 2018

These numbers are related to the taxicab number 1729, which has multiplier 1. This is why they might be called "multiplicative Hardy-Ramanujan numbers".

If a(n) is in the sequence, then 10 * a(n) is also in the sequence, with the multiplier 10 times larger. We could call primitive the terms not of this form. Primitive terms which end in 0 are 40, 640, 1300, 2430, 3640, 12070, 12100, 16120, 27010, ... - M. F. Hasler, May 27 2018

			For k = 1729 the sum of the digits is 19 and M = 1: 19 * 91 = 1729.
For k = 122512 the sum of the digits is 13 and M = 31: 13 * 31 = 403 and 403 * 304 = 122512.

Viorel Nitica, About some relatives of the taxicab numbers, arXiv:1805.10739 [math.NT], 2018; J. of Int. Seq., 21 (2018), Article 18.9.4. [Where these numbers are introduced.]
Viorel Nitica, Andrei Török, About Some Relatives of Palindromes, arXiv:1908.00713 [math.NT], 2019.
Viorel Niţică, Jeroz Makhania, About the Orbit Structure of Sequences of Maps of Integers, Symmetry (2019), Vol. 11, No. 11, 1374.

Subsequence of A005349 (Niven numbers).

Cf. A004086, A011541, A061205, A305130.

select( is(n,s=sumdigits(n))=n&&!frac(n/=s)&&fordiv(n,M,fromdigits(Vecrev(digits(s*M)))*M==n&&return(1)), [0..10^5]) \\ M. F. Hasler, May 27 2018

A323190 Integers k for which there exists an integer j such that s(k) + j + reversal(s(k) + j) = k where s(k) is the sum of digits of k.

Original entry on oeis.org

0, 10, 11, 12, 14, 16, 18, 22, 33, 44, 55, 66, 77, 88, 99, 101, 110, 121, 132, 141, 143, 154, 161, 165, 176, 181, 187, 198, 201, 202, 221, 222, 241, 242, 261, 262, 281, 282, 302, 303, 322, 323, 342, 343, 362, 363, 382, 383, 403, 404, 423, 424, 443, 444, 463

Offset: 1

Viorel Nitica, Jan 06 2019

Theorem: all palindromes that have an even number of digits and all palindromes that have an odd number of digits and the digit in the middle is even are in this sequence.

			k=10 is a term because a solution exists with j=4: s(10)=1, s(k) + j + reversal(s(k) + j) = 1 + 4 + reversal(1 + 4) = 10.

Viorel Niţică, Jeroz Makhania, About the Orbit Structure of Sequences of Maps of Integers, Symmetry (2019), Vol. 11, No. 11, 1374.

Cf. A007953 (sum of digits), A004086 (reversal).
A305130 is a subsequence.
This sequence is a subsequence of A067030.

package com.company;
public class Main {
public static void main(String args[]) {
int counter=1;
for (int i = 0; i < 10000; i++) {
for (int j = 0; j < 10000; j++) {
int sumPlus = sumDigits(i) + j;
int check = sumPlus + reverse(sumPlus);
if (check == i) {
System.out.println(String.format("n(%d)=%d,a=%d", counter,i, j));
counter++;
break;
}
}
}
System.out.println(String.format("%d", counter));
}
public static int reverse(int x) {
String s = String.valueOf(x);
StringBuilder sb = new StringBuilder();
sb.append(s);
sb.reverse();
return Integer.parseInt(sb.toString());
}
public static int sumDigits(int x) {
int result = 0;
while (x > 0) {
result += x % 10;
x = x / 10;
}
return result;
}
}

ok[n_] := Block[{s = Total@ IntegerDigits@ n}, Select[Range[0, n], s + # + FromDigits@ Reverse@ IntegerDigits[s + #] == n &, 1] != {}]; Select[ Range[0, 1000], ok] (* Giovanni Resta, Feb 19 2019 *)

a(30)-a(55) from Giovanni Resta, Feb 19 2019

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A305131 Numbers k with the property that there exists a positive integer multiplier M such that M times the sum of the digits of k, multiplied further by the reversal of this product, gives k.

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