Viorel Nitica - cp's OEIS frontend

A306830 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, 10, 40, 81, 90, 100, 121, 160, 250, 252, 360, 400, 403, 484, 490, 574, 640, 736, 765, 810, 900, 976, 1000, 1008, 1089, 1207, 1210, 1300, 1458, 1462, 1600, 1612, 1729, 1855, 1936, 1944, 2268, 2296, 2430, 2500, 2520, 2668, 2701, 2944, 3025, 3154, 3478, 3600, 3627, 3640, 4000, 4030, 4032, 4275

Offset: 1

Viorel Nitica, Mar 12 2019

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

			The sum of the digits of 90 is 9 and (9+21)*reversal(9+21) = 30*3 = 90, so 90 is in the sequence.
The sum of the digits of 2268 is 18 and (18 + 18)*reversal(18 + 18) = 36*63 = 2268, so 2268 is in the sequence.

David A. Corneth, Table of n, a(n) for n = 1..17624
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.

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

okQ[k_] := Module[{s, j}, s = Total[IntegerDigits[k]]; For[j = 0, jJean-François Alcover, Mar 17 2019 *)

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

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

Name clarified by David A. Corneth, Apr 15 2019

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

A305130 Numbers k with the property that there exists a positive integer M, called multiplier, such that the sum of the digits of k times the multiplier added to the reversal of this product gives k.

Original entry on oeis.org

10, 11, 12, 18, 22, 33, 44, 55, 66, 77, 88, 99, 101, 110, 121, 132, 141, 165, 181, 201, 202, 221, 222, 261, 262, 282, 302, 303, 322, 323, 342, 343, 363, 403, 404, 423, 424, 444, 463, 483, 504, 505, 525, 545, 564, 584, 585, 605, 606, 645, 646, 666, 686, 706

Offset: 1

Viorel Nitica, May 26 2018

These numbers are related to the taxicab number 1729. This is why they might be called "additive Hardy-Ramanujan numbers".

			For k = 11 the sum of the digits is 2 and the multiplier is 5: 2 * 5 = 10 and 10 + 01 = 11.
For k = 747 the sum of the digits is 18 and the multiplier is 7: 18 * 7 = 126 and 126 + 621 = 747.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Viorel Nitica, About some relatives of the taxicab numbers, submitted to Journal of Integer Sequences, (2018). [Where these numbers are introduced.]
Viorel Niţică, Jeroz Makhania, About the Orbit Structure of Sequences of Maps of Integers, Symmetry (2019), Vol. 11, No. 11, 1374.

Subsequence of A067030.

Cf. A004086, A011541, A305131.

Block[{k, d, j}, Reap[Do[k = 1; d = Total@ IntegerDigits[i]; While[Nor[k > i, Set[j, # + IntegerReverse@ #] == i &[d k]], k++]; If[j == i, Sow[{i, k}]], {i, 720}]][[-1, 1, All, 1]] ] (* Michael De Vlieger, Jan 28 2020 *)

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.

Original entry on oeis.org

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

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A306830 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.

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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.

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A305130 Numbers k with the property that there exists a positive integer M, called multiplier, such that the sum of the digits of k times the multiplier added to the reversal of this product gives k.

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Comments

Examples

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Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI