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.

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

Author

Viorel Nitica, May 26 2018

Keywords

• nonn
• base

Comments

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

Examples

			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.
		

Links

• 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.

Crossrefs

Subsequence of A067030.

Cf. A004086, A011541, A305131.

Programs

Mathematica

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 *)