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

Author

Viorel Nitica, May 26 2018

Keywords

• nonn
• base

Comments

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

Examples

			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.
		

Links

• 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.

Crossrefs

Subsequence of A005349 (Niven numbers).

Cf. A004086, A011541, A061205, A305130.

Programs

PARI

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