A019566 The differences 1-1, 21-12, 321-123, ..., 10987654321-12345678910, 1110987654321-1234567891011, etc.

0, 9, 198, 3087, 41976, 530865, 6419754, 75308643, 864197532, -1358024589, -123580236690, -2345801446791, 775432077543108, 178553219976533007, 27956332009875522906, 3805734210999774512805, 481583522109989673502704, 58259362312008979572492603

Offset: 1

Author

Robert Dickau

Keywords

• sign
• base
• easy

Comments

All terms are divisible by 9, cf. A083449. There is an increasingly longer subsequence of negative terms starting at each power of 10, namely for indices n = 10..12, 100..123, 1000..1234, etc. - M. F. Hasler, Nov 02 2016

Gupta (1988) calls these "unique numbers".

References

• S. S. Gupta, Unique Numbers, Science Today, Jan 01 1988, India.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..350
• Shyam Sunder Gupta, Unique Numbers
• Shyam Sunder Gupta, On Some Marvellous Numbers of Kaprekar, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 9, 275-315. See Section 9.4.

Crossrefs

Cf. A000422, A007908.

Programs

Maple

u:= proc(n) u(n):= `if`(n=1, 1, parse(cat(u(n-1), n))) end:
d:= proc(n) d(n):= `if`(n=1, 1, parse(cat(n, d(n-1)))) end:
a:= n-> d(n)-u(n):
seq(a(n), n=1..20);  # Alois P. Heinz, Dec 06 2014

Mathematica

f[n_] := Block[ {a = "", k = 1}, While[k < n + 1, a = StringJoin[ ToString[k], a]; k++ ]; Return[ ToExpression[a] - ToExpression[ StringReverse[a]]]]; Table[ f[n], {n, 1, 17} ]

PARI

A = vector(25); c = 1; f = 1; for (i = 2, 9, c = 10*c + i; f = f + i*10^(i - 1); A[i] = (f - c)); for (i = 10, 25, c = 100*c + i; f = f + i*10^(2*i - 11);; A[i] = (f - c)); A \\ David Wasserman, Nov 09 2004

PARI

apply( {A019566(n)=A000422(n)-A007908(n)}, [1..22]) \\ Replacing code from Jan 13 2013, following a comment from Nov 02 2016. - M. F. Hasler, Nov 07 2020

Formula

a(n) = A000422(n) - A007908(n) = 9*A083449(n).

Extensions

More terms from Robert G. Wilson v, Jan 11 2002

More terms from David Wasserman, Nov 09 2004

Edited by N. J. A. Sloane, Nov 22 2020