A010353 Base-9 Armstrong or narcissistic numbers (written in base 10).

1, 2, 3, 4, 5, 6, 7, 8, 41, 50, 126, 127, 468, 469, 1824, 8052, 8295, 9857, 1198372, 3357009, 3357010, 6287267, 156608073, 156608074, 403584750, 403584751, 586638974, 3302332571, 42256814922, 42256814923, 114842637961, 155896317510, 552468844242, 552468844243, 647871937482, 686031429775

Offset

1,2

Comments

From M. F. Hasler, Nov 20 2019: (Start)

Like the other single-digit terms, zero would satisfy the definition (n = Sum_{i=1..k} d[i]^k when d[1..k] are the base 9 digits of n), but here only positive numbers are considered.

Terms a(n+1) = a(n) + 1 (n = 11, 13, 20, 23, 25, 29, 33, 48, 51, 57) correspond to solutions a(n) that are multiples of 9, in which case a(n) + 1 is also a solution. (End)

Links

Joseph Myers, Table of n, a(n) for n = 1..58 (the full list of terms, from Winter)

René-Louis Clerc, Perfect r-narcissistic numbers in any base, hal-04376934, 2024.

Eric Weisstein's World of Mathematics, Narcissistic Number

D. T. Winter, Table of Armstrong Numbers

Example

126 = 150_9 (= 1*9^2 + 5*9^1 + 0*9^0) = 1^3 + 5^3 + 0^3. It is easy to see that 126 + 1 then also satisfies this relation, as for all other terms that are multiples of 9. - M. F. Hasler, Nov 20 2019

Mathematica

Select[Range[9^7], # == Total[IntegerDigits[#, 9]^IntegerLength[#, 9]] &] (* Michael De Vlieger, Jan 17 2024 *)

Prog

(PARI) select( {is_A010353(n)=n==vecsum([d^#n|d<-n=digits(n, 9)])}, [0..10^4]) \\ This gives only terms < 10^6, for illustration of is_A010353(). - M. F. Hasler, Nov 20 2019

Crossrefs

Cf. A010352 (a(n) written in base 9).

In other bases: A010344 (base 4), A010346 (base 5), A010348 (base 6), A010350 (base 7), A010354 (base 8), A005188 (base 10), A161948 (base 11), A161949 (base 12), A161950 (base 13), A161951 (base 14), A161952 (base 15), A161953 (base 16).

Sequence in context: A004848 A183531 A024652 * A010352 A024653 A024654

Adjacent sequences: A010350 A010351 A010352 * A010354 A010355 A010356

Keyword

base,fini,full,nonn

Author

N. J. A. Sloane

Extensions

Edited by Joseph Myers, Jun 28 2009

Status

approved