A161948 Base-11 Armstrong or narcissistic numbers (written in base 10).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 61, 72, 126, 370, 855, 1161, 1216, 1280, 10657, 16841, 16842, 17864, 17865, 36949, 36950, 63684, 66324, 71217, 90120, 99594, 99595, 141424, 157383, 1165098, 1165099, 5611015, 11959539, 46478562, 203821954, 210315331, 397800208, 826098079, 1308772162, 1399714480

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 11 digits of n), but here only positive numbers are considered.

Terms a(n+1) = a(n) + 1 (n = 20, 22, 24, 30, 34, 56, 67, 57, 195, ...) correspond to solutions a(n) that are multiples of 11, in which case a(n) + 1 is also a solution. (End)

Links

Joseph Myers, Table of n, a(n) for n = 1..134 (the full list of terms, from Winter)

Henk Koppelaar and Peyman Nasehpour, On Hardy's Apology Numbers, arXiv:2008.08187 [math.NT], 2020.

Eric Weisstein's World of Mathematics, Narcissistic Number

D. T. Winter, Table of Armstrong Numbers

Example

16841 = 11720_11 (= 1*11^4 + 1*11^3 + 7*11^2 + 2*11^1 + 0*11^0) = 1^5 + 1^5 + 7^5 + 2^5 + 0^5. It's easy to see that 16841 + 1 then also satisfies this relation, as for all terms that are multiples of 11. - M. F. Hasler, Nov 20 2019

Mathematica

Select[Range[10^7], # == Total[IntegerDigits[#, 11]^IntegerLength[#, 11]] &] (* Michael De Vlieger, Nov 04 2020 *)

Prog

(PARI) select( {is_A161948(n)=n==vecsum([d^#n|d<-n=digits(n, 11)])}, [0..10^5]) \\ This gives only terms < 10^5, for illustration of is_A161948(). - M. F. Hasler, Nov 20 2019

Crossrefs

In other bases: A010344 (base 4), A010346 (base 5), A010348 (base 6), A010350 (base 7), A010354 (base 8), A010353 (base 9), A005188 (base 10), A161949 (base 12), A161950 (base 13), A161951 (base 14), A161952 (base 15), A161953 (base 16).

Sequence in context: A161980 A183533 A132576 * A258785 A004861 A178404

Adjacent sequences: A161945 A161946 A161947 * A161949 A161950 A161951

Keyword

base,fini,full,nonn

Author

Joseph Myers, Jun 22 2009

Status

approved