A010352 -id:A010352 - cp's OEIS frontend

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

N. J. A. Sloane

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)

			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

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

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).

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

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

Edited by Joseph Myers, Jun 28 2009

A010343 Base-4 Armstrong or narcissistic positive numbers.

Original entry on oeis.org

1, 2, 3, 130, 131, 203, 223, 313, 332, 1103, 3303

Offset: 1

N. J. A. Sloane

Eric Weisstein's World of Mathematics, Narcissistic Number
D. T. Winter, Table of Armstrong Numbers

Cf. A010344 (a(n) written in base 10).

In other bases: A010345 (base 5), A010347 (base 6), A010349 (base 7), A010351 (base 8), A010352 (base 9), A005188 (base 10).

[fromdigits(digits(n,4))|n<-A010344] \\ M. F. Hasler, Nov 18 2019

a(n) = A007090(A010344(n)). - M. F. Hasler, Nov 18 2019

Edited by Joseph Myers, Jun 28 2009

"Positive" added to definition. - N. J. A. Sloane, Nov 18 2019

A010345 Base-5 Armstrong or narcissistic numbers, written in base 5.

Original entry on oeis.org

1, 2, 3, 4, 23, 33, 103, 433, 2124, 2403, 3134, 124030, 124031, 242423, 434434444, 1143204434402, 14421440424444

Offset: 1

N. J. A. Sloane

Also called Perfect Digital Invariant (PDI). When a(n) ends in 0, then a(n+1) = a(n) + 1 is also in the sequence, but in this base this only happens once. Zero would also satisfy the definition (n = Sum_{i=1..k} d[i]^k where d[1..k] are the base-5 digits of n), like the other single-digit terms. - M. F. Hasler, Nov 18 2019

The property of being an Armstrong number is an arithmetic property (like the number of divisors function) and is usually restricted to positive numbers. - N. J. A. Sloane, Nov 29 2019

Gordon L. Miller and Mary T. Whalen, Armstrong Numbers: 153 = 1^3 + 5^3 + 3^3, Fibonacci Quarterly, 30-3 (1992), 221-224. See Table 4 p. 223.
Eric Weisstein's World of Mathematics, Narcissistic Number
D. T. Winter, Table of Armstrong Numbers (latest backup on web.archive.org from Jan. 2010; page no longer available), published not later than Aug. 2003.

Cf. A010346 (a(n) written in base 10).

In other bases: A010343 (base 4), A010347 (base 6), A010349 (base 7), A010351 (base 8), A010352 (base 9), A005188 (base 10).

A010345=[fromdigits(digits(n,5))|n<-A010346] \\ Assumes the vector A010346 defined, see there for code. - M. F. Hasler, Nov 18 2019

Edited by Joseph Myers, Jun 28 2009

A010347 Base-6 Armstrong or narcissistic numbers, written in base 6.

Original entry on oeis.org

1, 2, 3, 4, 5, 243, 514, 14340, 14341, 14432, 23520, 23521, 44405, 435152, 5435254, 12222215, 555435035, 1053025020422, 1053122514003, 1435403205450, 1435403205451, 1450005114454, 2135254510352, 2145555022413, 2500150125455, 133024510545125

Offset: 1

N. J. A. Sloane

From M. F. Hasler, Nov 18 2019: (Start)
Whenever a(n) ends in 0 (n = 8, 11, 20, 28), then a(n+1) = a(n) + 1 also satisfies the definition.
Like the other single-digit terms, zero would satisfy the definition (n = Sum_{i=1..k} d[i]^k where d[1..k] are the base 6 digits of n), but here only positive numbers are considered. (End)

Joseph Myers, Table of n, a(n) for n = 1..30 (the full list of terms, from Winter)
Gordon L. Miller and Mary T. Whalen, Armstrong Numbers: 153 = 1^3 + 5^3 + 3^3, Fibonacci Quarterly, 30-3 (1992), 221-224.
Eric Weisstein's World of Mathematics, Narcissistic Number
D. T. Winter, Table of Armstrong Numbers (latest backup on web.archive.org from Jan. 2010; page no longer available), published not later than Aug. 2003.

Cf. A010348 (a(n) written in base 10).

In other bases: A010343 (base 4), A010345 (base 5), A010349 (base 7), A010351 (base 8), A010352 (base 9), A005188 (base 10).

A010347=[fromdigits(digits(n,6))|n<-A010348] \\ M. F. Hasler, Nov 18 2019

select( is_A010347(n)={vecmax(n=digits(n+!n))<6 && vecsum([d^#n|d<-n])==fromdigits(n,6)}, [0..10^5]) \\ M. F. Hasler, Nov 20 2019

Edited by Joseph Myers, Jun 28 2009

A010349 Base-7 Armstrong or narcissistic numbers, written in base 7.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 13, 34, 44, 63, 250, 251, 305, 505, 12205, 12252, 13350, 13351, 15124, 36034, 205145, 1424553, 1433554, 3126542, 4355653, 6515652, 125543055, 161340144, 254603255, 336133614, 542662326, 565264226, 13210652042, 13213641035, 13261421245, 23662020022, 52112660266

Offset: 1

N. J. A. Sloane

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 where d[1..k] are the base 7 digits of n), but here only positive numbers are considered.
Whenever a(n) ends in zero (n = 11, 17, 22, 38, 57), then a(n+1) = a(n) + 1 is also a solution to the above equation. (End)

Joseph Myers, Table of n, a(n) for n = 1..59 (the full list of terms, from Winter)
Gordon L. Miller and Mary T. Whalen, Armstrong Numbers: 153 = 1^3 + 5^3 + 3^3, Fibonacci Quarterly, 30-3 (1992), 221-224.
Eric Weisstein's World of Mathematics, Narcissistic Number
D. T. Winter, Table of Armstrong Numbers (latest backup on web.archive.org from Jan. 2010; page no longer available), published not later than Aug. 2003.

Cf. A010350 (a(n) written in base 10).

In other bases: A010343 (base 4), A010345 (base 5), A010347 (base 6), A010351 (base 8), A010352 (base 9), A005188 (base 10).

[fromdigits(digits(n,7))|n<-A010350] \\ M. F. Hasler, Nov 18 2019

Edited by Joseph Myers, Jun 28 2009

A010351 Base-8 Armstrong or narcissistic numbers, written in base 8.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 24, 64, 134, 205, 463, 660, 661, 40663, 42710, 42711, 60007, 62047, 636703, 3352072, 3352272, 3451473, 4217603, 7755336, 16450603, 63717005, 233173324, 3115653067, 4577203604, 61777450236, 147402312024

Offset: 1

N. J. A. Sloane

Whenever a term ends in 0, then a(n+1) = a(n) + 1 is also a term. 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-8 digits of n), but here only positive numbers are considered. - M. F. Hasler, Nov 18 2019

			432 = 660_8 (= 6*8^2 + 6*8^1 + 0*8^0), and 6^3 + 6^3 + 0^3 = 432, therefore 660 is in the sequence. It's easy to see that 432 + 1 then also satisfies the equation, as for any term that is a multiple of 8. - _M. F. Hasler_, Nov 21 2019

Joseph Myers, Table of n, a(n) for n = 1..62 (the full list of terms, from Winter)
Gordon L. Miller and Mary T. Whalen, Armstrong Numbers: 153 = 1^3 + 5^3 + 3^3, Fibonacci Quarterly, 30-3 (1992), 221-224.
Eric Weisstein's World of Mathematics, Narcissistic Number
D. T. Winter, Table of Armstrong Numbers (latest backup on web.archive.org from Jan. 2010; page no longer available), published not later than Aug. 2003.

Cf. A010354 (a(n) written in base 10).

In other bases: A010343 (base 4), A010345 (base 5), A010347 (base 6), A010349 (base 7), A010352 (base 9), A005188 (base 10).

[fromdigits(digits(n,8))|n<-A010354] \\ M. F. Hasler, Nov 18 2019

Edited by Joseph Myers, Jun 28 2009

A329817 a(n) is the maximum number of digits that must be checked to obtain all Armstrong numbers in base n.

Original entry on oeis.org

2, 7, 13, 20, 28, 35, 43, 52, 60, 69, 78, 87, 97, 106, 116, 126, 136, 146, 156, 167, 177, 188, 199, 209, 220, 231, 242, 253, 264, 276, 287, 298, 310, 321, 333, 345, 356, 368, 380, 392, 404, 416, 428, 440, 452, 464, 476, 489, 501, 513, 526, 538, 551, 563, 576, 588

Offset: 2

Michel Marcus, Nov 22 2019

Nick Hobson, Table of n, a(n) for n = 2..10000
Nick Hobson, C program
Gordon L. Miller and Mary T. Whalen, Armstrong Numbers: 153 = 1^3 + 5^3 + 3^3, Fibonacci Quarterly, 30-3 (1992), 221-224. See Table 3 p. 222.

Cf. A010343 (base 4), A010345 (base 5), A010347 (base 6), A010349 (base 7), A010351 (base 8), A010352 (base 9), A005188 (base 10).

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a[b_] := Floor[x /. NSolve[(b-1)^x x == b^(x-1) && x>1, x, Reals][[1]]]; a /@ Range[2, 57] (* Giovanni Resta, Nov 22 2019 *)

More terms from Giovanni Resta, Nov 22 2019

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

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A010343 Base-4 Armstrong or narcissistic positive numbers.

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A010345 Base-5 Armstrong or narcissistic numbers, written in base 5.

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A010347 Base-6 Armstrong or narcissistic numbers, written in base 6.

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A010349 Base-7 Armstrong or narcissistic numbers, written in base 7.

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A010351 Base-8 Armstrong or narcissistic numbers, written in base 8.

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A329817 a(n) is the maximum number of digits that must be checked to obtain all Armstrong numbers in base n.

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