A010344 -id:A010344 - cp's OEIS frontend

A010346 Base-5 Armstrong or narcissistic numbers (written in base 10).

1, 2, 3, 4, 13, 18, 28, 118, 289, 353, 419, 4890, 4891, 9113, 1874374, 338749352, 2415951874

Offset: 1

Zero would also satisfy the definition as the other single-digit terms, but here only positive numbers are considered. - M. F. Hasler, Nov 20 2019

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

Cf. A010345 (a(n) written in base 5).

In other bases: A010344 (base 4), A010348 (base 6), A010350 (base 7), A010354 (base 8), A010353 (base 9), A005188 (base 10), A161948 (base 11), A161949 (base 12), A161950 (base 13), A161951 (base 14), A161952 (base 15), A161953 (base 16).

A010346=select( is_A010346(n)={n==vecsum([d^#n|d<-n=digits(n,5)])}, [0..9999]) \\ This yields only terms < 10^4 (i.e., all but the last 3 terms), for illustration of is_A010346(). In older versions of PARI, use {n==sum(i=1,#n=digits(n,5),n[i]^#n)}. - M. F. Hasler, Nov 20 2019

Edited by Joseph Myers, Jun 28 2009

A010354 Base-8 Armstrong or narcissistic numbers (written in base 10).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 20, 52, 92, 133, 307, 432, 433, 16819, 17864, 17865, 24583, 25639, 212419, 906298, 906426, 938811, 1122179, 2087646, 3821955, 13606405, 40695508, 423056951, 637339524, 6710775966, 13892162580, 32298119799, 97095152738, 98250308556, 98317417420, 125586038802

Offset: 1

N. J. A. Sloane

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 20 2019

			From _M. F. Hasler_, Nov 20 2019: (Start)
20 = 24_8 (in base 8), and 2^2 + 4^2 = 20.
432 = 660_8, and 6^3 + 6^3 + 0^3 = 432; it's easy to see that 432 + 1 then also satisfies the equation, as for any term that is a multiple of 8. (End)

Joseph Myers, Table of n, a(n) for n = 1..62 (the full list of terms, from Winter)
Eric Weisstein's World of Mathematics, Narcissistic Number
D. T. Winter, Table of Armstrong Numbers

Cf. A010351 (a(n) written in base 8).

In other bases: A010344 (base 4), A010346 (base 5), A010348 (base 6), A010350 (base 7), A010353 (base 9), A005188 (base 10), A161948 (base 11), A161949 (base 12), A161950 (base 13), A161951 (base 14), A161952 (base 15), A161953 (base 16).

select( {is_A010354(n)=n==vecsum([d^#n|d<-n=digits(n,8)])}, [0..10^6]) \\ This gives only terms < 10^6, for illustration of is_A010354(). - M. F. Hasler, Nov 20 2019

from itertools import islice, combinations_with_replacement
def A010354_gen(): # generator of terms
    for k in range(1,30):
        a = tuple(i**k for i in range(8))
        yield from (x[0] for x in sorted(filter(lambda x:x[0] > 0 and tuple(int(d,8) for d in sorted(oct(x[0])[2:])) == x[1], \
                          ((sum(map(lambda y:a[y],b)),b) for b in combinations_with_replacement(range(8),k)))))
A010354_list = list(islice(A010354_gen(),20)) # Chai Wah Wu, Apr 20 2022

Edited by Joseph Myers, Jun 28 2009

A010348 Base-6 Armstrong or narcissistic numbers (written in base 10).

Original entry on oeis.org

1, 2, 3, 4, 5, 99, 190, 2292, 2293, 2324, 3432, 3433, 6197, 36140, 269458, 391907, 10067135, 2510142206, 2511720147, 3866632806, 3866632807, 3930544834, 4953134588, 5018649129, 6170640875, 124246559501, 4595333541803, 5341093125744, 5341093125745, 19418246235419

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 digits of n), but here only positive numbers are considered.
Terms a(n+1) = a(n) + 1 (n = 8, 11, 20, 28) correspond to solutions a(n) ending in a digit 0 in base 6, in which case a(n) + 1 also is a solution. (End)

Joseph Myers, Table of n, a(n) for n = 1..30 (the full list of terms, from Winter)
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. A010347 (a(n) written in base 6).

In other bases: A010344 (base 4), A010346 (base 5), A010350 (base 7), A010354 (base 8), A010353 (base 9), A005188 (base 10), A161948 (base 11), A161949 (base 12), A161950 (base 13), A161951 (base 14), A161952 (base 15), A161953 (base 16).

select( {is_A010348(n)=n==vecsum([d^#n|d<-n=digits(n,6)])}, [0..4e5\1]) \\ Note: this yields only terms < 10^6, for illustration of is_A010348(). - M. F. Hasler, Nov 20 2019

Edited by Joseph Myers, Jun 28 2009

A010350 Base-7 Armstrong or narcissistic numbers (written in base 10).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 10, 25, 32, 45, 133, 134, 152, 250, 3190, 3222, 3612, 3613, 4183, 9286, 35411, 191334, 193393, 376889, 535069, 794376, 8094840, 10883814, 16219922, 20496270, 32469576, 34403018, 416002778, 416352977, 420197083, 725781499, 1500022495, 15705029375, 15705029376, 28700208851

Offset: 1

N. J. A. Sloane

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 digits of n), but here only positive numbers are considered. - M. F. Hasler, Nov 20 2019

Joseph Myers, Table of n, a(n) for n = 1..59 (the full list of terms, from Winter)
Eric Weisstein's World of Mathematics, Narcissistic Number
D. T. Winter, Table of Armstrong Numbers

Cf. A010349 (a(n) written in base 7).

In other bases: A010344 (base 4), A010346 (base 5), A010348 (base 6), A010354 (base 8), A010353 (base 9), A005188 (base 10), A161948 (base 11), A161949 (base 12), A161950 (base 13), A161951 (base 14), A161952 (base 15), A161953 (base 16).

select( {is_A010350(n)=n==vecsum([d^#n|d<-n=digits(n,7)])}, [0..10^6]) \\ This yields only terms < 10^6, for illustration of is_A010350(). - M. F. Hasler, Nov 20 2019

Edited by Joseph Myers, Jun 28 2009

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

Original entry on oeis.org

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

A161949 Base-12 Armstrong or narcissistic numbers (written in base 10).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 29, 125, 811, 944, 1539, 28733, 193084, 887690, 2536330, 6884751, 17116683, 5145662993, 25022977605, 39989277598, 294245206529, 301149802206, 394317605931, 429649124722, 446779986586

Offset: 1

Joseph Myers, Jun 22 2009

Joseph Myers, Table of n, a(n) for n = 1..87 (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

In other bases: A010344 (base 4), A010346 (base 5), A010348 (base 6), A010350 (base 7), A010354 (base 8), A010353 (base 9), A005188 (base 10), A161948 (base 11), A161950 (base 13), A161951 (base 14), A161952 (base 15), A161953 (base 16).

Select[Range[10^7], # == Total[IntegerDigits[#, 12]^IntegerLength[#, 12]] &] (* Michael De Vlieger, Nov 04 2020 *)

A161953 Base-16 Armstrong or narcissistic numbers (written in base 10).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 342, 371, 520, 584, 645, 1189, 1456, 1457, 1547, 1611, 2240, 2241, 2458, 2729, 2755, 3240, 3689, 3744, 3745, 47314, 79225, 177922, 177954, 368764, 369788, 786656, 786657, 787680, 787681, 811239, 812263, 819424, 819425, 820448, 820449, 909360

Offset: 1

Joseph Myers, Jun 22 2009

Whenever 16|a(n) (n = 22, 26, 33, 41, 43, 47, 49, 51, 53, 61, 116, 149, 157, 196, 198, 204, 206, 243, 247), then a(n+1) = a(n) + 1. Zero also satisfies the definition (n = Sum_{i=1..k} d[i]^k where d[1..k] are the base-16 digits of n), but this sequence only considers positive terms. - M. F. Hasler, Nov 22 2019

			645 is in the sequence because 645 is 285 in hexadecimal and 2^3 + 8^3 + 5^3 = 645. (The exponent 3 is the number of hexadecimal digits.)

Joseph Myers, Table of n, a(n) for n=1..293 (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

In other bases: A010344 (base 4), A010346 (base 5), A010348 (base 6), A010350 (base 7), A010354 (base 8), A010353 (base 9), A005188 (base 10), A161948 (base 11), A161949 (base 12), A161950 (base 13), A161951 (base 14), A161952 (base 15).

Select[Range[10^7], # == Total[IntegerDigits[#, 16]^IntegerLength[#, 16]] &] (* Michael De Vlieger, Nov 04 2020 *)

isok(n) = {my(b=16, d=digits(n, b), e=#d); sum(k=1, #d, d[k]^e) == n;} \\ Michel Marcus, Feb 25 2019

select( is_A161953(n)={n==vecsum([d^#n|d<-n=digits(n,16)])}, [1..10^5]) \\ M. F. Hasler, Nov 22 2019

from itertools import islice, combinations_with_replacement
def A161953_gen(): # generator of terms
    for k in range(1,74):
        a = tuple(i**k for i in range(16))
        yield from (x[0] for x in sorted(filter(lambda x:x[0] > 0 and tuple(int(d,16) for d in sorted(hex(x[0])[2:])) == x[1], \
                          ((sum(map(lambda y:a[y],b)),b) for b in combinations_with_replacement(range(16),k)))))
A161953_list = list(islice(A161953_gen(),30)) # Chai Wah Wu, Apr 21 2022

Terms sorted in increasing order by Pontus von Brömssen, Mar 03 2019

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

Original entry on oeis.org

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

Joseph Myers, Jun 22 2009

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)

			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

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

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).

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

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

A161950 Base-13 Armstrong or narcissistic numbers (written in base 10).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 17, 45, 85, 98, 136, 160, 793, 794, 854, 1968, 8194, 62481, 167544, 167545, 294094, 320375, 323612, 325471, 325713, 350131, 365914, 2412003, 4861352, 21710514, 43757311, 43757312, 46299414, 51798568, 52994053

Offset: 1

Joseph Myers, Jun 22 2009

Joseph Myers, Table of n, a(n) for n = 1..201 (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

In other bases: A010344 (base 4), A010346 (base 5), A010348 (base 6), A010350 (base 7), A010354 (base 8), A010353 (base 9), A005188 (base 10), A161948 (base 11), A161949 (base 12), A161951 (base 14), A161952 (base 15), A161953 (base 16).

Select[Range[10^7], # == Total[IntegerDigits[#, 13]^IntegerLength[#, 13]] &] (* Michael De Vlieger, Nov 04 2020 *)

A161951 Base-14 Armstrong or narcissistic numbers (written in base 10).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 244, 793, 282007, 10362564, 1445712420, 29546248981, 164159496751, 342515735622, 359057049845, 216210334578515, 324075236456868, 338527182572746, 338609726265795, 382789516519507, 435198066019184, 526088332647250

Offset: 1

Joseph Myers, Jun 22 2009

Whenever 14|a(n) (n = 36, 46, 75, 77), then a(n+1) = a(n) + 1. Zero also satisfies the definition (n = Sum_{i=1..k} d[i]^k where d[1..k] are the base-14 digits of n), but this sequence only considers positive terms. - M. F. Hasler, Nov 22 2019

Joseph Myers, Table of n, a(n) for n=1..103 (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

In other bases: A010344 (base 4), A010346 (base 5), A010348 (base 6), A010350 (base 7), A010354 (base 8), A010353 (base 9), A005188 (base 10), A161948 (base 11), A161949 (base 12), A161950 (base 13), A161952 (base 15), A161953 (base 16).

Select[Range[2 * 10^7], # == Total[IntegerDigits[#, 14]^IntegerLength[#, 14]] &] (* Michael De Vlieger, Nov 04 2020 *)

select( is_A161951(n)={n==vecsum([d^#n|d<-n=digits(n,14)])}, [1..10^6\3]) \\ M. F. Hasler, Nov 22 2019

cp's OEIS Frontend

A010346 Base-5 Armstrong or narcissistic numbers (written in base 10).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Extensions

A010354 Base-8 Armstrong or narcissistic numbers (written in base 10).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Python

Extensions

A010348 Base-6 Armstrong or narcissistic numbers (written in base 10).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Extensions

A010350 Base-7 Armstrong or narcissistic numbers (written in base 10).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A161949 Base-12 Armstrong or narcissistic numbers (written in base 10).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A161953 Base-16 Armstrong or narcissistic numbers (written in base 10).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Python

Extensions

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

Views

Author