A005188 -id:A005188 - cp's OEIS frontend

A256360 Numbers that are multiple-digit narcissistic numbers in exactly one base.

5, 8, 10, 13, 18, 20, 25, 26, 32, 35, 37, 40, 41, 43, 52, 53, 55, 58, 61, 62, 65, 68, 72, 80, 82, 83, 90, 92, 97, 98, 99, 101, 104, 109, 113, 117, 118, 122, 127, 128, 134, 146, 148, 152, 162, 170, 173, 178, 180, 181, 185, 190, 197, 205, 221, 225

Offset: 1

Tim Johannes Ohrtmann, Mar 26 2015

See A258273 for the corresponding bases.

			a(1) = 5 because this is the first number that is a multiple-digit narcissistic number in exactly one base (3).

Tim Johannes Ohrtmann, Table of n, a(n) for n = 1..49479
W. Schneider, Perfect Digital Invariants: Pluperfect Digital Invariants(PPDIs)
Eric Weisstein's World of Mathematics, Narcissistic Number
Wikipedia, Narcissistic number

Cf. A005188.
Cf. A256359 (every number of bases).
Cf. A256361, A256362, A256363, A256364, A256365 (2 to 6 bases).
Cf. A256459 (first occurrences).

for(n=3, 1000000, k=0; for(z=2, n, y=n; j=0; L=List(); until(y==0, x=y%z; j++; listinsert(L, x, j); while(!((y%z)==0), y--); y=y/z); t=0; for(p=1, j, t+=L[p]^j); if(n==t, k++)); if(k==1, print1(n, ", ")))

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

A010352 Base-9 Armstrong or narcissistic numbers, written in base 9.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 45, 55, 150, 151, 570, 571, 2446, 12036, 12336, 14462, 2225764, 6275850, 6275851, 12742452, 356614800, 356614801, 1033366170, 1033366171, 1455770342, 8463825582, 131057577510, 131057577511

Offset: 1

N. J. A. Sloane

From M. F. Hasler, Nov 18 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 21 2019

Joseph Myers, Table of n, a(n) for n = 1..58 (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. A010353 (a(n) written in base 10).

In other bases: A010343 (base 4), A010345 (base 5), A010347 (base 6), A010349 (base 7), A010351 (base 8), A005188 (base 10).

[fromdigits(digits(n,9))|n<-A010353] \\ M. F. Hasler, Nov 18 2019

Edited by Joseph Myers, Jun 28 2009

A256361 Numbers that are multiple-digit narcissistic numbers in exactly two bases.

Original entry on oeis.org

17, 28, 29, 45, 50, 85, 126, 133, 136, 145, 153, 160, 200, 245, 250, 260, 261, 265, 353, 365, 371, 405, 425, 442, 450, 490, 514, 520, 533, 585, 605, 650, 666, 680, 738, 800, 855, 925, 936, 1000, 1025, 1105, 1225, 1233, 1250, 1280

Offset: 1

Tim Johannes Ohrtmann, Mar 26 2015

			a(1) = 17 because this is the first number that is a multiple-digit narcissistic number in exactly two bases (3 and 13).

Tim Johannes Ohrtmann, Table of n, a(n) for n = 1..3541
W. Schneider, Perfect Digital Invariants: Pluperfect Digital Invariants(PPDIs)
Eric Weisstein's World of Mathematics, Narcissistic Number
Wikipedia, Narcissistic number

Cf. A005188.
Cf. A256359 (every number of bases).
Cf. A256360, A256362, A256363, A256364, A256365 (1, 3 to 6 bases).
Cf. A256459 (first occurrences).

for(n=3, 1000000, k=0; for(z=2, n, y=n; j=0; L=List(); until(y==0, x=y%z; j++; listinsert(L, x, j); while(!((y%z)==0), y--); y=y/z); t=0; for(p=1, j, t+=L[p]^j); if(n==t, k++)); if(k==2, print1(n, ", ")))

A256362 Numbers that are multiple-digit narcissistic numbers in exactly three bases.

Original entry on oeis.org

125, 325, 370, 793, 845, 1125, 1445, 2080, 2125, 2925, 3125, 3200, 3725, 3757, 5050, 5265, 6125, 6250, 6845, 7605, 8125, 8405, 10125, 10261, 10440, 10625, 11250, 13005, 13690, 14365, 15125, 15925, 18785, 22100

Offset: 1

Tim Johannes Ohrtmann, Mar 26 2015

			a(1) = 125 because this is the first number that is a multiple-digit narcissistic number in exactly three bases (12, 23 and 57).

Tim Johannes Ohrtmann, Table of n, a(n) for n = 1..445
W. Schneider, Perfect Digital Invariants: Pluperfect Digital Invariants(PPDIs)
Eric Weisstein's World of Mathematics, Narcissistic Number
Wikipedia, Narcissistic number

Cf. A005188.
Cf. A256359 (every number of bases).
Cf. A256360, A256361, A256363, A256364, A256365 (1, 2, 4, 5 and 6 bases).
Cf. A256459 (first occurrences).

for(n=3, 1000000, k=0; for(z=2, n, y=n; j=0; L=List(); until(y==0, x=y%z; j++; listinsert(L, x, j); while(!((y%z)==0), y--); y=y/z); t=0; for(p=1, j, t+=L[p]^j); if(n==t, k++)); if(k==3, print1(n, ", ")))

A306360 Numbers k such that A101337(k)/k is an integer.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 459, 1634, 8208, 9474, 13598, 48495, 54748, 92727, 93084, 119564, 174961, 306979, 548834, 1741725, 3194922, 4210818, 9800817, 9926315, 12720569, 24678050, 24678051, 88593477, 144688641, 146511208

Offset: 1

Ctibor O. Zizka, Feb 10 2019

A005188 is a subsequence of this sequence.

Sequence is finite. In particular, a(n) < 10^60. If k >= 10^60, then A101337(k) < k. - Chai Wah Wu, Feb 26 2019

			For k = 1, (1^1)/1 = 1;
for k = 459, (4^3 + 5^3 + 9^3) / 459 = 2.

Giovanni Resta, Table of n, a(n) for n = 1..109 (full sequence; first 56 terms from Chai Wah Wu)
Barry Fagin, Idempotent Factorizations of Square-free Integers, Preprints (2019).

Cf. A005188, A101337, A306354, A306361.

Select[Range[10^6], IntegerQ[Total[IntegerDigits[#]^IntegerLength[#]]/#] &] (* Michael De Vlieger, Aug 01 2019 *)

isok(n) = frac(A101337(n)/n) == 0; \\ Michel Marcus, Feb 11 2019

select( is(n)=!(A101337(n)%n), [0..999]) \\ M. F. Hasler, Nov 17 2019

A306360_list, k = [], 1
while k < 10**9:
    s = str(k)
    l, c = len(s), 0
    for i in range(l):
        c = (c + int(s[i])**l) % k
    if c == 0:
        A306360_list.append(k)
    k += 1 # Chai Wah Wu, Feb 26 2019

a(22)-a(37) from Daniel Suteu, Feb 10 2019

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A256360 Numbers that are multiple-digit narcissistic numbers in exactly one base.

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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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A010352 Base-9 Armstrong or narcissistic numbers, written in base 9.

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A256361 Numbers that are multiple-digit narcissistic numbers in exactly two bases.

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