A045503 -id:A045503 - cp's OEIS frontend

A166024 Define dsf(n) = A045503(n) = n_1^{n_1}+n_2^{n_2}+n_3^{n_3} + n_m^{n_m}, where {n_1,n_2,n_3,...n_m} is the list of the decimal digits of n. Starting with a(1) = 421845123, a(n+1) = dsf(a(n)).

421845123, 16780890, 421845123, 16780890, 421845123, 16780890, 421845123, 16780890, 421845123, 16780890, 421845123, 16780890, 421845123, 16780890, 421845123, 16780890, 421845123, 16780890, 421845123, 16780890, 421845123, 16780890, 421845123, 16780890

Offset: 1

Ryohei Miyadera, Satoshi Hashiba and Koichiro Nishimura, Oct 04 2009

In fact there are only 8 loops among all the nonnegative integers for the "dsf" function that we defined.

Periodic with period 2.

			dsf(421845123) = 16780890 and dsf(16780890) = 421845123, so these 2 numbers make a loop for the function dsf.

Ryohei Miyadera, Curious Properties of an Iterative Process, Mathsource, Wolfram Library Archive.
Shoei Takahashi, Unchone Lee, Hikaru Manabe, Aoi Murakami, Daisuke Minematsu, Kou Omori, and Ryohei Miyadera, Curious Properties of Iterative Sequences, arXiv:2308.06691 [math.GM], 2023.
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A165942, A045503.

dsf[n_] := Block[{m = n, t}, t = IntegerDigits[m]; Sum[Max[1, t[[k]]]^t[[k]], {k, Length[t]}]]; NestList[dsf, 421845123,4]
LinearRecurrence[{0, 1},{421845123, 16780890},24] (* Ray Chandler, Aug 25 2015 *)

a(n+1) = dsf(a(n)).

Comment and editing by Charles R Greathouse IV, Aug 02 2010

Second sentence of Name moved to Example by Michael De Vlieger, Aug 24 2023

A108302 Concatenate n and the sum of the digits of n raised to their own power (A045503).

Original entry on oeis.org

1, 11, 24, 327, 4256, 53125, 646656, 7823543, 816777216, 9387420489, 102, 112, 125, 1328, 14257, 153126, 1646657, 17823544, 1816777217, 19387420490, 205, 215, 228, 2331, 24260, 253129, 2646660, 27823547, 2816777220, 29387420493, 3028, 3128

Offset: 0

Jason Earls, Jun 29 2005

			a(15)=153126 because 1^1 + 5^5 = 3126.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

cnsd[n_]:=Module[{idn=IntegerDigits[n]/.(0->1),c},c=Total[idn^idn];n*10^IntegerLength[c]+c]; Join[{1},cnsd/@Range[40]] (* Harvey P. Dale, Nov 25 2016 *)

A166072 Define dsf(n) = A045503(n) = n_1^{n_1}+n_2^{n_2}+n_3^{n_3} + n_m^{n_m}, where {n_1,n_2,n_3,...n_m} is the list of the decimal digits of n. dsf(809265896) = 808491852 and dsf(808491852) = 437755524,...,dsf(792488396) = 809265896, so these 8 numbers make a loop for the function dsf.

Original entry on oeis.org

809265896, 808491852, 437755524, 1657004, 873583, 34381154, 16780909, 792488396, 809265896, 808491852, 437755524, 1657004, 873583, 34381154, 16780909, 792488396, 809265896, 808491852, 437755524, 1657004, 873583

Offset: 1

Ryohei Miyadera, Satoshi Hashiba and Koichiro Nishimura, Oct 06 2009

In fact there are only 8 loops among all the nonnegative integers for the "dsf" function that we defined. We have discovered this fact through calculations using Mathematica and general-purpose languages.

Periodic with period 8.

Ryohei Miyadera, Curious Properties of an Iterative Process, Mathsource, Wolfram Library Archive.
Shoei Takahashi, Unchone Lee, Hikaru Manabe, Aoi Murakami, Daisuke Minematsu, Kou Omori, and Ryohei Miyadera, Curious Properties of Iterative Sequences, arXiv:2308.06691 [math.GM], 2023.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,1).

Cf. A165942, A166024.

dsf[n_] := Block[{m = n, t}, t = IntegerDigits[m]; Sum[Max[1, t[[k]]]^t[[k]], {k, Length[t]}]]; NestList[dsf, 809265896,16]
LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 1},{809265896, 808491852, 437755524, 1657004, 873583, 34381154, 16780909, 792488396},24] (* Ray Chandler, Aug 25 2015 *)

a(n+1) = dsf(a(n)).

Edited by Charles R Greathouse IV, Aug 02 2010

Extended by Ray Chandler, Aug 25 2015

A108303 Numbers n such that concatenating n and the sum of the digits of n raised to their own power (A045503) produces a prime.

Original entry on oeis.org

1, 34, 41, 43, 52, 60, 67, 85, 101, 110, 113, 122, 126, 128, 146, 148, 150, 155, 168, 175, 184, 186, 191, 202, 208, 212, 234, 241, 252, 267, 287, 300, 355, 397, 404, 423, 432, 445, 469, 511, 602, 606, 620, 627, 634, 641, 656, 680, 685, 750, 762, 793, 806, 919

Offset: 1

Jason Earls, Jun 29 2005

2^2560 produces a 782 digit prime (certified).

For purposes of this sequence, zero raised to the zero power = 1. - Harvey P. Dale, Feb 16 2020

			a(5)=52 because 5^5 + 2^2 = 3129 and 523129 is prime.

Cf. A045503.

cnsdQ[n_]:=Module[{idn=IntegerDigits[n]/.(0->1),c},c=Total[idn^idn];PrimeQ[n*10^IntegerLength[c]+c]]; Select[Range[1000],cnsdQ] (* Harvey P. Dale, Feb 16 2020 *)

A108406 Numbers k such that concatenating k and the sum of the digits of k raised to their own power (A045503) produces a square.

Original entry on oeis.org

0, 211, 220, 235, 20403, 111416, 1011231, 3444142, 10003400, 22303600, 31151021, 53231032, 121542025, 126423126, 202032110, 243425212, 302434003, 311544033, 324231521, 334130241, 375607602, 406221650, 561620561, 662033363, 1053045074

Offset: 1

Jason Earls, Jul 04 2005

			235 is a term because 2^2 + 3^3 + 5^5 = 3156 and 2353156 = 1534^2.

Cf. A045503, A108302.

f(n)=if(n, n=digits(n); sum(i=1, #n, n[i]^n[i]), 1); \\ A045503
isok(k) = issquare(fromdigits(concat(digits(k), digits(f(k))))); \\ Michel Marcus, Mar 05 2024

More terms from Ryan Propper, Jul 07 2005

A046253 Equal to the sum of its nonzero digits raised to its own power.

Original entry on oeis.org

0, 1, 3435, 438579088

Offset: 1

Patrick De Geest, May 15 1998

A variant of Münchausen numbers, cf. A166623.

The sequence is finite, because the sum can't exceed 9^9*L < 10^9*L, where L is the number of digits, and for L > 10 this is less than the number N >= 10^(L-1). - M. F. Hasler, Oct 01 2024

			3435 = 3^3 + 4^4 + 3^3 + 5^5.

J. S. Madachy, "Madachy's Mathematical Recreations", Dover N.Y., pp. 163-175.
C. A. Pickover, "Keys to Infinity", Wiley 1995, Ch. 22, pp. 169-171.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 37.
David Wells, "Curious and Interesting Numbers", Penguin 1988, pp. 169, 190.

Devin Akman, Munchausen Numbers Redux, Missouri J. Math. Sci. 30 (2018), no. 1, 1--4.
Geoff Bailey, C program for the sequence (cf. Hutchens link for more info), Aug. 1998.
Daan van Berkel, On a curious property of 3435, arXiv:0911.3038 [math.HO], 2009.
Jason Hutchens, power summation (originally at ciips.ee.uwa.edu.au/~hutch), 1997.
Eric Weisstein's World of Mathematics, Münchhausen Number.

Cf. A032799, A166623.

Fixed points of A045512. See also A045503 (includes zero digits).

C
```
// See Bailey and Hutchens links
```

Select[Range[0,10000],Total[#^#&/@DeleteCases[IntegerDigits@#,0]]==#&]  (* Giorgos Kalogeropoulos, May 08 2019 *)

select( {is_A046253(n)=n==A045512(n)}, [0..10^4]) \\ To find the 4th solution, multiply the set by 51817. - M. F. Hasler, Oct 01 2024

A045512 If decimal expansion of n is ab...d, a(n) = a^a + b^b + ... + d^d (ignoring any 0's).

Original entry on oeis.org

0, 1, 4, 27, 256, 3125, 46656, 823543, 16777216, 387420489, 1, 2, 5, 28, 257, 3126, 46657, 823544, 16777217, 387420490, 4, 5, 8, 31, 260, 3129, 46660, 823547, 16777220, 387420493, 27, 28, 31, 54, 283, 3152, 46683, 823570, 16777243, 387420516, 256, 257, 260

Offset: 0

N. J. A. Sloane, Jeff Burch

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A045503.

See A046253 for fixed points.

a:= n-> add(`if`(i=0, 0, i^i), i=convert(n,base,10)):
seq(a(n), n=0..42);  # Alois P. Heinz, Apr 29 2022

p[n_]:=Module[{idn=Select[IntegerDigits[n],#>0&]},Total[idn^idn]]; Array[p, 40,0] (* Harvey P. Dale, Dec 13 2012 *)

apply( {A045512(n)=vecsum([d^d|d<-digits(n),d])}, [0..44]) \\ M. F. Hasler, Oct 01 2024

Checked by Neven Juric (neven.juric(AT)apis-it.hr), Feb 04 2008

A165942 For a nonnegative integer n, define dsf(n) = n_1^{n_1}+n_2^{n_2}+n_3^{n_3} + n_m^{n_m}, where {n_1,n_2,n_3,...n_m} lists digits of n. Then starting with a(1) = 3418, a(n+1) = dsf(a(n)).

Original entry on oeis.org

3418, 16777500, 2520413, 3418, 16777500, 2520413, 3418, 16777500, 2520413, 3418, 16777500, 2520413, 3418, 16777500, 2520413, 3418, 16777500, 2520413, 3418, 16777500, 2520413, 3418, 16777500, 2520413, 3418, 16777500, 2520413, 3418, 16777500, 2520413

Offset: 1

Ryohei Miyadera, Daisuke Minematsu and Taishi Inoue, Oct 01 2009

Period 3. In fact there are only 8 such loops among all the nonnegative integers for the "dsf" function that we defined.

			a(2) = dsf(a(1)) = dsf(3418) = 3^3+4^4+1^1+8^8 = 16777500; a(3) = dsf(16777500) = 1^1+6^6+7^7+7^7+7^7+5^5+0^0+0^0 = 2520413; a(4) = dsf(2520413) = 2^2+5^5+2^2+0^0+4^4+1^1+3^3 = 3418.
This is an iterative process that starts with 3418.

Ryohei Miyadera, Curious Properties of an Iterative Process, Mathsource, Wolfram Library Archive.
Shoei Takahashi, Unchone Lee, Hikaru Manabe, Aoi Murakami, Daisuke Minematsu, Kou Omori, and Ryohei Miyadera, Curious Properties of Iterative Sequences, arXiv:2308.06691 [math.GM], 2023.
Index entries for linear recurrences with constant coefficients, signature (0,0,1).

dsf is A045503.

dsf[n_] := Block[{m = n, t}, t = IntegerDigits[m]; Sum[Max[1, t[[k]]]^t[[k]], {k, Length[t]}]]; NestList[dsf, 3418, 6]
LinearRecurrence[{0, 0, 1},{3418, 16777500, 2520413},30] (* Ray Chandler, Aug 25 2015 *)

Cross-reference from Charles R Greathouse IV, Nov 01 2009
Edited by Charles R Greathouse IV, Mar 18 2010
Extended by Ray Chandler, Aug 25 2015

A109372 Numbers k such that k * (sum of the digits of k raised to their own power) + 1 is prime.

Original entry on oeis.org

1, 11, 12, 20, 33, 34, 35, 36, 52, 64, 75, 79, 84, 94, 95, 102, 104, 110, 112, 121, 138, 163, 167, 170, 174, 184, 192, 200, 217, 231, 235, 246, 250, 255, 256, 321, 336, 343, 352, 354, 365, 390, 394, 414, 415, 420, 422, 438, 440, 446, 450, 455, 462, 471, 474

Offset: 1

Jason Earls, Aug 24 2005

A zero digit raised to the zeroth power is treated as equaling one. - Harvey P. Dale, Feb 19 2013

			a(7)=35 because 35*(3^3 + 5^5) + 1 = 110321 is prime.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000

Cf. A045503.

ndnQ[n_]:=Module[{idn=IntegerDigits[n]/.{0->1}},PrimeQ[n*Total[idn^idn]+1]]; Select[Range[500],ndnQ] (* Harvey P. Dale, Feb 19 2013 *)

isok(n) = my(d = digits(n)); isprime(n*sum(i=1,#d, d[i]^d[i])+1); \\ Michel Marcus, Sep 16 2018

A055207 Sum of n-th powers of digits of n.

Original entry on oeis.org

1, 1, 4, 27, 256, 3125, 46656, 823543, 16777216, 387420489, 1, 2, 4097, 1594324, 268435457, 30517578126, 2821109907457, 232630513987208, 18014398509481985, 1350851717672992090, 1048576, 2097153, 8388608, 94151567435, 281474993487872, 298023223910507557

Offset: 0

Henry Bottomley, Jun 19 2000

			a(12) = 1^12 + 2^12 = 1 + 4096 = 4097.

Seiichi Manyama, Table of n, a(n) for n = 0..1048

Cf. A045503, A003132, A055012, A055013, A055014.

a:= n-> add(i^n, i=convert(n, base, 10)):
seq(a(n), n=0..29);  # Alois P. Heinz, Dec 18 2022

Join[{1},Table[Total[IntegerDigits[n]^n],{n,25}]] (* Harvey P. Dale, Jul 16 2011 *)

cp's OEIS Frontend

A166024 Define dsf(n) = A045503(n) = n_1^{n_1}+n_2^{n_2}+n_3^{n_3} + n_m^{n_m}, where {n_1,n_2,n_3,...n_m} is the list of the decimal digits of n. Starting with a(1) = 421845123, a(n+1) = dsf(a(n)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A108302 Concatenate n and the sum of the digits of n raised to their own power (A045503).

Views

Author

Keywords

Examples

Links

Programs

Mathematica

A166072 Define dsf(n) = A045503(n) = n_1^{n_1}+n_2^{n_2}+n_3^{n_3} + n_m^{n_m}, where {n_1,n_2,n_3,...n_m} is the list of the decimal digits of n. dsf(809265896) = 808491852 and dsf(808491852) = 437755524,...,dsf(792488396) = 809265896, so these 8 numbers make a loop for the function dsf.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A108303 Numbers n such that concatenating n and the sum of the digits of n raised to their own power (A045503) produces a prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A108406 Numbers k such that concatenating k and the sum of the digits of k raised to their own power (A045503) produces a square.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Extensions

A046253 Equal to the sum of its nonzero digits raised to its own power.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

C

Mathematica

PARI

A045512 If decimal expansion of n is ab...d, a(n) = a^a + b^b + ... + d^d (ignoring any 0's).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A165942 For a nonnegative integer n, define dsf(n) = n_1^{n_1}+n_2^{n_2}+n_3^{n_3} + n_m^{n_m}, where {n_1,n_2,n_3,...n_m} lists digits of n. Then starting with a(1) = 3418, a(n+1) = dsf(a(n)).

Views

Author

Keywords

Comments

Examples