A055014 -id:A055014 - cp's OEIS frontend

A228542 Numbers that are sums of two coprime positive fifth powers.

2, 33, 244, 275, 1025, 1267, 3126, 3157, 3368, 4149, 7777, 10901, 16808, 16839, 17050, 17831, 19932, 24583, 32769, 33011, 35893, 49575, 59050, 59081, 60073, 62174, 75856, 91817, 100001, 100243, 116807, 159049, 161052, 161083, 161294, 162075, 164176, 168827

Offset: 1

Arkadiusz Wesolowski, Aug 25 2013

nonn

			244 is in the sequence since 1^5 + 3^5 = 244 and (1, 3) = 1.

Supersequence of A228556. Cf. A002561, A055014.

lst:=[]; for m in [2..168827] do f:=func; if f(m)[1] gt 0 and GCD(f(m)[1], f(m)[2]) eq 1 then Append(~lst, m); end if; end for; lst; // Arkadiusz Wesolowski, Dec 19 2020

A362945 Numbers k such that k + the sum of the fifth powers of its digits is again a fifth power.

Original entry on oeis.org

0, 210, 66374, 76933, 294137, 722571, 741300, 1023716, 2412593, 3959235, 4022940, 5090402, 6351886, 7821198, 9653864, 11808061, 11814811, 20427955, 24190287, 24239133, 24245187, 33518020, 33532714, 33536551, 39060306, 52476271, 52480534, 69255266, 69265142, 79091461, 89980491, 90147222

Offset: 0

M. F. Hasler, Jun 15 2023

Cf. A000584 (4th powers), A055014 (sum of 5th powers of decimal digits).

Cf. A362953 (same for 3rd powers), A362954 (same for 4th powers).

is_A362945(n,p=5)=ispower(vecsum([d^p|d<-digits(n)])+n,p)
for(n=0,1e8, is_A362945(n) && print1(n", "))

ispower5 = lambda n: n==round(n**.2)**5
is_A362945 = lambda n: ispower5(sum(int(d)**5 for d in str(n))+n)
def A362945(n, A=[0]):
    while len(A) < n:
        A.append(A[-1]+1)
        while not is_A362945(A[-1]): A[-1]+=1
    return A[n]

A031193 Numbers having period-22 5-digitized sequences.

Original entry on oeis.org

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 138, 141, 144, 147, 150, 153, 156, 159, 162, 165, 168, 171, 174

Offset: 1

Eric W. Weisstein

This sequence consists of those n whose trajectory under the "sum of fifth power of digits" function has period 22. All elements are multiples of 3, but some multiples of 3, the least being 0 and 888, are not in the sequence. - David W. Wilson, Aug 03 2005

Multiples of 3 have period 22 (like 3), period 1 (like 888), or period 2 (like 38889). The absence of 888 from this sequence is enough to demonstrate that it is not a Beatty sequence. - Charles R Greathouse IV, Jan 27 2022

Tanya Khovanova, Non Recursions
Eric Weisstein's World of Mathematics, Digitaddition

Cf. A055014 (sum of 5th powers of digits).

A055208 Table read by ascending antidiagonals: T(n,k) (n >= 1, k >= 1) is the sum of k-th powers of digits of n.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 9, 8, 1, 5, 16, 27, 16, 1, 6, 25, 64, 81, 32, 1, 7, 36, 125, 256, 243, 64, 1, 8, 49, 216, 625, 1024, 729, 128, 1, 9, 64, 343, 1296, 3125, 4096, 2187, 256, 1, 1, 81, 512, 2401, 7776, 15625, 16384, 6561, 512, 1, 2, 1, 729, 4096, 16807, 46656, 78125

Offset: 1

Henry Bottomley, Jun 19 2000

			T(34,2) = 3^2 + 4^2 = 25.
From _Georg Fischer_, Mar 01 2022: (Start)
Array T(n, k) (n >= 1, k >= 1) begins:
  1,   1,   1,   1, ...
  2,   4,   8,  16, ...
  3,   9,  27,  64, ...
  4,  16,  64, 256, ...
  ...
(End)

Cf. A051128. Rows include A003132, A055012, A055013, A055014. Columns include A000012, A000079, A000244, A000302, A000351, A000400, A000420, A001018, A001019, A000012, A007395, A000051, A034472, A052539, A034474.

Definition clarified by Georg Fischer, Mar 01 2022

A270540 Numbers that are equal to the number of their digits multiplied by the sum of the fifth powers of the digits.

Original entry on oeis.org

0, 1, 1232, 4100, 268542

Offset: 1

José de Jesús Camacho Medina, Mar 18 2016

Terms up to 10^8.

No further terms after 10^7 since 10^k > k^2*9^5 beyond that point. - Ray Chandler, Apr 01 2016

			4100 is a term because 4100 = 4*(4^5+1^5+0^5+0^5).

Cf. A055014.

Position[ Table[ IntegerLength[ k] Sum[( Floor[k/10^n] - 10 Floor[k/10^(n + 1)])^5, {n, 0, IntegerLength@ k}] - k, {k, 1, 10^6}], 0] // Flatten = {1, 1232, 4100, 268542}
Select[Range[10^7],With[{id=IntegerDigits[#]},#==Length[id]*Plus@@(id^5)]&] (* Ray Chandler, Apr 01 2016 *)

isok(n) = my(d=digits(n)); n == #d*sum(k=1, #d, d[k]^5); \\ Michel Marcus, Mar 25 2016

a(5) from Michel Marcus, Mar 25 2016

A341286 Numbers k such that k plus the sum of the fifth powers of the digits of k is a cube.

Original entry on oeis.org

0, 2435, 3403, 5625, 8781, 11140, 22664, 23325, 32908, 33346, 34822, 41332, 58555, 99180, 103925, 109272, 133118, 136386, 145263, 170740, 180105, 182142, 194261, 207459, 208813, 228224, 249945, 251991, 266080, 305840, 341539, 351824, 359720, 372287, 380064, 415434

Offset: 1

Will Gosnell, Feb 08 2021

			2435 is a term since 2435 + 2^5 + 4^5 + 3^5 + 5^5 = 19^3;
3403 is a term since 3403 + 3^5 + 4^5 + 0^5 + 3^5 = 17^3.

Cf. A055014 (sum of 5th powers of digits).

filter:= proc(n) local x, d;
  x:= n + add(d^5, d = convert(n, base, 10));
  surd(x, 3)::integer
end proc:
select(filter, [$0..10^5]); # Robert Israel, Feb 09 2021

Select[Range[0, 500000], IntegerQ@ Power[# + Total[IntegerDigits[#]^5], 1/3] &] (* Michael De Vlieger, Feb 22 2021 *)
Select[Range[0,416000],IntegerQ[Surd[#+Total[IntegerDigits[#]^5],3]]&] (* Harvey P. Dale, Jul 19 2022 *)

isok(k) = ispower(k+vecsum(apply(x->x^5, digits(k))), 3); \\ Michel Marcus, Feb 09 2021

from sympy import integer_nthroot
def powsum(n): return sum(int(d)**5 for d in str(n))
def ok(n): return integer_nthroot(n + powsum(n), 3)[1]
def aupto(lim):
  alst = []
  for k in range(lim+1):
    if ok(k): alst.append(k)
  return alst
print(aupto(415434)) # Michael S. Branicky, Feb 22 2021

More terms from Michel Marcus, Feb 09 2021

a(1)=0 prepended by Michael S. Branicky, Feb 22 2021

A355708 Irregular triangle read by rows in which row n lists the possible periods for the iterations of the map sum of n-th powers of digits.

Original entry on oeis.org

1, 1, 8, 1, 2, 3, 1, 2, 7, 1, 2, 4, 6, 10, 12, 22, 28, 1, 2, 3, 4, 10, 30, 1, 2, 3, 6, 12, 14, 21, 27, 30, 56, 92, 1, 25, 154, 1, 2, 3, 4, 8, 10, 19, 24, 28, 30, 80, 93, 1, 6, 7, 17, 81, 123

Offset: 1

Mohammed Yaseen, Jul 14 2022

			Triangle begins:
  1;
  1, 8;
  1, 2, 3;
  1, 2, 7;
  1, 2, 4, 6, 10, 12, 22, 28;
  1, 2, 3, 4, 10, 30;
  1, 2, 3, 6, 12, 14, 21, 27, 30, 56, 92;
  1, 25, 154;
  1, 2, 3, 4, 8, 10, 19, 24, 28, 30, 80, 93;
  1, 6, 7, 17, 81, 123;
  ...

Eric Weisstein's World of Mathematics, Digitaddition

Periods of sum of m-th powers of digits iterated: A031176 (m=2), A031178 (m=3), A031182 (m=4), A031186 (m=5), A031195 (m=6), A031200 (m=7), A031211 (m=8), A031212 (m=9), A031213 (m=10).

Sum of m-th powers of digits: A007953 (m=1), A003132 (m=2), A055012 (m=3), A055013 (m=4), A055014 (m=5), A055015 (m=6), A123253 (m=7), A210840 (m=8).

A359610 Numbers k such that the sum of the 5th powers of the digits of k is prime.

Original entry on oeis.org

11, 101, 110, 111, 119, 128, 133, 182, 188, 191, 218, 223, 227, 229, 232, 247, 272, 274, 281, 292, 313, 322, 331, 337, 346, 359, 364, 368, 373, 377, 379, 386, 395, 397, 427, 436, 463, 472, 478, 487, 539, 557, 568, 575, 577, 586, 593, 634, 638, 643, 658, 667

Offset: 1

José Hernández, Jan 06 2023

It is easy to establish that the sequence is infinite: if x is in the sequence, so is 10*x.

Alternatively: the sequence is infinite as the sequence contains all numbers consisting of a prime number of 1s and an arbitrary number of 0s. - Charles R Greathouse IV, Jan 06 2023

			11 is a term since 1^5 + 1^5 = 2 is prime.

A031974 is a subsequence.
Cf. A055014 (sum of the 5th powers of digits).
Cf. A028834, A108662, A225534, A210767, A245358.

top = 999; (* Find all terms <= top *)
For[t = 11, t <= top, t++, k = IntegerLength[t]; sum = 0;
   For[e = 0, e <= k - 1, e++, sum = sum + NumberDigit[t, e]^5];
      If[PrimeQ[sum] == True, Print[t]]]
Select[Range[670],PrimeQ[Total[IntegerDigits[#]^5]] &] (* Stefano Spezia, Jan 08 2023 *)

isok(k) = isprime(vecsum(apply(x->x^5, digits(k)))); \\ Michel Marcus, Jan 07 2023

cp's OEIS Frontend

A228542 Numbers that are sums of two coprime positive fifth powers.

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Magma

A362945 Numbers k such that k + the sum of the fifth powers of its digits is again a fifth power.

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PARI

Python

A031193 Numbers having period-22 5-digitized sequences.

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A055208 Table read by ascending antidiagonals: T(n,k) (n >= 1, k >= 1) is the sum of k-th powers of digits of n.

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A270540 Numbers that are equal to the number of their digits multiplied by the sum of the fifth powers of the digits.

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Mathematica

PARI

Extensions

A341286 Numbers k such that k plus the sum of the fifth powers of the digits of k is a cube.

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Maple

Mathematica

PARI

Python

Extensions

A355708 Irregular triangle read by rows in which row n lists the possible periods for the iterations of the map sum of n-th powers of digits.

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A359610 Numbers k such that the sum of the 5th powers of the digits of k is prime.

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Mathematica

PARI