A001370 -id:A001370 - cp's OEIS frontend

A330822 a(n) = digsum(2^a(n-1)) with a(0) = 0.

0, 1, 2, 4, 7, 11, 14, 22, 25, 29, 41, 50, 76, 106, 142, 214, 313, 380, 508, 691, 875, 1184, 1687, 2243, 3164, 4126, 5578, 7855, 10676, 13981, 18659, 25421, 34277, 46409, 63023, 85658, 116248, 157660, 213892, 290554, 393145, 532838, 723451, 981020, 1328449, 1799363

Offset: 0

Haris Ziko, Jan 02 2020

			Starting with a(0) = 0.
a(1) = digsum(2^0) = digsum(1) = 1.
a(2) = digsum(2^1) = digsum(2) = 2.
a(3) = digsum(2^2) = digsum(4) = 4.
a(4) = digsum(2^4) = digsum(16) = 7.
etc.

Chai Wah Wu, Table of n, a(n) for n = 0..78

Cf. A001370, A007953, A131577.

a:=[0,1]; [n le 2 select a[n] else &+Intseq(2^Self(n-1)):n in [1..50]]; // Marius A. Burtea, Jan 03 2020

a[0] = 0; a[n_] := a[n] = Total @ IntegerDigits[2^a[n-1]]; Array[a, 45, 0] (* Amiram Eldar, Jan 03 2020 *)
NestList[Total[IntegerDigits[2^#]]&,0,50] (* Harvey P. Dale, Jul 19 2025 *)

a(n)={my(x=0); for(i=0, n-1, x=sumdigits(2^x)); x};
lista(nn)={my(x=0, v=vector(nn+1)); v[1]=0; for(i=1, nn, x=sumdigits(2^x); v[i+1]=x); v};

A364606 Numbers k such that the average digit of 2^k is an integer.

Original entry on oeis.org

0, 1, 2, 3, 6, 13, 16, 26, 46, 51, 56, 73, 122, 141, 166, 313, 383

Offset: 1

Jon E. Schoenfield, Jul 29 2023

			2^26 = 67108864 is an 8-digit number; its average digit is (6+7+1+0+8+8+6+4)/8 = 40/8 = 5, an integer, so 26 is a term.

Cf. A000079, A001370, A034887, A061383.

q:= n-> (l-> irem(add(i, i=l), nops(l))=0)(convert(2^n, base, 10)):
select(q, [$0..400])[];  # Alois P. Heinz, Jul 29 2023

Select[Range[0, 2^12], IntegerQ@ Mean@ IntegerDigits[2^#] &] (* Michael De Vlieger, Jul 29 2023 *)

isok(k) = my(d=digits(2^k)); !(vecsum(d) % #d); \\ Michel Marcus, Jul 29 2023

from itertools import count, islice
from gmpy2 import mpz, digits
def A364606_gen(startvalue=0): # generator of terms >= startvalue
    m = mpz(1)<A364606_list = list(islice(A364606_gen(),10)) # Chai Wah Wu, Jul 31 2023

{ k : A001370(k) mod A034887(k) = 0 }.

A375976 Sum of squares of the decimal digits of 2^n.

Original entry on oeis.org

1, 4, 16, 64, 37, 13, 52, 69, 65, 30, 21, 84, 133, 150, 126, 162, 131, 64, 77, 177, 191, 164, 139, 301, 225, 113, 266, 197, 231, 269, 209, 275, 404, 450, 443, 371, 426, 332, 461, 487, 413, 288, 266, 396, 346, 382, 426, 404, 463, 393, 514, 528, 517, 569, 584

Offset: 0

Luca Khan, Sep 04 2024

			For n=4, 2^4 = 16 and those digits 1^2 + 6^2 = 37 = a(4).

Cf. A000079, A003132, A001370, A110892.

a[n_]:=Norm[IntegerDigits[2^n]]^2; Array[a,55,0] (* Stefano Spezia, Sep 06 2024 *)

a(n) = norml2(digits(2^n)); \\ Michel Marcus, Sep 06 2024

def A375976(n): return sum((0, 1, 4, 9, 16, 25, 36, 49, 64, 81)[int(d)] for d in str(1<'0') # Chai Wah Wu, Sep 30 2024

a(n) = A003132(A000079(n)).

A112436 Next term is the sum of the last 10 digits in the sequence, beginning with a(10) = 5.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 5, 10, 11, 13, 17, 25, 22, 25, 30, 29, 32, 30, 29, 33, 36, 34, 36, 42, 37, 41, 37, 40, 35, 37, 37, 42, 38, 45, 46, 46, 46, 50, 44, 43, 40, 34, 31, 30, 25, 25, 28, 31, 31, 32, 30, 26, 24, 26, 30, 28, 35, 35, 37, 39, 48, 50, 47, 50, 45, 42, 36, 40, 33

Offset: 1

Alexandre Wajnberg, Dec 11 2005

Variation on Angelini's A112395. The sequence cycles at a(28)=42 and the loop has 312 terms. Computed by Gilles Sadowski.

			a(28)=42 because 2+9 + 3+3 + 3+6 + 3+4 + 3+6 = 42

Cf. A112395, A004207, A065075, A001370.

A112438 Next term is the sum of the last 10 digits in the sequence, beginning with a(10) = 7.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 7, 14, 19, 29, 40, 44, 38, 44, 42, 37, 43, 42, 37, 39, 45, 44, 45, 48, 50, 43, 41, 38, 40, 32, 32, 30, 28, 27, 32, 32, 32, 34, 31, 26, 29, 35, 38, 42, 44, 44, 41, 38, 38, 43, 42, 40, 39, 40, 33, 32, 31, 31, 23, 24, 24, 25, 28, 34, 36, 39, 45, 47, 48

Offset: 1

Alexandre Wajnberg, Dec 11 2005

Variation on Angelini's A112395. The sequence cycles at a(18)=44 and the loop has 312 terms. Computed by Gilles Sadowski.

			a(18)=44 because 1+9 + 2+9 + 4+0 + 4+4 + 3+8 = 44

Cf. A112395, A004207, A065075, A001370.

A112439 Next term is the sum of the last 10 digits in the sequence, beginning with a(10) = 8.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 8, 16, 23, 28, 38, 49, 46, 49, 57, 59, 62, 57, 59, 60, 54, 49, 54, 51, 43, 44, 43, 37, 38, 43, 43, 42, 41, 36, 34, 34, 34, 35, 38, 40, 37, 40, 37, 39, 40, 40, 34, 37, 37, 35, 39, 47, 51, 47, 48, 52, 47, 47, 52, 48, 48, 53, 50, 44, 45, 42, 36, 37, 42

Offset: 1

Alexandre Wajnberg, Dec 11 2005

Variation on Angelini's A112395. The sequence cycles at a(32)=37 and the loop has 312 terms. Computed by Gilles Sadowski.

			a(32)=37 because 5+4 + 5+1 + 4+3 + 4+4 + 4+3 = 37

Cf. A112395, A004207, A065075, A001370.

A112440 Next term is the sum of the last 10 digits in the sequence, beginning with a(10) = 9.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 9, 18, 27, 36, 45, 54, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45

Offset: 1

Alexandre Wajnberg, Dec 11 2005

Variation on Angelini's A112395. The sequence cycles at a(17)=45 and the loop has one term. Computed by Gilles Sadowski.

			a(17)=45 because 1+8 + 2+7 + 3+6 + 4+5 + 5+4 = 45

Cf. A112395, A004207, A065075, A001370.

A117771 Numbers n such that digit sum of 2^n is less than or equal to n.

Original entry on oeis.org

5, 9, 10, 17, 70

Offset: 1

Joshua Zucker, Jul 24 2006

Probably there are no more terms.
Probably 5 and 70 are the only cases when digit sum of 2^n is equal to n. - Tanya Khovanova, Jul 23 2006
Probably 1, 2, 5 and 70 are the only cases when digit sum of 2^n is divisible by n. - Zak Seidov, Jul 24 2006

Cf. A001370.

Select[Range[80],#>=Total[IntegerDigits[2^#]]&] (* Harvey P. Dale, Sep 22 2019 *)

isok(n) = d = digits(2^n); sum(i=1, #d, d[i]) <= n; \\ Michel Marcus, Aug 17 2013

A286512 Numbers N for which there is k > 0 such that sum of digits(N^k) = N, but the least such k is larger than the least k for which sum of digits(N^k) > N*11/10.

Original entry on oeis.org

17, 31, 63, 86, 91, 103, 118, 133, 155, 157, 211, 270, 290, 301, 338, 352, 421, 432, 440, 441, 450, 478, 513, 533, 693, 853, 1051, 1237, 1363, 1459, 1526, 1665, 2781

Offset: 1

M. F. Hasler, May 18 2017

The set of these numbers appears to be finite, and probably 2781 is its largest element.
The motivation for this sequence is the study of the behavior of the sum of digits of powers of a given number. Statistically, sumdigits(n^k) ~ 4.5*log_10(n')*k (where n' = n without trailing 0's), but typically fluctuations of some percent persist up to large values of k. (Cf. the graph of sequences n^k cited in the cross-references.)
The ratio of 11/10 is somewhat arbitrary, but larger ratios of the simple form (1 + 1/m) yield quite small subsets of this sequence (for m=2 the only element is 118, for m=3 the set is {31, 86, 118}, for m=1 it is empty), and smaller ratios yield much larger (possibly infinite?) sets. Also, the condition can be written sumdigits(N^k)-N > N/10, and 10 is the base we are using.
To compute the sequence A247889 we would like to have a rule telling us when we can stop the search for an exponent. It appears that sumdigits(N^k) >= 2*N is a limit that works for all N; the present sequence gives counterexamples to the (r.h.s.) limit of 1.1*N. The above comment mentioned the counterexamples {118} resp. {31, 86, 118}) for limits N*3/2 and N*4/3.

Cf. A247889, A007953.

Cf. sum of digits of k^n: A001370 (k=2), A004166 (k=3), A065713 (k=4), A066001 (k=5), A066002 (k=6), A066003 (k=7), A066004 (k=8), A065999 (k=9), A066005 (k=11), A066006 (k=12). (In these sequences, k is fixed and n is the index/exponent; in the present sequence it's the opposite and therefore the names k <-> n are exchanged.)

for(n=1,5000,A247889(n)&&!A247889(n,n*11\10)&&print1(n",")) \\ Here, A247889() is a variant of the function computing that sequence which accepts as second optional argument a limit m, stopping the search for the exponent as soon as the digital sum of n^k exceeds m.

A287058 Sum of decimal digits of 118^n.

Original entry on oeis.org

1, 10, 19, 19, 55, 64, 55, 64, 82, 91, 109, 100, 109, 181, 118, 145, 127, 163, 154, 172, 154, 190, 226, 190, 208, 217, 271, 289, 253, 280, 298, 307, 334, 289, 334, 280, 361, 343, 334, 379, 406, 406, 379, 424, 379, 424, 415, 406, 523, 433, 478

Offset: 0

M. F. Hasler, May 18 2017

118 is exceptional in the sense that it appears to be the only number m for which the smallest k such that sumdigits(m^k) = m occurs after the smallest k such that sumdigits(m^k) > m*3/2. If this last limit is decreased to m*4/3, then 31 and 86 also have this property. It appears that no number has this property if the limit is increased to 2m, see also A247889.

It is also remarkable that many values in the sequence are repeated (19, 55, 64, 109, 190, 154, 280, 289, 334 (3 times), 379, 406, 424, ...), while most other numbers never appear.

Cf. A007953, A247889.

Cf. sum of digits of k^n: A001370 (k=2), A004166 (k=3), A065713 (k=4), A066001(k=5), A066002 (k=6), A066003 (k=7), A066004 (k=8), A065999 (k=9), A066005 (k=11), A066006 (k=12).

Total[IntegerDigits[#]]&/@NestList[118#&,1,50] (* Harvey P. Dale, Feb 24 2022 *)

PARI
```
a(n)=sumdigits(118^n)
```

cp's OEIS Frontend

A330822 a(n) = digsum(2^a(n-1)) with a(0) = 0.

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Magma

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A364606 Numbers k such that the average digit of 2^k is an integer.

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Maple

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A375976 Sum of squares of the decimal digits of 2^n.

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A112436 Next term is the sum of the last 10 digits in the sequence, beginning with a(10) = 5.

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A112438 Next term is the sum of the last 10 digits in the sequence, beginning with a(10) = 7.

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A112439 Next term is the sum of the last 10 digits in the sequence, beginning with a(10) = 8.

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A112440 Next term is the sum of the last 10 digits in the sequence, beginning with a(10) = 9.

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A117771 Numbers n such that digit sum of 2^n is less than or equal to n.

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A286512 Numbers N for which there is k > 0 such that sum of digits(N^k) = N, but the least such k is larger than the least k for which sum of digits(N^k) > N*11/10.

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