A001370 -id:A001370 - cp's OEIS frontend

A135076 Primes appearing in A001370.

2, 7, 5, 11, 13, 7, 19, 19, 29, 31, 41, 37, 29, 43, 41, 37, 47, 61, 59, 67, 71, 61, 73, 79, 89, 109, 103, 89, 109, 107, 107, 113, 139, 151, 127, 137, 107, 113, 167, 173, 167, 181, 191, 173, 193, 223, 233, 211, 199, 229, 251, 239, 281, 251, 277, 281, 239, 241, 239, 269

Offset: 1

Zak Seidov, Nov 18 2007

			a(1)=2 because with s=A076203(1)=1, 2^s=2 and sod(2)=2; sod(x)=sum of digits of x;
a(2)=7 because with s=A076203(2)=4, 2^s=16 and sod(16)=7.
a(7)=9 because with s=A076203(7)=12, 2^s=4096 and sod(4096)=19.
a(8)=9 because with s=A076203(8)=18, 2^s=262144 and sod(262144)=19.

Zak Seidov, Table of n, a(n) for n = 1..1000.

Cf. A001370, A076203.

lista(nn) = {for (n=1, nn, sd = sumdigits(2^n); if (isprime(sd), print1(sd, ", ")););} \\ Michel Marcus, Oct 13 2013

a(n)=A007953(2^A076203(n)).

Name corrected by Michel Marcus, Oct 13 2013

A004207 a(0) = 1, a(n) = sum of digits of all previous terms.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 23, 28, 38, 49, 62, 70, 77, 91, 101, 103, 107, 115, 122, 127, 137, 148, 161, 169, 185, 199, 218, 229, 242, 250, 257, 271, 281, 292, 305, 313, 320, 325, 335, 346, 359, 376, 392, 406, 416, 427, 440, 448, 464, 478, 497, 517, 530, 538

Offset: 0

N. J. A. Sloane

If the leading 1 is omitted, this is the important sequence b(1)=1, for n >= 2, b(n) = b(n-1) + sum of digits of b(n-1). Cf. A016052, A016096, etc. - N. J. A. Sloane, Dec 01 2013
Same digital roots as A065075 (Sum of digits of the sum of the preceding numbers) and A001370 (Sum of digits of 2^n); they end in the cycle {1 2 4 8 7 5}. - Alexandre Wajnberg, Dec 11 2005
More precisely, mod 9 this sequence is 1 (1 2 4 8 7 5)*, the parenthesized part being repeated indefinitely. This shows that this sequence is disjoint from A016052. - N. J. A. Sloane, Oct 15 2013
There are infinitely many even terms (Belov 2003).
a(n) = A007618(n-5) for n > 57; a(n) = A006507(n-4) for n > 15. - Reinhard Zumkeller, Oct 14 2013

N. Agronomof, Problem 4421, L'Intermédiaire des mathématiciens, v. 21 (1914), p. 147.
D. R. Kaprekar, Puzzles of the Self-Numbers. 311 Devlali Camp, Devlali, India, 1959.
D. R. Kaprekar, The Mathematics of the New Self Numbers, Privately printed, 311 Devlali Camp, Devlali, India, 1963.
J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 65.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
G. E. Stevens and L. G. Hunsberger, A Result and a Conjecture on Digit Sum Sequences, J. Recreational Math. 27, no. 4 (1995), pp. 285-288.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 37.

T. D. Noe, Table of n, a(n) for n = 0..10000
A. Ya. Belov (ed.), Collection of monster problems in mathematics (in Russian), 2003. Problem 39.
D. R. Kaprekar, The Mathematics of the New Self Numbers [annotated and scanned]
J. Laroche & N. J. A. Sloane, Correspondence, 1977
Project Euler, Problem 551: Sum of digits sequence.
Kenneth B. Stolarsky, The sum of a digitaddition series, Proc. Amer. Math. Soc. 59 (1976), no. 1, 1--5. MR0409340 (53 #13099)
Index entries for Colombian or self numbers and related sequences

Cf. A016052, A016096, A033298, A007612, A007953, A229527, A230107.
For the base-2 analog see A010062.
A065075 gives sum of digits of a(n).
See A219675 for an essentially identical sequence.

a004207 n = a004207_list !! n
a004207_list = 1 : iterate a062028 1
-- Reinhard Zumkeller, Oct 14 2013, Sep 12 2011

read("transforms") :
A004207 := proc(n)
    option remember;
    if n = 0 then
        1;
    else
        add( digsum(procname(i)),i=0..n-1) ;
    end if;
end proc: # R. J. Mathar, Apr 02 2014
# second Maple program:
a:= proc(n) option remember; `if`(n<2, 1, (t->
     t+add(i, i=convert(t, base, 10)))(a(n-1)))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Jul 31 2022

f[s_] := Append[s, Plus @@ Flatten[IntegerDigits /@ s]]; Nest[f, {1}, 55] (* Robert G. Wilson v, May 26 2006 *)
f[n_] := n + Plus @@ IntegerDigits@n; Join[{1}, NestList[f, 1, 80]] (* Alonso del Arte, May 27 2006 *)

a(n) = { my(f(d, i) = d+vecsum(digits(d)), S=vector(n)); S[1]=1; for(k=1, n-1, S[k+1] = fold(f, S[1..k])); S } \\ Satish Bysany, Mar 03 2017

a = 1; print1(a, ", "); for(i = 1, 50, print1(a, ", "); a = a + sumdigits(a)); \\ Nile Nepenthe Wynar, Feb 10 2018

from itertools import islice
def agen():
    yield 1; an = 1
    while True: yield an; an += sum(map(int, str(an)))
print(list(islice(agen(), 54))) # Michael S. Branicky, Jul 31 2022

For n>1, a(n) = a(n-1) + sum of digits of a(n-1).

For n > 1: a(n) = A062028(a(n-1)). - Reinhard Zumkeller, Oct 14 2013

Errors from 25th term on corrected by Leonid Broukhis, Mar 15 1996

Typo in definition fixed by Reinhard Zumkeller, Sep 14 2011

A004166 Sum of digits of 3^n.

Original entry on oeis.org

1, 3, 9, 9, 9, 9, 18, 18, 18, 27, 27, 27, 18, 27, 45, 36, 27, 27, 45, 36, 45, 27, 45, 54, 54, 63, 63, 81, 72, 72, 63, 81, 63, 72, 99, 81, 81, 90, 90, 81, 90, 99, 90, 108, 90, 99, 108, 126, 117, 108, 144, 117, 117, 135, 108, 90, 90, 108, 126, 117, 99

Offset: 0

N. J. A. Sloane

All terms a(n), n > 1, are divisible by 9. - M. F. Hasler, Sep 27 2017

Michel Marcus, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vincenzo Librandi)
Albert Frank, Solutions of International Contest Of Logical Sequences, 2002 - 2003. (The original Contest page without solutions was removed but remains available on web.archive.org.)

Cf. A067500, A118872, A175435.

Cf. sum of digits of k^n: A001370 (k=2), this sequence (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), A175527 (k=13).

Total[IntegerDigits[#]]&/@(3^Range[0,60]) (* Harvey P. Dale, Mar 03 2013 *)
Table[Total[IntegerDigits[3^n]], {n, 0, 60}] (* Vincenzo Librandi, Oct 08 2013 *)

a(n)=sumdigits(3^n); \\ Michel Marcus, Nov 01 2013

def a(n): return sum(map(int, str(3**n)))
print([a(n) for n in range(61)]) # Michael S. Branicky, Apr 25 2022

a(n) = A007953(A000244(n)). - Michel Marcus, Nov 01 2013

Edited by M. F. Hasler, May 18 2017

A066001 Sum of digits of 5^n.

Original entry on oeis.org

1, 5, 7, 8, 13, 11, 19, 23, 25, 26, 40, 38, 28, 23, 34, 44, 58, 56, 64, 59, 61, 62, 67, 74, 82, 77, 79, 89, 85, 83, 91, 104, 106, 89, 103, 92, 109, 104, 124, 134, 130, 137, 145, 149, 151, 116, 112, 128, 145, 158, 151, 152, 130, 119, 127, 167, 196, 215, 211, 191

Offset: 0

N. J. A. Sloane, Dec 11 2001

We can expect and conjecture that a(n) ~ 4.5*log_10(5)*n, but for n ~ 10^3..10^4 there are still fluctuations of +- 1%, e.g., a(10^3)/log_10(5) ~ 4538, a(10^4)/log_10(5) ~ 44518. Modulo 9, the sequence is periodic with period (1, 5, 7, 8, 4, 2) of length 6. No term is divisible by 3, a(n) = (-1)^n (mod 3). - M. F. Hasler, May 18 2017

Harry J. Smith, Table of n, a(n) for n = 0..1000

Cf. A007953, A000351.

Cf. sum of digits of k^n: A001370 (k=2), A004166 (k=3), A065713 (k=4), this sequence (k=5), A066002 (k=6), A066003 (k=7), A066004 (k=8), A065999 (k=9), A066005 (k=11), A066006 (k=12), A175527 (k=13).

Table[ Total@ IntegerDigits[5^n], {n, 0, 60}] (* Robert G. Wilson v, Oct 25 2006 *)
Table[Total[IntegerDigits[5^n]], {n, 0, 60}] (* Vincenzo Librandi, Oct 08 2013 *)

a(n)=sumdigits(5^n); \\ Michel Marcus, Sep 04 2014

a(n) = A007953(A000351(n)). - Michel Marcus, Aug 05 2025

A065075 Sum of digits of the sum of the preceding numbers.

Original entry on oeis.org

1, 1, 2, 4, 8, 7, 5, 10, 11, 13, 8, 7, 14, 10, 2, 4, 8, 7, 5, 10, 11, 13, 8, 16, 14, 19, 11, 13, 8, 7, 14, 10, 11, 13, 8, 7, 5, 10, 11, 13, 17, 16, 14, 10, 11, 13, 8, 16, 14, 19, 20, 13, 8, 16, 14, 19, 20, 13, 8, 16, 14, 19, 20, 22, 17, 16, 14, 19, 20, 13, 17, 16, 14, 19, 20, 13

Offset: 1

Bodo Zinser, Nov 09 2001

This sequence has the same digital roots as A004207 (a(1) = 1, a(n) = sum of digits of all previous terms) and A001370 (Sum of digits of 2^n); the digital roots sequence ends in the cycle {1 2 4 8 7 5}. - Alexandre Wajnberg, Dec 11 2005
The missing digital roots are precisely the multiples of 3. - Alexandre Wajnberg, Dec 28 2005
Conjecture: every non-multiple of 3 does appear in the sequence. - Franklin T. Adams-Watters, Jun 29 2009. See A230289. - N. J. A. Sloane, Oct 17 2013
a(n) = sum of digits of A004207(n). - N. J. A. Sloane, Oct 18 2013

			a(6) = 7 because a(1)+a(2)+a(3)+a(4)+a(5) = 16 and 7 = 1+6.

Harry J. Smith and N. J. A. Sloane, Table of n, a(n) for n = 1..10000 (the first 1000 terms were computed by Harry J. Smith)
Index entries for Colombian or self numbers and related sequences

Cf. A004207, A001370, A230289, A001651.

a065075 n = a065075_list !! (n-1)
a065075_list = 1 : 1 : f 2 where
   f x = y : f (x + y) where y = a007953 x
-- Reinhard Zumkeller, Nov 13 2014

read transforms;
sp:=1;
lprint(1,sp);
s:=sp;
for n from 2 to 10000 do
sp:=digsum(s);
lprint(n,sp);
s:=s+sp;
od:
# N. J. A. Sloane, Oct 17 2013

a065075(m) = local(a,j,s); a=1; print1(a,", "); s=a; for(j=1,m,a=sumdigits(s); print1(a,", "); s=s+a)
a065075(80)

a(1) = 1, a(2) = 1, a(n) = sum of digits of (a(1)+a(2)+...+a(n-1)).

More terms from Larry Reeves (larryr(AT)acm.org) and Klaus Brockhaus, Nov 13 2001

Edited by Franklin T. Adams-Watters, Jun 29 2009

A066002 Sum of digits of 6^n.

Original entry on oeis.org

1, 6, 9, 9, 18, 27, 27, 36, 36, 36, 36, 45, 45, 36, 54, 63, 54, 72, 72, 63, 72, 81, 63, 72, 90, 90, 99, 99, 90, 135, 117, 99, 126, 126, 135, 135, 126, 135, 135, 162, 171, 126, 153, 153, 153, 162, 180, 162, 153, 162, 171, 216, 171, 216, 171, 162

Offset: 0

N. J. A. Sloane, Dec 11 2001

Harry J. Smith, Table of n, a(n) for n = 0..1000

Cf. sum of digits of k^n: A001370 (k=2), A004166 (k=3), A065713 (k=4), A066001 (k=5), this sequence (k=6), A066003(k=7), A066004 (k=8), A065999 (k=9), A066005 (k=11), A066006 (k=12), A175527 (k=13).

Cf. A007953.

Table[Total[IntegerDigits[6^n]], {n, 0, 60}] (* Vincenzo Librandi, Oct 08 2013 *)

a(n) = sumdigits(6^n); \\ Michel Marcus, Nov 01 2013

a(n) = A007953(A000400(n)). - Michel Marcus, Nov 01 2013 [corrected by Georg Fischer, Dec 28 2020]

A066003 Sum of digits of 7^n.

Original entry on oeis.org

1, 7, 13, 10, 7, 22, 28, 25, 31, 28, 43, 49, 37, 52, 58, 64, 52, 58, 73, 79, 76, 82, 97, 85, 73, 97, 112, 91, 133, 121, 118, 115, 103, 127, 142, 157, 136, 115, 130, 136, 142, 148, 136, 169, 175, 163, 187, 175, 136, 178, 184, 217, 196, 220, 217

Offset: 0

N. J. A. Sloane, Dec 11 2001

Harry J. Smith, Table of n, a(n) for n=0,...,1000

Cf. A000420 (7^n), A007953 (sum of digits).

Cf. sum of digits of k^n: A001370 (k=2), A004166 (k=3), A065713 (k=4), A066001 (k=5), A066002 (k=6), this sequence (k=7), A066004 (k=8), A065999 (k=9), A066005 (k=11), A066006 (k=12), A175527 (k=13).

Magma
```
[ &+Intseq(7^n): n in [0..60] ];
```

Table[Total[IntegerDigits[7^n]],{n,55}] (* Harvey P. Dale, Nov 22 2010 *)

a(n) = sumdigits(7^n); \\ Michel Marcus, Nov 01 2013

a(n) = A007953(A000420(n)). - Michel Marcus, Nov 01 2013

A175527 Digit sum of 13^n.

Original entry on oeis.org

1, 4, 16, 19, 22, 25, 37, 40, 34, 46, 67, 52, 55, 58, 97, 73, 85, 88, 91, 85, 115, 91, 121, 106, 109, 121, 133, 118, 121, 133, 163, 184, 169, 181, 193, 169, 172, 175, 178, 199, 193, 214, 226, 238, 169, 190, 247, 241, 208, 247, 232, 253, 292, 241, 316, 292, 268, 271, 301, 286, 298, 337, 304, 325

Offset: 0

N. J. A. Sloane, Dec 03 2010

It is surprising that many values repeat twice (for 85, 91, 121, 133, 169 this happens at a(n) = a(n+3) (but 169 occurs later for a third time), for 193, 241, 292, ... the second occurrence comes later) while many other values never occur. Is there a simple explanation? - M. F. Hasler, May 18 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A001022, A175528, A175525.

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), this sequence (k=13).

Table[Total[IntegerDigits[13^k]], {k,0,1000}]

a(n)=sumdigits(13^n); \\ Michel Marcus, Nov 01 2013

a(n) = A007953(A001022(n)). - Michel Marcus, Nov 01 2013

a(n) ~ 4.5*log_10(13)*n ~ 5.0127*n (conjectured). - M. F. Hasler, May 18 2017

A002114 Glaisher's H' numbers.

Original entry on oeis.org

1, 11, 301, 15371, 1261501, 151846331, 25201039501, 5515342166891, 1538993024478301, 533289474412481051, 224671379367784281901, 113091403397683832932811, 67032545884354589043714301, 46211522130188693681603906171

Offset: 1

N. J. A. Sloane

a(n) mod 9 = 1,2,4,8,7,5 repeated period 6 (A153130, see also A001370). a(n) mod 10 = 1. - Paul Curtz, Sep 10 2009

A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 76.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..100
J. W. L. Glaisher, On a set of coefficients analogous to the Eulerian numbers, Proc. London Math. Soc., 31 (1899), 216-235.
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010. [Author's named corrected by _N. J. A. Sloane_, Dec 12 2021]
Index entries for sequences related to Glaisher's numbers

a := n -> (-1)^n*6^(2*n)*(Zeta(0,-n*2,1/3)-Zeta(0,-n*2, 5/6)):
seq(a(n), n=1..14);

Select[Rest[With[{nn=28},CoefficientList[Series[1/(2 (2Cos[x]-1)), {x,0,nn}], x]Range[0,nn]!]],#!=0&] (* Harvey P. Dale, Jul 27 2011 *)
FullSimplify[Table[(-1)^(s+1) * BernoulliB[2*s] * (Zeta[2*s + 1, 1/6] - Zeta[2*s + 1, 5/6]) / (4*Pi*Sqrt[3]*Zeta[2*s]), {s, 1, 20}]]  (* Vaclav Kotesovec, May 05 2020 *)

a(n) := sum(sum(binomial(k,j)*(-1)^(k-j+1)*1/2^(j-1)*sum((-1)^(n)*binomial(j,i)*(2*i-j)^(2*n),i,0,floor((j-1)/2)),j,0,k)*(-2)^(k-1),k,1,2*n); /* Vladimir Kruchinin, Aug 05 2010 */

H'(n) = H(n)/3, where H(n)=2^(2n+1)*I(n) (see A002112) and e.g.f. for (-1)^n*I(n) is (3/2)/(1+exp(x)+exp(-x)) (see A047788, A047789).
H'(n) = A000436(n)/2^(2n+1). - Philippe Deléham, Jan 17 2004
For n > 0, H'(n) = Sum{k = 0..n, T(n, k)*9^(n-k)*2^(k-1) }; where DELTA is the operator defined in A084938, T(n, k) is the triangle, read by rows, given by :[0, 1, 0, 4, 0, 9, 0, 16, 0, 25, ...] DELTA [1, 0, 10, 0, 28, 0, 55, 0, 90, ..]= {1}; {0, 1}; {0, 1, 1}; {0, 1, 12, 1}; {0, 1, 63, 123, 1}; {0, 1, 274, 2366, 1234, 1}; ... For 1, 10, 28, 55, 90, 136, ... see A060544 or A060544. - Philippe Deléham, Jan 17 2004
E.g.f. 1/2*1/(2*cos(x)-1). a(n)=sum(sum(binomial(k,j)*(-1)^(k-j+1)*1/2^(j-1)*sum((-1)^(n)*binomial(j,i)*(2*i-j)^(2*n),i,0,floor((j-1)/2)),j,0,k)*(-2)^(k-1),k,1,2*n), n>0. - Vladimir Kruchinin, Aug 05 2010
E.g.f.: E(x)= x^2/(G(0)-x^2) ; G(k)= 2*(2*k+1)*(k+1) - x^2 + 2*x^2*(2*k+1)*(k+1)/G(k+1); (continued fraction Euler's kind, 1-step ). - Sergei N. Gladkovskii, Jan 03 2012
If E(x)=Sum(k=0,1,..., a(k+1)*x^(2k+2)), then A002114(k) = a(k+1)*(2*k+2)!. - Sergei N. Gladkovskii, Jan 09 2012
a(n) ~ (2*n)! * 3^(2*n+1/2) / Pi^(2*n+1). - Vaclav Kotesovec, Feb 26 2014
a(n) = (-1)^n*6^(2*n)*(zeta(-n*2,1/3)-zeta(-n*2,5/6)), where zeta(a, z) is the generalized Riemann zeta function.
From Vaclav Kotesovec, May 05 2020: (Start)
a(n) = (2*n)! * (zeta(2*n+1, 1/6) - zeta(2*n+1, 5/6)) / (sqrt(3)*(2*Pi)^(2*n+1)).
a(n) = (-1)^(n+1) * Bernoulli(2*n) * (zeta(2*n+1, 1/6) - zeta(2*n+1, 5/6)) / (4*Pi*sqrt(3)*zeta(2*n)). (End)
Conjectural e.g.f.: Sum_{n >= 1} (-1)^n*Product_{k = 1..n} (1 - exp(A007310(k)*z) ) =  z + 11*z^2/2! + 301*z^3/3! + .... - Peter Bala, Dec 09 2021

A065999 Sum of digits of 9^n.

Original entry on oeis.org

1, 9, 9, 18, 18, 27, 18, 45, 27, 45, 45, 45, 54, 63, 72, 63, 63, 99, 81, 90, 90, 90, 90, 108, 117, 144, 117, 108, 90, 126, 99, 153, 144, 117, 153, 144, 162, 171, 153, 153, 153, 198, 162, 171, 198, 216, 171, 198, 198, 225, 153, 252, 216, 234, 207

Offset: 0

N. J. A. Sloane, Dec 11 2001

a(n) mod 9 = 0 for n > 0. - Reinhard Zumkeller, May 14 2011

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
M. Sapir et al., The Decimal Expansions of Powers of 9: Problem 10758, Amer. Math. Monthly, 108 (Dec., 2001), 977-978.
H. G. Senge and E. G. Straus, PV-numbers and sets of multiplicity, Periodica Math. Hungar., 3 (1971), 93-100.
C. L. Stewart, On the representation of an integer in two different bases, J. Reine Angew. Math., 319 (1980), 63-72.

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), this sequence (k=9), A066005 (k=11), A066006 (k=12), A175527 (k=13).

Cf. also A056888, A001019.

Table[Total[IntegerDigits[9^n]], {n, 0, 60}] (* Vincenzo Librandi, Oct 08 2013 *)

a(n) = sumdigits(9^n); \\ Michel Marcus, Nov 01 2013

a(n) = A007953(A001019(n)). - Michel Marcus, Nov 01 2013

cp's OEIS Frontend

A135076 Primes appearing in A001370.

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A004207 a(0) = 1, a(n) = sum of digits of all previous terms.

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A004166 Sum of digits of 3^n.

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A066001 Sum of digits of 5^n.

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A065075 Sum of digits of the sum of the preceding numbers.

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A066002 Sum of digits of 6^n.

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A066003 Sum of digits of 7^n.

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