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a(n) = Sum_{i=0..n} digsum(i)^4, where digsum(i) = A007953(i).
5

%I #16 Jan 20 2022 10:31:22

%S 0,1,17,98,354,979,2275,4676,8772,15333,15334,15350,15431,15687,16312,

%T 17608,20009,24105,30666,40666,40682,40763,41019,41644,42940,45341,

%U 49437,55998,65998,80639,80720,80976,81601,82897,85298,89394,95955,105955,120596,141332,141588,142213,143509,145910,150006,156567,166567,181208,201944,230505,231130,232426,234827,238923

%N a(n) = Sum_{i=0..n} digsum(i)^4, where digsum(i) = A007953(i).

%H Amiram Eldar, <a href="/A231689/b231689.txt">Table of n, a(n) for n = 0..10001</a>

%H Jean Coquet, <a href="https://doi.org/10.1016/0022-314X(86)90067-3">Power sums of digital sums</a>, J. Number Theory, Vol. 22, No. 2 (1986), pp. 161-176.

%H P. J. Grabner, P. Kirschenhofer, H. Prodinger and R. F. Tichy, <a href="http://math.sun.ac.za/~hproding/abstract/abs_80.htm">On the moments of the sum-of-digits function</a>, <a href="http://math.sun.ac.za/~hproding/pdffiles/st_andrews.pdf">PDF</a>, Applications of Fibonacci numbers, Vol. 5 (St. Andrews, 1992), Kluwer Acad. Publ., Dordrecht, 1993, pp. 263-271.

%H Robert E. Kennedy and Curtis N. Cooper, <a href="http://www.fq.math.ca/Scanned/29-2/kennedy.pdf">An extension of a theorem by Cheo and Yien concerning digital sums</a>, Fibonacci Quarterly, Vol. 29, No. 2 (1991), pp. 145-149.

%H J.-L. Mauclaire and Leo Murata, <a href="https://dx.doi.org/10.3792/pjaa.59.274">On q-additive functions. I</a>, Proc. Japan Acad. Ser. A Math. Sci., Vol. 59, No. 6 (1983), pp. 274-276.

%H J.-L. Mauclaire and Leo Murata, <a href="https://dx.doi.org/10.3792/pjaa.59.441">On q-additive functions. II</a>, Proc. Japan Acad. Ser. A Math. Sci., Vol. 59, No. 9 (1983), pp. 441-444.

%H Harald Riede, <a href="http://www.fq.math.ca/Scanned/36-1/riede.pdf">Asymptotic estimation of a sum of digits</a>, Fibonacci Quarterly, Vol. 36, No. 1 (1998), pp. 72-75.

%H J. R. Trollope, <a href="http://www.jstor.org/stable/2687954">An explicit expression for binary digital sums</a>, Math. Mag., Vol. 41, No. 1 (1968), pp. 21-25.

%p See A037123.

%t Accumulate[Table[Total[IntegerDigits[n]]^4,{n,0,60}]] (* _Harvey P. Dale_, May 12 2014 *)

%o (PARI) a(n) = sum(i=0, n, sumdigits(i)^4); \\ _Michel Marcus_, Sep 20 2017

%Y Cf. A007953, A037123, A074784, A231688.

%K nonn,base

%O 0,3

%A _N. J. A. Sloane_, Nov 13 2013