A231502 - OEIS

%I #26 Mar 06 2023 18:13:42

%S 0,1,2,18,19,35,51,132,133,149,165,246,262,343,424,680,681,697,713,

%T 794,810,891,972,1228,1244,1325,1406,1662,1743,1999,2255,2880,2881,

%U 2897,2913,2994,3010,3091,3172,3428,3444,3525,3606,3862,3943,4199,4455,5080,5096,5177,5258,5514,5595,5851,6107,6732,6813,7069

%N a(n) = Sum_{i=0..n} wt(i)^4, where wt() = A000120().

%H Amiram Eldar, <a href="/A231502/b231502.txt">Table of n, a(n) for n = 0..10000</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; <a href="https://math.sun.ac.za/prodinger/pdffiles/st_andrews.pdf">alternative link</a>.

%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 Kennth B. Stolarsky, <a href="http://dx.doi.org/10.1137/0132060">Power and exponential sums of digital sums related to binomial coefficient parity</a>, SIAM J. Appl. Math., Vol. 32, No. 4 (1977), pp. 717-730.

%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.

%F a(n) ~ n * (log(n)/(2*log(2)))^4 + O(n*log(n)^3) (Stolarsky, 1977). - _Amiram Eldar_, Jan 20 2022

%F a(n) = Sum_{k=0..floor(log_2(n+1))} k^4 * A360189(n,k). - _Alois P. Heinz_, Mar 06 2023

%t Accumulate @ (Table[DigitCount[n, 2, 1], {n, 0, 60}]^4) (* _Amiram Eldar_, Jan 20 2022 *)

%o (PARI) a(n) = sum(i=0, n, hammingweight(i)^4); \\ _Michel Marcus_, Nov 12 2013

%Y Cf. A000120, A000788, A231501, A231500, A360189.

%K nonn,base

%O 0,3

%A _N. J. A. Sloane_, Nov 12 2013