A231502 a(n) = Sum_{i=0..n} wt(i)^4, where wt() = A000120().
0, 1, 2, 18, 19, 35, 51, 132, 133, 149, 165, 246, 262, 343, 424, 680, 681, 697, 713, 794, 810, 891, 972, 1228, 1244, 1325, 1406, 1662, 1743, 1999, 2255, 2880, 2881, 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
Offset: 0
Links
- Amiram Eldar, Table of n, a(n) for n = 0..10000
- Jean Coquet, Power sums of digital sums, J. Number Theory, Vol. 22, No. 2 (1986), pp. 161-176.
- P. J. Grabner, P. Kirschenhofer, H. Prodinger and R. F. Tichy, On the moments of the sum-of-digits function, PDF, Applications of Fibonacci numbers, Vol. 5 (St. Andrews, 1992), Kluwer Acad. Publ., Dordrecht, 1993, pp. 263-271; alternative link.
- J.-L. Mauclaire and Leo Murata, On q-additive functions. I, Proc. Japan Acad. Ser. A Math. Sci., Vol. 59, No. 6 (1983), pp. 274-276.
- J.-L. Mauclaire and Leo Murata, On q-additive functions. II, Proc. Japan Acad. Ser. A Math. Sci., Vol. 59, No. 9 (1983), pp. 441-444.
- Kennth B. Stolarsky, Power and exponential sums of digital sums related to binomial coefficient parity, SIAM J. Appl. Math., Vol. 32, No. 4 (1977), pp. 717-730.
- J. R. Trollope, An explicit expression for binary digital sums, Math. Mag., Vol. 41, No. 1 (1968), pp. 21-25.
Programs
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Mathematica
Accumulate @ (Table[DigitCount[n, 2, 1], {n, 0, 60}]^4) (* Amiram Eldar, Jan 20 2022 *) -
PARI
a(n) = sum(i=0, n, hammingweight(i)^4); \\ Michel Marcus, Nov 12 2013
Formula
a(n) ~ n * (log(n)/(2*log(2)))^4 + O(n*log(n)^3) (Stolarsky, 1977). - Amiram Eldar, Jan 20 2022
a(n) = Sum_{k=0..floor(log_2(n+1))} k^4 * A360189(n,k). - Alois P. Heinz, Mar 06 2023