A360189 -id:A360189 - cp's OEIS frontend

A231500 a(n) = Sum_{i=0..n} wt(i)^2, where wt(i) = A000120(i).

0, 1, 2, 6, 7, 11, 15, 24, 25, 29, 33, 42, 46, 55, 64, 80, 81, 85, 89, 98, 102, 111, 120, 136, 140, 149, 158, 174, 183, 199, 215, 240, 241, 245, 249, 258, 262, 271, 280, 296, 300, 309, 318, 334, 343, 359, 375, 400, 404, 413, 422, 438, 447, 463, 479, 504, 513, 529, 545, 570, 586, 611, 636, 672, 673, 677, 681, 690, 694

Offset: 0

N. J. A. Sloane, Nov 12 2013

Stolarsky (1977) has an extensive bibliography.

Amiram Eldar, Table of n, a(n) for n = 0..10000 (terms 0..1024 from Ivan Neretin)
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.
Kenneth 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.

Cf. A000120, A000788, A231501, A231502, A360189.

digsum:=proc(n,B) local a; a := convert(n, base, B):
add(a[i], i=1..nops(a)): end;
f:=proc(n,k,B) global digsum; local i;
add( digsum(i,B)^k,i=0..n); end;
[seq(f(n,1,2),n=0..100)]; #A000788
[seq(f(n,2,2),n=0..100)]; #A231500
[seq(f(n,3,2),n=0..100)]; #A231501
[seq(f(n,4,2),n=0..100)]; #A231502

FoldList[#1 + DigitCount[#2, 2, 1]^2 &, 0, Range[1, 68]] (* Ivan Neretin, May 21 2015 *)

a(n) = sum(i=0, n, hammingweight(i)^2); \\ Michel Marcus, Sep 20 2017

Stolarsky (1977) studies the asymptotics.
a(n) ~ n * (log(n)/(2*log(2)))^2 + O(n*log(n)) (Stolarsky, 1977). - Amiram Eldar, Jan 20 2022
a(n) = Sum_{k=0..floor(log_2(n+1))} k^2 * A360189(n,k). - Alois P. Heinz, Mar 06 2023

A231501 a(n) = Sum_{i=0..n} wt(i)^3, where wt() = A000120().

Original entry on oeis.org

0, 1, 2, 10, 11, 19, 27, 54, 55, 63, 71, 98, 106, 133, 160, 224, 225, 233, 241, 268, 276, 303, 330, 394, 402, 429, 456, 520, 547, 611, 675, 800, 801, 809, 817, 844, 852, 879, 906, 970, 978, 1005, 1032, 1096, 1123, 1187, 1251, 1376, 1384, 1411, 1438, 1502, 1529, 1593, 1657, 1782, 1809, 1873, 1937, 2062, 2126, 2251

Offset: 0

N. J. A. Sloane, Nov 12 2013

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.
Kenneth 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.

Cf. A000120, A000788, A231500, A231502, A360189.

Accumulate @ (Table[DigitCount[n, 2, 1], {n, 0, 60}]^3) (* Amiram Eldar, Jan 20 2022 *)

a(n) = sum(i=0, n, hammingweight(i)^3); \\ Michel Marcus, Sep 20 2017

a(n) ~ n * (log(n)/(2*log(2)))^3 + O(n*log(n)^2) (Stolarsky, 1977). - Amiram Eldar, Jan 20 2022

a(n) = Sum_{k=0..floor(log_2(n+1))} k^3 * A360189(n,k). - Alois P. Heinz, Mar 06 2023

A231502 a(n) = Sum_{i=0..n} wt(i)^4, where wt() = A000120().

Original entry on oeis.org

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

N. J. A. Sloane, Nov 12 2013

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.

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

Accumulate @ (Table[DigitCount[n, 2, 1], {n, 0, 60}]^4) (* Amiram Eldar, Jan 20 2022 *)

a(n) = sum(i=0, n, hammingweight(i)^4); \\ Michel Marcus, Nov 12 2013

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

A361257 a(n) = Sum_{j=0..n} n^wt(j), where wt = A000120.

Original entry on oeis.org

1, 2, 5, 16, 29, 66, 127, 512, 737, 1090, 1541, 3312, 4369, 7658, 12209, 65536, 83537, 105282, 130987, 167600, 203701, 254122, 313259, 649728, 766201, 912626, 1079027, 1778896, 2071469, 3081570, 4329151, 33554432, 39135425, 45436546, 52524221, 60511536

Offset: 0

Alois P. Heinz, Mar 06 2023

Alois P. Heinz, Table of n, a(n) for n = 0..16382

Cf. A000120, A002416, A059841, A360189.

b:= proc(n) option remember; `if`(n<0, 0,
      b(n-1)+x^add(i, i=Bits[Split](n)))
    end:
a:= n-> subs(x=n, b(n)):
seq(a(n), n=0..37);

def A361257(n): return sum([n**j.bit_count() for j in range(0,n+1)])
print(list(A361257(n) for n in range(0,37))) # Dumitru Damian, Mar 06 2023

from collections import Counter
def A361257(n): return sum(j*n**i for i, j in Counter(j.bit_count() for j in range(n+1)).items()) # Chai Wah Wu, Mar 06 2023

a(n) = Sum_{j=0..n} n^wt(j), where wt = A000120.
a(n) = Sum_{k>=0} n^k * A360189(n,k).
a(n) mod 2 = A059841(n).
a(2^n-1) = 2^(n^2) = A002416(n).

cp's OEIS Frontend

A231500 a(n) = Sum_{i=0..n} wt(i)^2, where wt(i) = A000120(i).

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A231501 a(n) = Sum_{i=0..n} wt(i)^3, where wt() = A000120().

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Mathematica

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A231502 a(n) = Sum_{i=0..n} wt(i)^4, where wt() = A000120().

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A361257 a(n) = Sum_{j=0..n} n^wt(j), where wt = A000120.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Python

Python

Formula