A053737 -id:A053737 - cp's OEIS frontend

A173524 a(n) = A053737(4^k+n-1) in the limit k->infinity, where k plays the role of a row index in A053737.

1, 2, 3, 4, 2, 3, 4, 5, 3, 4, 5, 6, 4, 5, 6, 7, 2, 3, 4, 5, 3, 4, 5, 6, 4, 5, 6, 7, 5, 6, 7, 8, 3, 4, 5, 6, 4, 5, 6, 7, 5, 6, 7, 8, 6, 7, 8, 9, 4, 5, 6, 7, 5, 6, 7, 8, 6, 7, 8, 9, 7, 8, 9, 10, 2, 3, 4, 5, 3, 4, 5, 6, 4, 5, 6, 7, 5, 6, 7, 8, 3, 4, 5, 6, 4, 5, 6, 7, 5, 6, 7, 8, 6, 7, 8, 9, 4, 5, 6, 7

Offset: 1

Omar E. Pol, Feb 20 2010

nonn

It appears that if A053737 is written as a triangle then the rows are initial segments of the present sequence; see the conjecture in A000120.

The comments in A173525 (base b=5 there) apply here with base b=4. The base b=3 is considered in A173523.

Cf. A000120, A053737, A063787.

Cf. A173523, A173525, A173526, A173527, A173528, A173529.

A053737 := proc(n) add(d, d=convert(n,base,4)) ; end proc:
A173524 := proc(n) local b; b := 4 ; if n < b then n; else k := n/(b-1); k := ceil(log(k)/log(b)) ; A053737(b^k+n-1) ; end if; end proc:
seq(A173524(n),n=1..100) ; # R. J. Mathar, Dec 09 2010

a(n) = A053737(4^k+n-1) where k >= ceiling(log_4(n/3)). [R. J. Mathar, Dec 09 2010]

Conjecture: Fixed point of the morphism 1->{1,2,3,...,b}, 2->{2,3,4,...,b+1}, j->{j,j+1,...,j+b-1} for b=4. [Joerg Arndt, Dec 08 2010]

A231664 a(n) = Sum_{i=0..n} digsum_4(i), where digsum_4(i) = A053737(i).

Original entry on oeis.org

0, 1, 3, 6, 7, 9, 12, 16, 18, 21, 25, 30, 33, 37, 42, 48, 49, 51, 54, 58, 60, 63, 67, 72, 75, 79, 84, 90, 94, 99, 105, 112, 114, 117, 121, 126, 129, 133, 138, 144, 148, 153, 159, 166, 171, 177, 184, 192, 195, 199, 204, 210, 214, 219, 225, 232, 237, 243, 250, 258, 264, 271, 279, 288, 289, 291, 294, 298, 300, 303, 307, 312, 315, 319, 324, 330, 334, 339, 345, 352, 354, 357

Offset: 0

N. J. A. Sloane, Nov 13 2013

Jean-Paul Allouche and Jeffrey Shallit, Automatic sequences, Cambridge University Press, 2003, p. 94.

Reinhard Zumkeller, 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), pp. 263-271, Kluwer Acad. Publ., Dordrecht, 1993.
Hsien-Kuei Hwang, Svante Janson and Tsung-Hsi Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, Vol. 13, No. 4 (2017), Article #47; ResearchGate link; preprint, 2016.
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.
J. R. Trollope, An explicit expression for binary digital sums, Math. Mag., Vol. 41, No. 1 (1968), pp. 21-25.

Cf. A053737, A231665, A231666, A231667.

[(&+[&+Intseq(j, 4): j in [0..n]]): n in [0..100]]; // G. C. Greubel, Feb 16 2019

ListTools:-PartialSums([seq(convert(convert(n,base,4),`+`), n=0..200)]); # Robert Israel, Sep 20 2017

Table[Sum[Total[IntegerDigits[j, 4]], {j,0,n}], {n, 0, 100}] (* G. C. Greubel, Feb 16 2019 *)

a(n) = sum(i=0, n, sumdigits(i, 4)); \\ Michel Marcus, Sep 20 2017

G.f.: g(x) satisfies g(x) = (1+x+x^2+x^3)^2*g(x^4) + (x+2*x^2+3*x^3)/(1-x-x^4+x^5). - Robert Israel, Sep 20 2017

a(n) ~ 3*n*log(n)/(4*log(2)). - Amiram Eldar, Dec 09 2021

A231667 a(n) = Sum_{i=0..n} digsum_4(i)^4, where digsum_4(i) = A053737(i).

Original entry on oeis.org

0, 1, 17, 98, 99, 115, 196, 452, 468, 549, 805, 1430, 1511, 1767, 2392, 3688, 3689, 3705, 3786, 4042, 4058, 4139, 4395, 5020, 5101, 5357, 5982, 7278, 7534, 8159, 9455, 11856, 11872, 11953, 12209, 12834, 12915, 13171, 13796, 15092, 15348, 15973, 17269, 19670, 20295, 21591, 23992, 28088, 28169, 28425, 29050, 30346, 30602, 31227, 32523, 34924, 35549, 36845, 39246, 43342

Offset: 0

N. J. A. Sloane, Nov 13 2013

Marius A. Burtea, Table of n, a(n) for n = 0..10000
Jean Coquet, Power sums of digital sums, J. Number Theory 22 (1986), no. 2, 161-176.
P. J. Grabner, P. Kirschenhofer, H. Prodinger, R. F. Tichy, On the moments of the sum-of-digits function, PDF, Applications of Fibonacci numbers, Vol. 5 (St. Andrews, 1992), 263-271, Kluwer Acad. Publ., Dordrecht, 1993.
J.-L. Mauclaire, Leo Murata, On q-additive functions. I, Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 6, 274-276.
J.-L. Mauclaire, Leo Murata, On q-additive functions. II, Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 9, 441-444.
J. R. Trollope, An explicit expression for binary digital sums, Math. Mag. 41 1968 21-25.

Cf. A053737, A231664, A231665, A231666.

a(n) = sum(i=0, n, sumdigits(i, 4)^4); \\ Michel Marcus, Sep 20 2017

A231665 a(n) = Sum_{i=0..n} digsum_4(i)^2, where digsum_4(i) = A053737(i).

Original entry on oeis.org

0, 1, 5, 14, 15, 19, 28, 44, 48, 57, 73, 98, 107, 123, 148, 184, 185, 189, 198, 214, 218, 227, 243, 268, 277, 293, 318, 354, 370, 395, 431, 480, 484, 493, 509, 534, 543, 559, 584, 620, 636, 661, 697, 746, 771, 807, 856, 920, 929, 945, 970, 1006, 1022, 1047, 1083, 1132, 1157, 1193, 1242, 1306, 1342, 1391, 1455, 1536, 1537, 1541, 1550, 1566, 1570, 1579, 1595, 1620, 1629

Offset: 0

N. J. A. Sloane, Nov 13 2013

Marius A. Burtea, Table of n, a(n) for n = 0..10000
Jean Coquet, Power sums of digital sums, J. Number Theory 22 (1986), no. 2, 161-176.
P. J. Grabner, P. Kirschenhofer, H. Prodinger, R. F. Tichy, On the moments of the sum-of-digits function, PDF, Applications of Fibonacci numbers, Vol. 5 (St. Andrews, 1992), 263-271, Kluwer Acad. Publ., Dordrecht, 1993.
J.-L. Mauclaire, Leo Murata, On q-additive functions. I, Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 6, 274-276.
J.-L. Mauclaire, Leo Murata, On q-additive functions. II, Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 9, 441-444.
J. R. Trollope, An explicit expression for binary digital sums, Math. Mag. 41 1968 21-25.

Cf. A053737, A231664, A231666, A231667.

for u=0:2000; v(u+1)=sum(dec2base(u,4)-'0');end
sol=cumsum(v.^2); % Marius A. Burtea, Jan 18 2019

a(n) = sum(i=0, n, sumdigits(i, 4)^2); \\ Michel Marcus, Sep 20 2017

A231666 a(n) = Sum_{i=0..n} digsum_4(i)^3, where digsum_4(i) = A053737(i).

Original entry on oeis.org

0, 1, 9, 36, 37, 45, 72, 136, 144, 171, 235, 360, 387, 451, 576, 792, 793, 801, 828, 892, 900, 927, 991, 1116, 1143, 1207, 1332, 1548, 1612, 1737, 1953, 2296, 2304, 2331, 2395, 2520, 2547, 2611, 2736, 2952, 3016, 3141, 3357, 3700, 3825, 4041, 4384, 4896, 4923, 4987, 5112, 5328, 5392, 5517, 5733, 6076, 6201, 6417, 6760, 7272, 7488, 7831, 8343, 9072, 9073, 9081, 9108, 9172

Offset: 0

N. J. A. Sloane, Nov 13 2013

Marius A. Burtea, Table of n, a(n) for n = 0..10000
Jean Coquet, Power sums of digital sums, J. Number Theory 22 (1986), no. 2, 161-176.
P. J. Grabner, P. Kirschenhofer, H. Prodinger, R. F. Tichy, On the moments of the sum-of-digits function, PDF, Applications of Fibonacci numbers, Vol. 5 (St. Andrews, 1992), 263-271, Kluwer Acad. Publ., Dordrecht, 1993.
J.-L. Mauclaire, Leo Murata, On q-additive functions. I, Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 6, 274-276.
J.-L. Mauclaire, Leo Murata, On q-additive functions. II, Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 9, 441-444.
J. R. Trollope, An explicit expression for binary digital sums, Math. Mag. 41 1968 21-25.

Cf. A053737, A231664, A231665, A231667.

for u=0:2000; v(u+1)=sum(dec2base(u,4)-'0'); end
sol=cumsum(v.^3); % Marius A. Burtea, Jan 18 2019

a(n) = sum(i=0, n, sumdigits(i, 4)^3); \\ Michel Marcus, Sep 20 2017

A194973 Fractalization of (A053737(n+4)), n>=0.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 5, 2, 3, 4, 1, 5, 6, 2, 3, 4, 1, 5, 6, 7, 2, 3, 4, 1, 5, 6, 7, 8, 2, 3, 4, 1, 5, 9, 6, 7, 8, 2, 3, 4, 1, 5, 9, 10, 6, 7, 8, 2, 3, 4, 1, 5, 9, 10, 11, 6, 7, 8, 2, 3, 4, 1, 5, 9, 10, 11, 12, 6, 7, 8, 2, 3, 4, 1, 5, 9, 13, 10, 11, 12, 6, 7, 8, 2, 3, 4, 1, 5, 9

Offset: 1

Clark Kimberling, Sep 07 2011

nonn

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence. The sequence (A053737(n+4)), n>=0 is formed by concatenating 4-tuples of the form (n,n+1,n+2, n+3) for n>=1: 1,2,3,4,2,3,4,5,3,4,5,6,...

Cf. A194959, A194974, A194975.

p[n_] := Floor[(n + 3)/4] + Mod[n - 1, 4]
Table[p[n], {n, 1, 90}]  (* A053737(n+4), n>=0 *)
g[1] = {1}; g[n_] := Insert[g[n - 1], n, p[n]]
f[1] = g[1]; f[n_] := Join[f[n - 1], g[n]]
f[20]  (* A194973 *)
row[n_] := Position[f[30], n];
u = TableForm[Table[row[n], {n, 1, 5}]]
v[n_, k_] := Part[row[n], k];
w = Flatten[Table[v[k, n - k + 1], {n, 1, 13},
{k, 1, n}]]  (* A194974 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
{n, 1, 80}]]  (* A194975 *)

A381837 k/16 is in this list if A053737(k) < A235127(k), i.e. if digitsum(k, 4) < valuation(k, 4).

Original entry on oeis.org

1, 4, 8, 16, 20, 32, 48, 64, 68, 80, 96, 128, 144, 192, 256, 260, 272, 288, 320, 336, 384, 448, 512, 528, 576, 640, 768, 832, 1024, 1028, 1040, 1056, 1088, 1104, 1152, 1216, 1280, 1296, 1344, 1408, 1536, 1600, 1792, 2048, 2064, 2112, 2176, 2304, 2368, 2560, 2816

Offset: 1

Peter Luschny, Mar 08 2025

Cf. A053737, A235127.

Cf. A371176 (base 2), A381838 (base 3), A381836 (base 5).

aList := upto -> local k; [seq(k/16, k in select(n -> add(convert(n, base, 4)) < padic[ordp](n, 4), [seq(16..upto,16)]))]: aList(46000);

Select[Range[46000],DigitSum[#,4]Stefano Spezia, Mar 08 2025 *)

def aList(upto, b): return [n/b^2 for n in srange(b^2, upto, b^2) if sum(n.digits(b)) < valuation(n, b)]
print(aList(46000, 4))

A218085 Let S_5(x) denote the difference in counts of multiples of 5 in the interval [0,x), those with even digit sums in base 4 in one set, those with odd digit sums in base 4 in the other. Then a(n) = (-1)^s_4(n) (S_5(n) -10S_5(floor(n/16)) +5*S_5(floor(n/256))), where s_4(n) = A053737(n).

Original entry on oeis.org

0, -1, 1, -1, -1, 1, -2, 2, 2, -2, 2, -3, -3, 3, -3, 3, 6, -6, 6, -6, -6, 5, -5, 5, 5, -5, 4, -4, -4, 4, -4, 3, -3, 3, -3, 3, 4, -4, 4, -4, -4, 3, -3, 3, 3, -3, 2, -2, 2, -2, 2, -3, -3, 3, -3, 3, 4, -4, 4, -4, -4, 3, -3, 3, 3, -3, 2, -2, -2, 2, -2, 1, 1, -1, 1, -1, 0, 0, 0, 0, 0

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Oct 20 2012

The sequence S_5(n) starts 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, ... for n >= 0. Apart from the initial 0, these are blocks of 5 repetitions of 1, 2, 3, 4, 5, 6, 7, 6, 7, 8, 7, 6, 7, 8, 9, 10, 11, 12, 13, 14, ...
Theorem. The sequence is periodic with period 2560.
The theorem allows us to write a recursion for S_5(n), considering n modulo 2560: S_5(n) = 10*S_5(floor(n/16)) - 5*S_5(floor(n/256)) + (-1)^s_4(n)*a(n).

			a(n)=-9 for n=2411, 2412, 2414, 2491, 2492, 2494 (mod 2560);
a(n)=9 for n=2413, 2415, 2493, 2495 (mod 2560).

Peter J. C. Moses, Table of n, a(n) for n = 0..2559
Vladimir Shevelev and Peter J. C. Moses, A family of digit functions with large periods, arXiv:1209.5705 [math.NT], 2012.

Cf. A214458, A217971, A053737.

S := proc(n,j,x)
    a := 0 ;
    for r from j to x-1 by n do
        add(d,d=convert(r,base,n-1)) ;
        a := a+(-1)^% ;
    end do:
    a ;
end proc:
A218085 := proc(n)
    S(5,0,n)-10*S(5,0,floor(n/16))+5*S(5,0,floor(n/256)) ;
    %*(-1)^A053737(n) ;
end proc:
seq(A218085(n),n=0..80) ; # R. J. Mathar, Oct 31 2012

-9 <= a(n) <= 9, all 19 values are actually achieved.

A374056 a(n) = max_{i=0..n} S_4(i) + S_4(n-i) where S_4(x) = A053737(x) is the base-4 digit sum of x.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 4, 5, 6, 7, 5, 6, 7, 8, 6, 7, 8, 9, 7, 8, 9, 10, 8, 9, 10, 11, 9, 10, 11, 12, 7, 8, 9, 10, 8, 9, 10, 11, 9, 10, 11, 12, 10, 11, 12, 13, 8, 9, 10, 11, 9, 10, 11, 12, 10, 11, 12, 13, 11, 12, 13, 14, 9, 10, 11, 12, 10, 11, 12, 13, 11, 12, 13, 14

Offset: 0

Hermann Gruber, Jun 26 2024

As shown in the proof of [Gruber and Holzer, lemma 9], the maximum is attained by choosing i as the largest number not exceeding n whose ternary representation is (33...3)_4. Also by lemma 6, for this choice of i we have A053737(i) = 3*floor(log_4(n+1)) and A053737(n-i) = A053737(n+1)-1, giving the formula below.

			For n=74, the maximum is attained by 63 + 11 = (333)_4 + (23)_4. Using 75=(1023)_4, comparing with the formula above, A053737(63) = 3*floor(log_4(n+1)) = 9 and A053737(11) = A053737(74+1)-1 = 5. Notice that other pairs attain the maximum as well. Namely, 43 + 31 = (223)_4 + (133)_4, as well as 47 + 27 = (233)_4 + (123)_4, and 59 + 15 = (323)_4 + (33)_4.

Hermann Gruber and Markus Holzer, Optimal Regular Expressions for Palindromes of Given Length. Extended journal version, in preparation, 2024.

Paolo Xausa, Table of n, a(n) for n = 0..10000
Hermann Gruber and Markus Holzer, Optimal Regular Expressions for Palindromes of Given Length, Proceedings of the 46th International Symposium on Mathematical Foundations of Computer Science, Article No. 53, pp. 53:1-53:15, 2021.

Cf. A053737, A102572, A014701, A374054.

Table[3*Floor[Log[4, k]] + DigitSum[k, 4] - 1, {k, 100}] (* Paolo Xausa, Aug 01 2024 *)

a(n) = 3*logint(n+1, 4) + sumdigits(n+1, 4) - 1;

a(n) = 3*floor(log_4(n+1)) + A053737(n+1) - 1 [Gruber and Holzer, lemma 9].

A381834 k/16 is in this list if k > 4 and A053737(k) = A235127(k), i.e. if digitsum(k, 4) = valuation(k, 4).

Original entry on oeis.org

2, 5, 12, 17, 24, 36, 65, 72, 84, 112, 132, 160, 208, 257, 264, 276, 304, 324, 352, 400, 516, 544, 592, 704, 784, 896, 1025, 1032, 1044, 1072, 1092, 1120, 1168, 1284, 1312, 1360, 1472, 1552, 1664, 1856, 2052, 2080, 2128, 2240, 2320, 2432, 2624, 3088, 3200, 3392

Offset: 1

Peter Luschny, Mar 09 2025

Cf. A053737, A235127, A381837.

Cf. A381835 (base = 3), A381833 (base = 5).

Select[Range[16, 60000, 16], DigitSum[#, 4] == IntegerExponent[#, 4] &] / 16

cp's OEIS Frontend

A173524 a(n) = A053737(4^k+n-1) in the limit k->infinity, where k plays the role of a row index in A053737.

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A231664 a(n) = Sum_{i=0..n} digsum_4(i), where digsum_4(i) = A053737(i).

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A231667 a(n) = Sum_{i=0..n} digsum_4(i)^4, where digsum_4(i) = A053737(i).

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A231665 a(n) = Sum_{i=0..n} digsum_4(i)^2, where digsum_4(i) = A053737(i).

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A231666 a(n) = Sum_{i=0..n} digsum_4(i)^3, where digsum_4(i) = A053737(i).

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A194973 Fractalization of (A053737(n+4)), n>=0.

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A381837 k/16 is in this list if A053737(k) < A235127(k), i.e. if digitsum(k, 4) < valuation(k, 4).

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A218085 Let S_5(x) denote the difference in counts of multiples of 5 in the interval [0,x), those with even digit sums in base 4 in one set, those with odd digit sums in base 4 in the other. Then a(n) = (-1)^s_4(n) *(S_5(n) -10*S_5(floor(n/16)) +5*S_5(floor(n/256))), where s_4(n) = A053737(n).

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A374056 a(n) = max_{i=0..n} S_4(i) + S_4(n-i) where S_4(x) = A053737(x) is the base-4 digit sum of x.

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A218085 Let S_5(x) denote the difference in counts of multiples of 5 in the interval [0,x), those with even digit sums in base 4 in one set, those with odd digit sums in base 4 in the other. Then a(n) = (-1)^s_4(n) (S_5(n) -10S_5(floor(n/16)) +5*S_5(floor(n/256))), where s_4(n) = A053737(n).