A053735 -id:A053735 - cp's OEIS frontend

A276623 The infinite trunk of ternary beanstalk: The only infinite sequence such that a(n-1) = a(n) - A053735(a(n)), where A053735(n) = base-3 digit sum of n.

0, 2, 4, 8, 10, 12, 16, 20, 26, 28, 30, 34, 38, 42, 46, 52, 56, 62, 68, 72, 80, 82, 84, 88, 92, 96, 100, 106, 110, 116, 122, 126, 134, 140, 144, 152, 160, 164, 170, 176, 180, 188, 194, 198, 204, 212, 216, 224, 232, 242, 244, 246, 250, 254, 258, 262, 268, 272, 278, 284, 288, 296, 302, 306, 314, 322, 326, 332, 338, 342, 350, 356, 360

Offset: 0

Antti Karttunen, Sep 11 2016

Antti Karttunen, Table of n, a(n) for n = 0..6250

Cf. A004128, A024023, A053735, A054861, A261231 (left inverse), A261233, A276622, A276624, A276603 (terms divided by 2), A276604 (first differences).
Cf. A179016, A219648, A219666, A255056, A259934, A276573, A276583, A276613 for similar constructions.
Cf. also A263273.

(define (A276623 n) (A276624 (A276622 n)))

a(n) = A276624(A276622(n)).
Other identities. For all n >= 0:
A261231(a(n)) = n.
a(A261233(n)) = A024023(n) = 3^n - 1.

A286585 a(n) = A053735(A048673(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 31 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A000079, A048673, A053735, A286582, A286583, A286584, A286586.

from sympy.ntheory.factor_ import digits
from sympy import factorint, nextprime
from operator import mul
def a053735(n): return sum(digits(n, 3)[1:])
def a048673(n):
    f = factorint(n)
    return 1 if n==1 else (1 + reduce(mul, [nextprime(i)**f[i] for i in f]))//2
def a(n): return a053735(a048673(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 12 2017

(define (A286585 n) (A053735 (A048673 n)))

a(n) = A053735(A048673(n)).

For all n >= 0, a(A000079(n)) = n+1.

A173523 1+A053735(n-1), where A053735 is the sum-of-digits function in base 3.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Feb 20 2010

A053735 can be obtained as 0 followed by the first 2 terms of this sequence, followed by the first 6 terms, followed by the first 18 terms, ..., followed by the first 2*3^n terms, etc.
Similar observations are possible for: A063787 (base-2 case), and generic comments have been gathered in A173525 (base-5 case).
Fixed point of morphism 1->123, 2->234, 3->345 etc. (start with 1).

			If written as a triangle, begins:
1,
2,3,
2,3,4,3,4,5,
2,3,4,3,4,5,4,5,6,3,4,5,4,5,6,5,6,7,
2,3,4,3,4,5,4,5,6,3,4,5,4,5,6,5,6,7,4,5,6,5,6,7,6,7,8,...

Cf. A000120, A053735, A063787, A173524 - A173529.

A286632 Base-3 digit sum of A254103: a(n) = A053735(A254103(n)).

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 2, 4, 1, 3, 3, 5, 3, 5, 2, 5, 4, 6, 3, 4, 2, 4, 2, 6, 2, 4, 3, 6, 1, 3, 4, 6, 3, 5, 4, 7, 4, 6, 4, 5, 5, 7, 2, 5, 3, 5, 3, 7, 5, 7, 3, 5, 4, 6, 3, 7, 6, 8, 4, 4, 3, 5, 5, 7, 6, 8, 4, 6, 3, 5, 6, 8, 3, 5, 5, 7, 5, 7, 3, 6, 4, 6, 5, 8, 1, 3, 5, 6, 2, 4, 4, 6, 2, 4, 2, 8, 4, 6, 6, 8, 2, 4, 2, 6, 3, 5, 4, 7, 4, 6, 5, 8, 5, 7, 5, 9, 5, 7, 4, 5

Offset: 0

Antti Karttunen, Jun 03 2017

Reflecting the structure of A254103 also this sequence can be represented as a binary tree:
                                     0
                                     |
                  ...................1...................
                 2                                       1
       3......../ \........2                   4......../ \........2
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
   4       1           3       3           5       3           5       2
  5 4     6 3         4 2     4 2         6 2     4 3         6 1     3 4
etc.

Antti Karttunen, Table of n, a(n) for n = 0..8191

Cf. A053735, A254103, A286585, A286633.

from sympy.ntheory.factor_ import digits
def a254103(n):
    if n==0: return 0
    if n%2==0: return 3*a254103(n/2) - 1
    else: return floor((3*(1 + a254103((n - 1)/2)))/2)
def a(n): return sum(digits(a254103(n), 3)[1:]) # Indranil Ghosh, Jun 06 2017

(define (A286632 n) (A053735 (A254103 n)))

a(n) = A053735(A254103(n)).
a(n) = A056239(A286633(n)).
For all n >= 0, a(A000079(n)) = n+1.

A231503 a(n) = Sum_{i=0..n} digsum_3(i)^2, where digsum_3(i) = A053735(i).

Original entry on oeis.org

0, 1, 5, 6, 10, 19, 23, 32, 48, 49, 53, 62, 66, 75, 91, 100, 116, 141, 145, 154, 170, 179, 195, 220, 236, 261, 297, 298, 302, 311, 315, 324, 340, 349, 365, 390, 394, 403, 419, 428, 444, 469, 485, 510, 546, 555, 571, 596, 612, 637, 673, 698, 734, 783, 787, 796, 812, 821, 837, 862, 878, 903, 939, 948, 964, 989, 1005, 1030, 1066, 1091, 1127, 1176, 1192, 1217, 1253, 1278

Offset: 0

N. J. A. Sloane, Nov 13 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.
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. A053735, A094345, A231504, A231505.

Accumulate @ (Table[Plus @@ IntegerDigits[n, 3], {n, 0, 75}]^2) (* Amiram Eldar, Jan 20 2022 *)

a(n) = sum(i=0, n, sumdigits(i, 3)^2); \\ Michel Marcus, Sep 20 2017

A231504 a(n) = Sum_{i=0..n} digsum_3(i)^3, where digsum_3(i) = A053735(i).

Original entry on oeis.org

0, 1, 9, 10, 18, 45, 53, 80, 144, 145, 153, 180, 188, 215, 279, 306, 370, 495, 503, 530, 594, 621, 685, 810, 874, 999, 1215, 1216, 1224, 1251, 1259, 1286, 1350, 1377, 1441, 1566, 1574, 1601, 1665, 1692, 1756, 1881, 1945, 2070, 2286, 2313, 2377, 2502, 2566, 2691, 2907, 3032, 3248, 3591, 3599, 3626, 3690, 3717, 3781, 3906, 3970, 4095, 4311, 4338, 4402, 4527, 4591, 4716

Offset: 0

N. J. A. Sloane, Nov 13 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.
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. A053735, A094345, A231503, A231505.

Accumulate @ Array[(Plus @@ IntegerDigits[#, 3])^3 &, 70, 0] (* Amiram Eldar, Jan 20 2022 *)

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

A231505 a(n) = Sum_{i=0..n} digsum_3(i)^4, where digsum_3(i) = A053735(i).

Original entry on oeis.org

0, 1, 17, 18, 34, 115, 131, 212, 468, 469, 485, 566, 582, 663, 919, 1000, 1256, 1881, 1897, 1978, 2234, 2315, 2571, 3196, 3452, 4077, 5373, 5374, 5390, 5471, 5487, 5568, 5824, 5905, 6161, 6786, 6802, 6883, 7139, 7220, 7476, 8101, 8357, 8982, 10278, 10359, 10615, 11240, 11496, 12121, 13417, 14042, 15338, 17739, 17755, 17836, 18092, 18173, 18429, 19054, 19310, 19935, 21231

Offset: 0

N. J. A. Sloane, Nov 13 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.
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. A053735, A094345, A231503, A231504.

Accumulate @ Array[(Plus @@ IntegerDigits[#, 3])^4 &, 60, 0] (* Amiram Eldar, Jan 20 2022 *)

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

A358975 Numbers that are coprime to their digital sum in base 3 (A053735).

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 17, 19, 23, 27, 29, 31, 37, 41, 43, 47, 49, 51, 53, 55, 59, 61, 67, 69, 71, 73, 79, 81, 83, 85, 89, 91, 97, 101, 103, 107, 109, 113, 119, 121, 123, 125, 127, 129, 131, 137, 139, 141, 143, 147, 149, 151, 153, 155, 157, 159, 161, 163, 167, 169

Offset: 1

Amiram Eldar, Dec 07 2022

Numbers k such that gcd(k, A053735(k)) = 1.
All the terms are odd since if k is even then A053735(k) is even and so gcd(k, A053735(k)) >= 2.
Olivier (1975, 1976) proved that the asymptotic density of this sequence is 4/Pi^2 = 0.40528... (A185199).
The powers of 3 (A000244) are terms. These are also the only ternary Niven numbers (A064150) in this sequence.
Includes all the odd prime numbers (A065091).

			3 is a term since A053735(3) = 1, and gcd(3, 1) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Christian Mauduit, Carl Pomerance, and András Sárközy, On the distribution in residue classes of integers with a fixed sum of digits, The Ramanujan Journal, Vol. 9, No. 1-2 (2005), pp. 45-62; alternative link.
Michel Olivier, Sur la probabilité que n soit premier à la somme de ses chiffres, C. R. Math. Acad. Sci. Paris, Série A, Vol. 280 (1975), pp. 543-545.
Michel Olivier, Fonctions g-additives et formule asymptotique pour la propriété (n, f(n)) = q, Acta Arithmetica, Vol. 31, No. 4 (1976), pp. 361-384; alternative link.

Cf. A064150, A053735, A185199, A332880.
Subsequences: A000244, A065091.
Similar sequences: A094387, A339076, A358976, A358977, A358978.

q[n_] := CoprimeQ[n, Plus @@ IntegerDigits[n, 3]]; Select[Range[200], q]

PARI
```
is(n) = gcd(n, sumdigits(n, 3)) == 1;
```

A381838 k/9 is in this list if A053735(k) < A007949(k), i.e. if digitsum(k, 3) < valuation(k, 3).

Original entry on oeis.org

1, 3, 6, 9, 12, 18, 27, 30, 36, 45, 54, 63, 81, 84, 90, 99, 108, 117, 135, 162, 171, 189, 216, 243, 246, 252, 261, 270, 279, 297, 324, 333, 351, 378, 405, 432, 486, 495, 513, 540, 567, 594, 648, 729, 732, 738, 747, 756, 765, 783, 810, 819, 837, 864, 891, 918, 972, 981, 999

Offset: 1

Peter Luschny, Mar 08 2025

Cf. A007949, A053735.

Cf. A371176 (base 2), A381837 (base 4), A381836 (base 5).

aList := upto -> local k; [seq(k/9, k in select(n -> add(convert(n, base, 3)) < padic[ordp](n, 3), [seq(9..upto,9)]))]: aList(9000);

Select[Range[9000],DigitSum[#,3]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(9000, 3))

A177511 A053735-perfect numbers.

Original entry on oeis.org

3, 26, 62, 74, 77, 133, 134, 143, 155, 161, 185, 203, 206, 209, 215, 218, 319, 323, 341, 386, 398, 458, 473, 542, 545, 551, 554, 562, 565, 581, 589, 611, 614, 629, 635, 662, 671, 695, 698, 703, 706, 707, 713, 718, 721, 889, 899, 913, 959, 965, 998

Offset: 1

Vladimir Shevelev, Dec 11 2010

For definition, see A175522.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A175522, A000396, A175807, A175853, A176234, A177084, A177050.

A053735 := proc(n) add(d, d=convert(n,base,3)) ;end proc:
isA177511 := proc(n) local a,d ; a := 0 ; for d in numtheory[divisors](n) minus {n} do a := a+A053735(d) ; end do: a = A053735(n) ;end proc:
for n from 1 to 1000 do if isA177511(n) then printf("%d,",n) ; end if; end do: # R. J. Mathar

isok(n) = sumdiv(n, d, (dMichel Marcus, Feb 06 2016

A053735 = lambda n: sum(n.digits(base=3))
is_A177511 = lambda n: sum(A053735(d) for d in divisors(n)) == 2*A053735(n)
# D. S. McNeil, Dec 11 2010

{n : sum_{d|n, dA053735(d) = A053735(n)}.

Extended by D. S. McNeil, Dec 11 2010

cp's OEIS Frontend

A276623 The infinite trunk of ternary beanstalk: The only infinite sequence such that a(n-1) = a(n) - A053735(a(n)), where A053735(n) = base-3 digit sum of n.

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A286585 a(n) = A053735(A048673(n)).

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A173523 1+A053735(n-1), where A053735 is the sum-of-digits function in base 3.

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A286632 Base-3 digit sum of A254103: a(n) = A053735(A254103(n)).

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A231503 a(n) = Sum_{i=0..n} digsum_3(i)^2, where digsum_3(i) = A053735(i).

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A231504 a(n) = Sum_{i=0..n} digsum_3(i)^3, where digsum_3(i) = A053735(i).

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A231505 a(n) = Sum_{i=0..n} digsum_3(i)^4, where digsum_3(i) = A053735(i).

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A358975 Numbers that are coprime to their digital sum in base 3 (A053735).

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A381838 k/9 is in this list if A053735(k) < A007949(k), i.e. if digitsum(k, 3) < valuation(k, 3).

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