A007094 -id:A007094 - cp's OEIS frontend

A309959 Product of digits of (n written in base 8).

0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 0, 2, 4, 6, 8, 10, 12, 14, 0, 3, 6, 9, 12, 15, 18, 21, 0, 4, 8, 12, 16, 20, 24, 28, 0, 5, 10, 15, 20, 25, 30, 35, 0, 6, 12, 18, 24, 30, 36, 42, 0, 7, 14, 21, 28, 35, 42, 49, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 0, 2, 4, 6, 8, 10, 12, 14, 0, 3, 6, 9, 12, 15, 18, 21, 0, 4, 8, 12, 16

Offset: 0

Ilya Gutkovskiy, Aug 24 2019

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Product of digits of (n written in base k): A309953 (k = 3), A309954 (k = 4), A309956 (k = 5), A309957 (k = 6), A309958 (k = 7), this sequence (k = 8), A309788 (k = 9), A007954 (k = 10).

Cf. A007094, A053829.

[0] cat [&*Intseq(n,8):n in [1..100]]; // Marius A. Burtea, Aug 25 2019

Table[Times @@ IntegerDigits[n, 8], {n, 0, 100}]

G.f. A(x) satisfies: A(x) = x * (1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + 6*x^5 + 7*x^6) * (1 + A(x^8)).

A346731 Replace 8^k with (-1)^k in base-8 expansion of n.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Jul 30 2021

If n has base-8 expansion abc..xyz with least significant digit z, a(n) = z - y + x - w + ...

			79 = 117_8, 7 - 1 + 1 = 7, so a(79) = 7.

Cf. A007094, A053829, A055017, A065359, A065368, A346688, A346689, A346690, A346691, A346732.

nmax = 104; A[] = 0; Do[A[x] = x (1 + 2 x + 3 x^2 + 4 x^3 + 5 x^4 + 6 x^5 + 7 x^6)/(1 - x^8) - (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7) A[x^8] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[n + 9 Sum[(-1)^k Floor[n/8^k], {k, 1, Floor[Log[8, n]]}], {n, 0, 104}]

from sympy.ntheory.digits import digits
def a(n):
    return sum(bi*(-1)**k for k, bi in enumerate(digits(n, 8)[1:][::-1]))
print([a(n) for n in range(105)]) # Michael S. Branicky, Jul 31 2021

G.f. A(x) satisfies: A(x) = x * (1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + 6*x^5 + 7*x^6) / (1 - x^8) - (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7) * A(x^8).

a(n) = n + 9 * Sum_{k>=1} (-1)^k * floor(n/8^k).

A037405 Numbers k such that every base-8 digit of k is a base-9 digit of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 18, 27, 36, 45, 54, 63, 73, 124, 137, 146, 177, 210, 219, 283, 284, 292, 356, 365, 429, 438, 502, 511, 568, 576, 586, 763, 768, 769, 772, 897, 906, 942, 945, 946, 950, 970, 1100, 1152, 1163, 1172, 1199, 1474

Offset: 1

Clark Kimberling

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A007094, A007095.

Cf. A037406.

import Data.List ((\\), nub)
a037405 n = a037405_list !! (n-1)
a037405_list = filter f [1..] where
   f x = null $ nub (ds 8 x) \\ nub (ds 9 x)
   ds b x = if x > 0 then d : ds b x' else []  where (x', d) = divMod x b
-- Reinhard Zumkeller, May 30 2013

bQ[n_]:=Module[{idn8=Union[IntegerDigits[n,8]],idn9=Union[ IntegerDigits[ n,9]]},And@@ (MemberQ[ idn9,#]&/@idn8)]; Select[Range[1500],bQ] (* Harvey P. Dale, Jul 17 2011 *)

A037442 Positive numbers having the same set of digits in base 8 and base 10.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 614, 1124, 1233, 1273, 1653, 2154, 2466, 3160, 3161, 3162, 3163, 3164, 3165, 3166, 3167, 3226, 4100, 4711, 5117, 5213, 5421, 6414, 6541, 7105, 7125, 7135, 7151, 7671, 10241, 11257, 11625, 11662, 11726, 13000, 13271, 13320

Offset: 1

Clark Kimberling

			1233 is in the sequence because 1233 in base 8 is 2321.

John Cerkan, Table of n, a(n) for n = 1..10000

Subsequence of A037406.

Cf. A007094

Select[Range@ 13400, Union@ IntegerDigits[#, 8] == Union@ IntegerDigits@ # &] (* Michael De Vlieger, Feb 18 2017 *)

isok(n) = Set(digits(n, 8)) == Set(digits(n)); \\ Michel Marcus, Feb 18 2017

More terms from Don Reble, Apr 28 2006

Edited by John Cerkan, Feb 17 2017

A037849 a(n) = Sum_{d(i) < d(i-1), i=1..m} (d(i-1) - d(i)), where Sum{d(i)*8^i: i=0,1,...,m} is the base-8 representation of n.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

This is the base-8 up-variation sequence; see A297330. - Clark Kimberling, Jan 18 2017

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A007094, A037858, A297330.

A037849 := proc(n)
    a := 0 ;
    dgs := convert(n,base,8);
    for i from 2 to nops(dgs) do
        if op(i,dgs)R. J. Mathar, Oct 19 2015

g[n_, b_] := Differences[IntegerDigits[n, b]]; b = 8; z = 120;
Table[-Total[Select[g[n, b], # < 0 &]], {n, 1, z}];  (* A037858 *)
Table[Total[Select[g[n, b], # > 0 &]], {n, 1, z}];   (* A037849 *)

Definition swapped with A037858 by R. J. Mathar, Oct 19 2015

Updated by Clark Kimberling, Jan 19 2018

A044879 Numbers having, in base 8, (sum of even run lengths)=(sum of odd run lengths).

Original entry on oeis.org

513, 514, 515, 516, 517, 518, 519, 521, 530, 539, 548, 557, 566, 575, 577, 578, 579, 580, 581, 582, 583, 592, 593, 595, 596, 597, 598, 599, 600, 601, 602, 604, 605, 606, 607, 608, 609, 610, 611, 613, 614, 615, 616, 617, 618

Offset: 1

Clark Kimberling

Cf. A007094.
Cf. A044873, A044874, A044875, A044876, A044877, A044878, A044880.
Cf. A044881, A044882, A044883, A044884, A044885, A044886, A044887.

from itertools import groupby
def ok(n):
    rl_sums = [0, 0]
    for k, g in groupby(oct(n)[2:]):
        rl = len(list(g))
        rl_sums[rl%2] += rl
    return rl_sums[0] == rl_sums[1]
print(list(filter(ok, range(619)))) # Michael S. Branicky, Sep 11 2021

A331565 The base 10 numbers with a digit product > 0 and which when written in bases 3,4,5,6,7,8,9 have two or more other base representations with the same digit product.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 91, 491, 921, 1138, 1234, 4853, 13581, 23568, 29242, 42161, 42162, 42163, 42164, 42991, 43365, 44313, 83342, 83651, 85226, 114382, 153881, 155462, 159422, 232868, 291862, 296183, 352486, 372642, 398543, 419563, 441194, 465326, 616146, 625431, 625523, 635813

Offset: 1

Scott R. Shannon, Jan 20 2020

For terms 10 < a(n) < 10^9 none have a base-3 representation whose digit product equals the base-10 product. The first such entry using the base-4 representation is 491.

			6 is a term as 6_10 = 6_7 = 6_8 = 6_9, so it has three other base representations where the digit product also equals 6.
91 is a term as 91_10 = 331_5 = 133_8, so it has two other base representations where the digit product also equals 9.
491 is a term as 491_10 = 13223_4 = 3431_5, so it has two other base representations where the digit product also equals 36.

Cf. A007954, A007089, A007090, A007091, A007092, A007093, A007094, A007095, A331561.

Subsequence of A052382 (zeroless numbers).

proDig[n_, b_] := Times @@ IntegerDigits[n, b]; seqQ[n_] := Module[{prod = proDig[n, 10], count = 0}, If[prod > 0, Do[If[proDig[n, b] == prod, count++]; If[count == 2, Break[]], {b, 3, 9}]]; count == 2]; Select[Range[650000], seqQ] (* Amiram Eldar, Jan 21 2020 *)

isok(n) = {my(p=vecprod(digits(n))); (p != 0) && (sum(k=3, 9, p==vecprod(digits(n,k))) >= 2);} \\ Michel Marcus, Jan 21 2020

A004691 Fibonacci numbers written in base 8.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 10, 15, 25, 42, 67, 131, 220, 351, 571, 1142, 1733, 3075, 5030, 10125, 15155, 25302, 42457, 67761, 132440, 222421, 355061, 577502, 1154563, 1754265, 3131050, 5105335, 10236405, 15343742

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

CF. A000045 (Fibonacci), A007094 (numbers in base 8).

[Seqint(Intseq(Fibonacci(n),8)): n in [0..50]]; // G. C. Greubel, Oct 09 2018

FromDigits[IntegerDigits[#, 8]] & / @ Fibonacci[Range[0, 40]] (* Vincenzo Librandi, Jun 08 2013 *)

vector(50, n, n--; fromdigits(digits(fibonacci(n), 8))) \\ G. C. Greubel, Oct 09 2018

A008557 Repeatedly convert from decimal to octal.

Original entry on oeis.org

8, 10, 12, 14, 16, 20, 24, 30, 36, 44, 54, 66, 102, 146, 222, 336, 520, 1010, 1762, 3342, 6416, 14420, 34124, 102514, 310162, 1135622, 4252006, 20160546, 114720042, 665476452, 4752456544, 43321135540, 502611010664, 7240574662150, 151272370273006

Offset: 1

sbv(AT)VNET.IBM.COM (Scott Vetter), msvetter(AT)eng.xyplex.com (Mark Vetter)

Reinhard Zumkeller, Table of n, a(n) for n = 1..75

Cf. A007094, A008558.

a008557 n = a008557_list !! (n-1)
a008557_list = iterate a007094 8  -- Reinhard Zumkeller, Aug 29 2013

NestList[ FromDigits[ IntegerDigits[ #, 8 ]] &, 8, 35 ]

def aupton(terms):
    alst = [8]
    for n in range(2, terms+1): alst.append(int(oct(alst[-1])[2:]))
    return alst
print(aupton(35)) # Michael S. Branicky, Aug 10 2021

a(n+1) = A007094(a(n)), a(1) = 8. - Reinhard Zumkeller, Aug 29 2013

A033021 Numbers whose base-8 expansion has no run of digits with length < 2.

Original entry on oeis.org

9, 18, 27, 36, 45, 54, 63, 73, 146, 219, 292, 365, 438, 511, 576, 585, 594, 603, 612, 621, 630, 639, 1152, 1161, 1170, 1179, 1188, 1197, 1206, 1215, 1728, 1737, 1746, 1755, 1764, 1773, 1782, 1791, 2304, 2313, 2322, 2331, 2340

Offset: 1

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 1..1100

Cf. A007094.

Select[Range[10000], Min[Length/@Split[IntegerDigits[#, 8]]]>1&] (* Vincenzo Librandi, Feb 05 2014 *)

cp's OEIS Frontend

A309959 Product of digits of (n written in base 8).

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A346731 Replace 8^k with (-1)^k in base-8 expansion of n.

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A037405 Numbers k such that every base-8 digit of k is a base-9 digit of k.

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A037442 Positive numbers having the same set of digits in base 8 and base 10.

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A037849 a(n) = Sum_{d(i) < d(i-1), i=1..m} (d(i-1) - d(i)), where Sum{d(i)*8^i: i=0,1,...,m} is the base-8 representation of n.

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Maple

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A044879 Numbers having, in base 8, (sum of even run lengths)=(sum of odd run lengths).

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Python

A331565 The base 10 numbers with a digit product > 0 and which when written in bases 3,4,5,6,7,8,9 have two or more other base representations with the same digit product.

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Mathematica

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A004691 Fibonacci numbers written in base 8.

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Magma

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