A007093 -id:A007093 - cp's OEIS frontend

A073788 Numbers in base -7.

0, 1, 2, 3, 4, 5, 6, 160, 161, 162, 163, 164, 165, 166, 150, 151, 152, 153, 154, 155, 156, 140, 141, 142, 143, 144, 145, 146, 130, 131, 132, 133, 134, 135, 136, 120, 121, 122, 123, 124, 125, 126, 110, 111, 112, 113, 114, 115, 116, 100, 101, 102, 103, 104, 105

Offset: 0

Robert G. Wilson v, Aug 11 2002

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 2, p. 189.

Chai Wah Wu, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Negabinary
Prepared and presented by Matthew Szudzik of Wolfram Research, A Mathematica programming contest

Cf. A007093, A039724, A073785, A007608, A073786, A073787, A073789, A073790 & A039723.

ToNegaBases[i_Integer, b_Integer] := FromDigits[ Rest[ Reverse[ Mod[ NestWhileList[(#1 - Mod[ #1, b])/-b &, i, #1 != 0 &], b]]]]; Table[ ToNegaBases[n, 7], {n, 0, 60}]

def A073788(n):
    s, q = '', n
    while q >= 7 or q < 0:
        q, r = divmod(q, -7)
        if r < 0:
            q += 1
            r += 7
        s += str(r)
    return int(str(q)+s[::-1]) # Chai Wah Wu, Apr 09 2016

A293660 Base-7 circular primes that are not base-7 repunits.

Original entry on oeis.org

11, 13, 17, 19, 23, 29, 37, 41, 43, 47, 79, 89, 97, 109, 131, 211, 233, 257, 263, 281, 307, 337, 439, 479, 509, 571, 619, 673, 677, 853, 941, 953, 977, 997, 1021, 1097, 1117, 1163, 1171, 1453, 1511, 1531, 1579, 1597, 1657, 1777, 1787, 1811, 1871, 1933, 1951

Offset: 1

Felix Fröhlich, Dec 30 2017

Conjecture: The sequence is finite, with 13143449029 being the last term. - [Comment extended by Felix Fröhlich, May 30 2019]

			109 written in base 7 is 214. The base-7 numbers 214, 142, 421 written in base 10 are 109, 79, 211, respectively, and all those numbers are prime, so 79, 109 and 211 are terms of the sequence.

Felix Fröhlich, Table of n, a(n) for n = 1..141

Cf. A007093, A293142.

Cf. base-b nonrepunit circular primes: A293657 (b=4), A293658 (b=5), A293659 (b=6), A293661 (b=8), A293662 (b=9), A293663 (b=10).

With[{b = 7}, Select[Prime@ Range[PrimePi@ b + 1, 300], Function[w, And[AllTrue[Array[FromDigits[RotateRight[w, #], b] &, Length@ w - 1], PrimeQ], Union@ w != {1} ]]@ IntegerDigits[#, b] &]] (* Michael De Vlieger, Dec 30 2017 *)

rot(n) = if(#Str(n)==1, v=vector(1), v=vector(#n-1)); for(i=2, #n, v[i-1]=n[i]); u=vector(#n); for(i=1, #n, u[i]=n[i]); v=concat(v, u[1]); v
decimal(v, base) = my(w=[]); for(k=0, #v-1, w=concat(w, v[#v-k]*base^k)); sum(i=1, #w, w[i])
is_circularprime(p, base) = my(db=digits(p, base), r=rot(db), i=0); if(vecmin(db)==0, return(0), while(1, dec=decimal(r, base); if(!ispseudoprime(dec), return(0)); r=rot(r); if(r==db, return(1))))
forprime(p=1, , if(vecmin(digits(p, 7))!=vecmax(digits(p, 7)), if(is_circularprime(p, 7), print1(p, ", "))))

A073794 Replace 7^k with (-7)^k in base 7 expansion of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, -7, -6, -5, -4, -3, -2, -1, -14, -13, -12, -11, -10, -9, -8, -21, -20, -19, -18, -17, -16, -15, -28, -27, -26, -25, -24, -23, -22, -35, -34, -33, -32, -31, -30, -29, -42, -41, -40, -39, -38, -37, -36, 49, 50, 51, 52, 53, 54, 55, 42, 43, 44, 45, 46, 47, 48, 35, 36, 37, 38, 39, 40, 41

Offset: 0

Robert G. Wilson v, Aug 12 2002

Base 7 representation for n (in lexicographic order) converted from base -7 to base 10.

Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625, Dec 08, 2020

Cf. A007093, A073788, A053985, A065369, A073791, A073792, A073793, A073795, A073796 & A073835.

f[n_Integer, b_Integer] := Block[{l = IntegerDigits[n]}, Sum[l[[ -i]]*(-b)^(i - 1), {i, 1, Length[l]}]]; a = Table[ FromDigits[ IntegerDigits[n, 7]], {n, 0, 80}]; b = {}; Do[b = Append[b, f[a[[n]], 7]], {n, 1, 80}]; b

a(7*k+m) = -7*a(k)+m for 0 <= m < 7. - Chai Wah Wu, Jan 16 2020

A097582 Base 7 representation of the concatenation of the first n numbers with the most significant digits first.

Original entry on oeis.org

1, 15, 234, 3412, 50664, 1022634, 13331215, 206636142, 3026236221, 614636352655, 155123512633260, 35001313215554565, 10403265603212022112, 2132066345452131466644, 434014101450663623262042

Offset: 1

Cino Hilliard, Aug 29 2004

Consider numbers of the form 1, 12, 123, 1234, ..., N. Find the highest power of 7^p such that 7^p < N. Then p = [log(N)/log(7)] and for 0 <= qi <= 6 [N/7^p] = q1 + r1 [r1/7^(p-1)] = q2 + r2 ........................ rp/7^1 = qp + rp+1 rp+1/7^0 = qp+1 0 For N = 1234, p = [log(1234)/log(7)] = 3 division quot rem 1234/7^3 = 3 205 205/7^2 = 4 9 9/7^1 = 1 2 2/7^0 = 2 0 The sequence of quotients, top down, forms the entry in the table for 1234. Obviously this algorithm works for any N.

Seiichi Manyama, Table of n, a(n) for n = 1..318

Cf. A007908, A050926, A097580, A097583.

Cf. A007093.

a(n) = A007093(A007908(n)). - Seiichi Manyama, Apr 23 2022

A346691 Replace 7^k with (-1)^k in base-7 expansion of n.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Jul 29 2021

If n has base-7 expansion abc..xyz with least significant digit z, a(n) = z - y + x - w + ...

			83 = 146_7, 6 - 4 + 1 = 3, so a(83) = 3.

Cf. A007093, A053828, A055017, A065359, A065368, A346688, A346689, A346690.

nmax = 104; A[] = 0; Do[A[x] = x (1 + 2 x + 3 x^2 + 4 x^3 + 5 x^4 + 6 x^5)/(1 - x^7) - (1 + x + x^2 + x^3 + x^4 + x^5 + x^6) A[x^7] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[n + 8 Sum[(-1)^k Floor[n/7^k], {k, 1, Floor[Log[7, 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, 7)[1:][::-1]))
print([a(n) for n in range(105)]) # Michael S. Branicky, Jul 29 2021

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

a(n) = n + 8 * Sum_{k>=1} (-1)^k * floor(n/7^k).

A037383 Numbers k such that every base-3 digit of k is a base-7 digit of k.

Original entry on oeis.org

1, 2, 13, 51, 63, 67, 68, 79, 84, 91, 99, 105, 134, 205, 211, 246, 252, 345, 351, 352, 354, 355, 357, 358, 359, 360, 361, 362, 363, 364, 366, 373, 380, 387, 394, 401, 406, 441, 442, 443, 444, 445, 446, 447, 448, 449, 454, 455, 457

Offset: 1

Clark Kimberling

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

Cf. A007089, A007093.

Cf. A037380, A037381, A037382, A037384, A037385, A037386.

import Data.List ((\\), nub)
a037383 n = a037383_list !! (n-1)
a037383_list = filter f [1..] where
   f x = null $ nub (ds 3 x) \\ nub (ds 7 x)
   ds b x = if x > 0 then d : ds b x' else []  where (x', d) = divMod x b
-- Reinhard Zumkeller, May 30 2013

Select[Range[500],Complement[Union[IntegerDigits[#,3]],Union[IntegerDigits[#,7]]]=={}&] (* Harvey P. Dale, Jan 28 2024 *)

upto(N) = my(s7); [n|n<-[1..N], setunion(Set(digits(n, 3)), s7=Set(digits(n, 7)))==s7]; \\ Ruud H.G. van Tol, May 09 2024

A102679 Number of digits >= 7 in decimal representation of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane, Feb 03 2005

a(n) = 0 iff n is in A007093 (numbers in base 7). - Bernard Schott, Feb 12 2023

Hieronymus Fischer, Table of n, a(n) for n = 0..10000

Cf. A007093, A027868, A054899, A055640, A055641, A102669-A102685, A117804, A122840, A122841, A160093, A160094, A196563, A196564. Partial sums see A102680.

Cf. A000120, A000788, A023416, A059015 (for base 2).

p:=proc(n) local b,ct,j: b:=convert(n,base,10): ct:=0: for j from 1 to nops(b) do if b[j]>=7 then ct:=ct+1 else ct:=ct fi od: ct: end: seq(p(n),n=0..125); # Emeric Deutsch, Feb 23 2005

From Hieronymus Fischer, Jun 10 2012: (Start)
a(n) = Sum_{j=1..m+1} (floor(n/10^j + 3/10) - floor(n/10^j)), where m = floor(log_10(n)).
G.f.: g(x) = (1/(1-x))*Sum_{j>=0} (x^(7*10^j) - x^(10*10^j))/(1 - x^10^(j+1)). (End)

More terms from Emeric Deutsch, Feb 23 2005

A262104 Pseudoprimes to base 7, written in base 7.

Original entry on oeis.org

6, 34, 643, 1431, 2023, 2245, 3136, 5215, 6061, 6601, 10121, 12361, 16123, 20032, 25345, 33155, 41545, 42601, 42652, 44122, 45406, 50026, 54561, 56035, 66522, 66666, 105403, 110254, 112612, 113345, 113356, 123616, 135206, 140011, 151142, 151354, 153022, 153101, 153352, 155554

Offset: 1

Abdul Gaffar Khan, Sep 11 2015

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

Cf. A005938 (pseudoprimes to base 7), A007093 (numbers in base 7).

Cf. A258189, A262101, A262102, A262103, A262105, A262154.

base = 7; t = {}; n = 1;
While[Length[t] < 40, n++;
If[! PrimeQ[n] && PowerMod[base, n - 1, n] == 1,
  AppendTo[t, FromDigits@IntegerDigits[n, 7]]]]; t
FromDigits[IntegerDigits[#,7]]&/@Select[Range[40000],CompositeQ[#] && PowerMod[ 7,#-1,#]==1&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Feb 24 2018 *)

lista(nn, b=7) = {for (n=1, nn, if (Mod(b, n)^(n-1)==1 && !ispseudoprime(n) && n>1, print1(subst(Pol(digits(n,b), x), x, 10), ", ");););} \\ Michel Marcus, Sep 30 2015

a(n) = A007093(A005938(n)).

A353115 Base-7 representation of A000422(n).

Original entry on oeis.org

1, 30, 636, 15412, 314241, 5363433, 122026543, 2113022646, 33321631443, 536166343420, 143160211151106, 34336660114345260, 11025520662032415631, 2346534662055361221261, 535040660151320030232114, 150206616416004416563301662

Offset: 1

Seiichi Manyama, Apr 24 2022

Cf. A000422, A353110, A353111, A353112, A353113, A353114, A353116, A353117.

Cf. A007093, A097582.

def A(k, n)
  (1..n).map{|i| (1..i).to_a.reverse.join.to_i.to_s(k).to_i}
end
p A(7, 20)

a(n) = A007093(A000422(n)).

A004053 For m=2,3,..., write m in bases 2,3,..,m.

Original entry on oeis.org

10, 11, 10, 100, 11, 10, 101, 12, 11, 10, 110, 20, 12, 11, 10, 111, 21, 13, 12, 11, 10, 1000, 22, 20, 13, 12, 11, 10, 1001, 100, 21, 14, 13, 12, 11, 10, 1010, 101, 22, 20, 14, 13, 12, 11, 10, 1011, 102, 23, 21, 15, 14, 13, 12, 11, 10, 1100, 110, 30, 22, 20, 15, 14, 13, 12, 11, 10

Offset: 2

Johan Boye (johbo(AT)ida.liu.se)

			Triangle begins:
    10;
    11,  10;
   100,  11, 10;
   101,  12, 11, 10;
   110,  20, 12, 11, 10;
   111,  21, 13, 12, 11, 10;
  1000,  22, 20, 13, 12, 11, 10;
  1001, 100, 21, 14, 13, 12, 11, 10;
  ...

Alois P. Heinz, Rows n = 2..20, flattened (row 21 would require digits > 9)

Cf. A007088, A007090, A007091, A007092, A007093, A007094, A007095.

FromDigits/@Flatten[Table[IntegerDigits[m,b],{m,2,20},{b,2,m}],1] (* Harvey P. Dale, Dec 01 2024 *)

T(n, k) = fromdigits(digits(n, k), 10);
tabl(nn) = for (n=2, nn, for (b=2, n, print1(T(n, b), ", "))); \\ Michel Marcus, Aug 30 2019

cp's OEIS Frontend

A073788 Numbers in base -7.

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A293660 Base-7 circular primes that are not base-7 repunits.

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A073794 Replace 7^k with (-7)^k in base 7 expansion of n.

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A097582 Base 7 representation of the concatenation of the first n numbers with the most significant digits first.

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A346691 Replace 7^k with (-1)^k in base-7 expansion of n.

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A037383 Numbers k such that every base-3 digit of k is a base-7 digit of k.

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Haskell

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A102679 Number of digits >= 7 in decimal representation of n.

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Maple

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A262104 Pseudoprimes to base 7, written in base 7.

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Mathematica