A062880 -id:A062880 - cp's OEIS frontend

A328849 Numbers in whose primorial base expansion only even digits appear.

0, 4, 12, 16, 24, 28, 60, 64, 72, 76, 84, 88, 120, 124, 132, 136, 144, 148, 180, 184, 192, 196, 204, 208, 420, 424, 432, 436, 444, 448, 480, 484, 492, 496, 504, 508, 540, 544, 552, 556, 564, 568, 600, 604, 612, 616, 624, 628, 840, 844, 852, 856, 864, 868, 900, 904, 912, 916, 924, 928, 960, 964, 972, 976, 984, 988, 1020, 1024

Offset: 1

Antti Karttunen, Oct 30 2019

Numbers for which the prime factor form (A276086) of their primorial base expansion is a square, A000290.

			144 is written as "4400" in primorial base (A049345), because 4*A002110(3) + 4*A002110(2) + 0*A002110(1) + 0*A002110(0) = 4*30 + 4*6 = 144, thus all the digits are even and 144 is included in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..9072
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..90720
Index entries for sequences related to primorial base.

Cf. A000290, A002110, A010052, A049345, A276086, A276156, A328770.
Cf. A328834, A328850 (squares in this sequence).
Similar sequences: A005823 (ternary), A014263 (decimal), A062880 (quaternary), A351893 (factorial base).

With[{max = 5}, bases = Prime@ Range[max, 1, -1]; nmax = Times @@ bases - 1; prmBaseDigits[n_] := IntegerDigits[n, MixedRadix[bases]]; Select[Range[0, nmax, 2], AllTrue[prmBaseDigits[#], EvenQ] &]] (* Amiram Eldar, May 23 2023 *)

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA328849(n) = issquare(A276086(n));

a(n) = 2*A328770(n).

A000196(A276086(a(n))) = A276086(a(n)/2) = A328834(n).

A071992 a(n) = 3n^2 + 2n - 4 * Sum_{k=1..n} A003159(k).

Original entry on oeis.org

1, 0, 1, 4, 5, 4, 1, 0, 1, 0, 1, 4, 5, 8, 13, 16, 17, 16, 17, 20, 21, 20, 17, 16, 17, 16, 13, 8, 5, 4, 1, 0, 1, 0, 1, 4, 5, 4, 1, 0, 1, 0, 1, 4, 5, 8, 13, 16, 17, 16, 17, 20, 21, 24, 29, 32, 37, 44, 49, 52, 53, 56, 61, 64, 65, 64, 65, 68, 69, 68, 65, 64, 65, 64, 65, 68, 69, 72, 77, 80

Offset: 1

Benoit Cloitre, Jun 17 2002

0 <= a(n) <= n for any n.

John Tyler Rascoe, Table of n, a(n) for n = 1..10241
Robert G. Wilson v, Illustration of initial terms

Cf. A000695, A003159, A062880.

def A003159(n): #see A003159
    return
def A071992_list(max_n):
    A,s = [],0
    for n in range(1,max_n+1):
        s += A003159(n)
        A.append(3*n**2 + 2*n - 4*s)
    return A # John Tyler Rascoe, Feb 24 2025

For any k, a(A062880(k)) = 0.

a(A000695(k)) = A000695(k).

A062879 Integers whose Zeckendorf expansion does not contain ones at even positions.

Original entry on oeis.org

0, 2, 5, 7, 13, 15, 18, 20, 34, 36, 39, 41, 47, 49, 52, 54, 89, 91, 94, 96, 102, 104, 107, 109, 123, 125, 128, 130, 136, 138, 141, 143, 233, 235, 238, 240, 246, 248, 251, 253, 267, 269, 272, 274, 280, 282, 285, 287, 322, 324, 327, 329, 335, 337, 340, 342, 356

Offset: 1

Antti Karttunen, Jun 26 2001

nonn

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

Bisection of A062877.
Subset of A022342.
Cf. A003714, A014417, A062880, A165276.

fibOddCount[n_] := Plus @@ (Reverse@IntegerDigits[n, 2])[[1 ;; -1 ;; 2]]; oddIndexed = fibOddCount /@ Select[Range[0, 10000], BitAnd[#, 2 #] == 0 &]; -1 + Position[oddIndexed, ?(# == 0 &)] // Flatten (* _Amiram Eldar, Jan 20 2020  *)

A062880(n) = A003714(a(n)).

A165276(a(n)) = 0. - Amiram Eldar, Jan 20 2020

Offset corrected by Amiram Eldar, Jan 20 2020

A351893 Numbers that contain only even digits in their factorial-base representation.

Original entry on oeis.org

0, 4, 12, 16, 48, 52, 60, 64, 96, 100, 108, 112, 240, 244, 252, 256, 288, 292, 300, 304, 336, 340, 348, 352, 480, 484, 492, 496, 528, 532, 540, 544, 576, 580, 588, 592, 1440, 1444, 1452, 1456, 1488, 1492, 1500, 1504, 1536, 1540, 1548, 1552, 1680, 1684, 1692, 1696

Offset: 1

Amiram Eldar, Feb 24 2022

All the terms are multiples of 4 (A008586).

			4 is a term since its factorial-base presentation, 20, has only even digits.
16 is a term since its factorial-base presentation, 220, has only even digits.

Amiram Eldar, Table of n, a(n) for n = 1..14400 (terms below 10!)
Wikipedia, Factorial number system.
Index entries for sequences related to factorial base representation.

Cf. A007623, A008586, A351894.
Subsequence: A052849 \ {2}.
Similar sequences: A005823 (ternary), A014263 (decimal), A062880 (quaternary).

max = 7; fctBaseDigits[n_] := IntegerDigits[n, MixedRadix[Range[max, 2, -1]]]; Select[Range[0, max!, 2], AllTrue[fctBaseDigits[#], EvenQ] &]

A004468 a(n) = Nim product 3 * n.

Original entry on oeis.org

0, 3, 1, 2, 12, 15, 13, 14, 4, 7, 5, 6, 8, 11, 9, 10, 48, 51, 49, 50, 60, 63, 61, 62, 52, 55, 53, 54, 56, 59, 57, 58, 16, 19, 17, 18, 28, 31, 29, 30, 20, 23, 21, 22, 24, 27, 25, 26, 32, 35, 33, 34, 44, 47, 45, 46, 36, 39, 37, 38, 40, 43, 41, 42, 192, 195, 193, 194, 204, 207, 205

Offset: 0

N. J. A. Sloane

From Jianing Song, Aug 10 2022: (Start)
Write n in quaternary (base 4), then replace each 1,2,3 by 3,1,2.
This is a permutation of the natural numbers; A006015 is the inverse permutation (since the nim product of 2 and 3 is 1). (End)

J. H. Conway, On Numbers and Games. Academic Press, NY, 1976, pp. 51-53.

Alois P. Heinz, Table of n, a(n) for n = 0..16383 (first 1001 terms from R. J. Mathar)
Index entries for sequences related to Nim-multiplication
Index entries for sequences that are permutations of the natural numbers

Row 3 of array in A051775.

Cf. A006015, A004469-A004480, A000695, A062880, A001196.

read("transforms") ;
# insert Maple procedures nimprodP2() and A051775() of the b-file in A051775 here.
A004468 := proc(n)
        A051775(3,n) ;
end proc:
L := [seq(A004468(n),n=0..1000)] ; # R. J. Mathar, May 28 2011
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 0,
      a(iquo(n, 4, 'r'))*4+[0, 3, 1, 2][r+1])
    end:
seq(a(n), n=0..70);  # Alois P. Heinz, Jan 25 2022

a[n_] := a[n] = If[n == 0, 0, {q, r} = QuotientRemainder[n, 4]; a[q]*4 + {0, 3, 1, 2}[[r + 1]]];
Table[a[n], {n, 0, 70}] (* Jean-François Alcover, May 20 2022, after Alois P. Heinz *)

a(n) = my(v=digits(n, 4), w=[0,3,1,2]); for(i=1, #v, v[i] = w[v[i]+1]); fromdigits(v, 4) \\ Jianing Song, Aug 10 2022

def a(n, D=[0, 3, 1, 2]):
    r, k = 0, 0
    while n>0: r+=D[n%4]*4**k; n//=4; k+=1
    return r
# Onur Ozkan, Mar 07 2023

a(n) = A051775(3,n).
From Jianing Song, Aug 10 2022: (Start)
a(n) = 3*n if n has only digits 0 or 1 in quaternary (n is in A000695). Otherwise, a(n) < 3*n.
a(n) = n/2 if n has only digits 0 or 2 in quaternary (n is in A062880). Otherwise, a(n) > n/2.
a(n) = 2*n/3 if and only if n has only digits 0 or 3 in quaternary (n is in A001196). Proof: let n = Sum_i d_i*4^i, d(i) = 0,1,2,3. Write A = Sum_{d_i=1} 4^i, B = Sum_{d_i=2} 4^i, then a(n) = 2*n/3 if and only if 3*A + B = 2/3*(A + 2*B), or B = 7*A. If A != 0, then A is of the form (4*s+1)*4^t, but 7*A is not of this form. So the only possible case is A = B = 0, namely n has only digits 0 or 3. (End)

More terms from Erich Friedman

A006015 Nim product 2*n.

Original entry on oeis.org

0, 2, 3, 1, 8, 10, 11, 9, 12, 14, 15, 13, 4, 6, 7, 5, 32, 34, 35, 33, 40, 42, 43, 41, 44, 46, 47, 45, 36, 38, 39, 37, 48, 50, 51, 49, 56, 58, 59, 57, 60, 62, 63, 61, 52, 54, 55, 53, 16, 18, 19, 17, 24, 26, 27, 25, 28, 30, 31, 29, 20, 22, 23, 21, 128, 130, 131, 129, 136, 138, 139

Offset: 0

N. J. A. Sloane

From Jianing Song, Aug 10 2022: (Start)
Write n in quaternary (base 4), then replace each 1,2,3 by 2,3,1.
This is a permutation of the natural numbers; A004468 is the inverse permutation (since the nim product of 2 and 3 is 1). (End)

J. H. Conway, On Numbers and Games. Academic Press, NY, 1976, pp. 51-53.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..16383 (first 1001 terms from R. J. Mathar)
Index entries for sequences related to Nim-multiplication
Index entries for sequences that are permutations of the natural numbers

Row 2 of array in A051775.

Cf. A004468-A004480, A000695, A062880, A001196.

a:= proc(n) option remember; `if`(n=0, 0,
      a(iquo(n, 4, 'r'))*4+[0, 2, 3, 1][r+1])
    end:
seq(a(n), n=0..70);  # Alois P. Heinz, Jan 25 2022

a[n_] := a[n] = If[n == 0, 0, {q, r} = QuotientRemainder[n, 4]; a[q]*4 + {0, 2, 3, 1}[[r + 1]]];
Table[a[n], {n, 0, 70}] (* Jean-François Alcover, May 20 2022, after Alois P. Heinz *)

a(n) = my(v=digits(n, 4), w=[0,2,3,1]); for(i=1, #v, v[i] = w[v[i]+1]); fromdigits(v, 4) \\ Jianing Song, Aug 10 2022

def a(n, D=[0, 2, 3, 1]):
    r, k = 0, 0
    while n>0: r+=D[n%4]*4**k; n//=4; k+=1
    return r
# Onur Ozkan, Mar 07 2023

From Jianing Song, Aug 10 2022: (Start)
a(n) = A051775(2,n).
a(n) = 2*n if n has only digits 0 or 1 in quaternary (n is in A000695). Otherwise, a(n) < 2*n.
a(n) = n/3 if n has only digits 0 or 3 in quaternary (n is in A001196). Otherwise, a(n) > n/3.
a(n) = 3*n/2 if and only if n has only digits 0 or 2 in quaternary (n is in A062880). Proof: let n = Sum_i d_i*4^i, d(i) = 0,1,2,3. Write A = Sum_{d_i=1} 4^i, B = Sum_{d_i=3} 4^i, then a(n) = 3*n/2 if and only if 2*A + B = 3/2*(A + 3*B), or A = 7*B. If B != 0, then B is of the form (4*s+1)*4^t, but 7*B is not of this form. So the only possible case is A = B = 0, namely n has only digits 0 or 2. (End)

More terms from Erich Friedman.

A054240 Bit-interleaved number addition table; like binary addition but carries shift 2 instead of 1; addition base sqrt(2).

Original entry on oeis.org

0, 1, 1, 2, 4, 2, 3, 3, 3, 3, 4, 6, 8, 6, 4, 5, 5, 9, 9, 5, 5, 6, 16, 6, 12, 6, 16, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 18, 12, 18, 16, 18, 12, 18, 8, 9, 9, 13, 13, 17, 17, 13, 13, 9, 9, 10, 12, 10, 24, 18, 20, 18, 24, 10, 12, 10, 11, 11, 11, 11, 19, 19, 19, 19, 11, 11, 11, 11, 12, 14, 32, 14

Offset: 0

Marc LeBrun, Feb 07 2000

			T(3,1)=6 because (0*2 + 1*sqrt(2) + 1*1) + (0*2 + 0*sqrt(2) + 1*1) = (1*2 + 1*sqrt(2) + 0*1) (i.e., base sqrt(2) addition).

Reinhard Zumkeller, Antidiagonals n=0..125 of square array, flattened

Cf. A000695, A054239, A057300, A062880, A352909 (pairs (i,j) such that A(i,j) = i+j).

Cf. A201651 (triangle read by rows).

import Data.Bits (xor, (.&.), shift)
a054240 :: Integer -> Integer -> Integer
a054240 x 0 = x
a054240 x y = a054240 (x `xor` y) (shift (x .&. y) 2)
a054240_adiag n =  map (\k -> a054240 (n - k) k) [0..n]
a054240_square = map a054240_adiag [0..]
-- Reinhard Zumkeller, Dec 03 2011

From Peter Munn, Dec 10 2019: (Start)
A(m,0) = A(0,m) = m.
A(n,k) = A(k,n).
A(n, A(m,k)) = A(A(n,m), k).
A(m,m) = 4*m.
A(2*n, 2*k) = 2*A(n,k).
A(A000695(n), A000695(k)) = A000695(n+k).
A(A000695(n), 2*A000695(k)) = A000695(n) + 2*A000695(k).
A(A000695(n) + 2*A000695(m), k) = A(A000695(n), k) + A(2*A000695(m), k) - k.
A(A057300(n), A057300(k)) = A057300(A(n,k)).
(End)

A269707 Decimal expansion of x = 3*Sum_{n in E} 1/10^n where E is the set of numbers whose base-4 representation consists of only 0's and 1's.

Original entry on oeis.org

3, 3, 0, 0, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 3, 0, 0, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 3, 0, 0, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 3, 0, 0, 3, 3, 0

Offset: 1

Michel Lagneau, Mar 10 2016

E = {0, 1, 4, 5, 16, 17, 20, 21, 64, ...} (A000695).
Among the real numbers it is exceptional for the decimal expansion of a real number to determine the decimal expansion of its reciprocal. The purpose of this sequence is to show an example of such a number.
x is irrational. Proof: For all n >= 1, the numbers 3*4^n, 3*4^n + 1, 3*4^n + 2, ..., 3*4^n + 4^(n - 1) each contain at least one base-4 digit different from 0 or 1. So, the decimal expansion of x contains sequences of consecutive zeros with an arbitrary length. Moreover, the decimal expansion also contains an infinite number of digits 3, which implies that x is not periodic, so irrational.
We obtain the following property: 1/x = 3*Sum_{n in 2*E} 1/10^(n + 1) where 2*E = {0, 2, 8, 10, 32, 34, 40, 42, ...} (A062880).

			x = 3.3003300000000003300330000000000000000000000000000...
1/x = 0.303000003030000000000000000000003030000030300000...

Daniel Duverney, Number Theory, World Scientific, 2010, 2.10 A striking number, pp. 19-20.

Index entries for transcendental numbers

Cf. A000695, A062880, A151666.

Digits:=200:nn:=5000:s:=0:
for n from 0 to nn do:
  x:=convert(n,base,4):n0:=nops(x):
  it:=0:ii:=0:
    for k from 1 to n0 while(ii=0) do:
     if x[k]=0 or x[k]=1
      then
      it:=it+1:
     else
    fi:
od:
if it=n0 then
s:= s+evalf(1/10^n):
else ii:=1:fi:
od:
print(3*s):
print(1/(3*s)):

a[n_] := 3 * Boole[Max @ IntegerDigits[n-1, 4] <= 1]; Array[a, 100] (* Amiram Eldar, Aug 06 2021 *)

Edited by Rick L. Shepherd, May 31 2016

A360613 Lexicographically earliest sequence of positive integers such that the products of the form a(2u-1) a(2*v) with u, v > 0 are all distinct.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 14, 17, 18, 19, 23, 24, 25, 29, 26, 31, 28, 33, 36, 37, 41, 40, 43, 47, 46, 49, 50, 51, 52, 53, 59, 55, 61, 57, 63, 64, 67, 71, 73, 79, 81, 83, 82, 85, 86, 87, 88, 89, 91, 93, 92, 95, 97, 101, 100, 103, 107, 109, 113, 111

Offset: 1

Rémy Sigrist, Feb 14 2023

nonn

In other words, the products of a term from the odd bisection by a term from the even bisection are all distinct.
If we consider the bitwise XOR operator instead of the multiplication then we obtain A000695 interleaved with A062880.
The value 1 is the only duplicate.
All prime numbers appear in this sequence, in ascending order.
For n = 1..50000, if m_n denotes the least positive value not in {a(2*u-1) * a(2*v), 1 <= 2*u-1 <= n and 1 <= 2*v <= n}, then a(n+1) = m_n or a(n+2) = m_n. Will this pattern last forever?

			The first terms, alongside the corresponding products, are:
  n   a(n)  Corresponding products
  --  ----  --------------------------
   1     1
   2     1   1
   3     2   2
   4     3   3,  6
   5     4   4, 12
   6     5   5, 10, 20
   7     7   7, 21, 35
   8     8   8, 16, 32,  56
   9     9   9, 27, 45,  72
  10    11  11, 22, 44,  77,  99
  11    13  13, 39, 65, 104, 143
  12    15  15, 30, 60, 105, 135, 195

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C program

Cf. A000695, A062880, A066724, A360627-A360628 (bisections), A360633 (products).

C
```
See Links section.
```

a(n) < a(n+2).

A033053 Numbers whose base-2 representation Sum_{i=0..m} d(i)*2^i has d(i)=1 when i != m mod 2.

Original entry on oeis.org

1, 3, 6, 7, 13, 15, 26, 27, 30, 31, 53, 55, 61, 63, 106, 107, 110, 111, 122, 123, 126, 127, 213, 215, 221, 223, 245, 247, 253, 255, 426, 427, 430, 431, 442, 443, 446, 447, 490, 491, 494, 495, 506, 507, 510, 511, 853, 855, 861, 863

Offset: 1

Clark Kimberling

Numbers 2^(2k)-1 - A062880(m) where 2^(2k-2) >= A062880(m) or 2^(2k+1)-1 - A000695(m) where 2^(2k-1) >= A000695(m). - Franklin T. Adams-Watters, Aug 30 2014

			26 = 11010_2 has m=4, and d(i) = 1 for i=1 and 3.
53 = 110101_2 has m=5, and d(i) = 1 for i=0, 2 and 4.

Robert Israel, Table of n, a(n) for n = 1..12286

Cf. A126684, A000695, A062880.

Disjoint with A032937 if more than 1 digit.

F:= proc(m)
   local n0,j,S;
   n0:= 2^m + add(2^(m-1-2*j),j=0..floor((m-1)/2));
   S:= combinat[powerset]({seq(2^(m-2*j),j=1..floor(m/2))});
   map(t -> convert(t,`+`)+n0,S);
end;
`union`(seq(F(m),m=0..24)}; # Robert Israel, Mar 30 2014

a(2j+2) = 4 a(j)+3,
a(2j+1) = 4 a(j) + 2 if j <= 3*2^(m-1)-2,
a(2j+1) = 4 a(j) + 1 otherwise, where m = floor(log_2(j+1)).

Definition corrected, incorrect cross-reference removed, and recurrence formulas by Robert Israel, Mar 30 2014

cp's OEIS Frontend

A328849 Numbers in whose primorial base expansion only even digits appear.

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A071992 a(n) = 3*n^2 + 2*n - 4 * Sum_{k=1..n} A003159(k).

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A062879 Integers whose Zeckendorf expansion does not contain ones at even positions.

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A351893 Numbers that contain only even digits in their factorial-base representation.

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Mathematica

A004468 a(n) = Nim product 3 * n.

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Maple

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A006015 Nim product 2*n.

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A054240 Bit-interleaved number addition table; like binary addition but carries shift 2 instead of 1; addition base sqrt(2).

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A071992 a(n) = 3n^2 + 2n - 4 * Sum_{k=1..n} A003159(k).

A360613 Lexicographically earliest sequence of positive integers such that the products of the form a(2u-1) a(2*v) with u, v > 0 are all distinct.