A014263 -id:A014263 - cp's OEIS frontend

A084545 Alternate number system in base 5.

1, 2, 3, 4, 5, 11, 12, 13, 14, 15, 21, 22, 23, 24, 25, 31, 32, 33, 34, 35, 41, 42, 43, 44, 45, 51, 52, 53, 54, 55, 111, 112, 113, 114, 115, 121, 122, 123, 124, 125, 131, 132, 133, 134, 135, 141, 142, 143, 144, 145, 151, 152, 153, 154, 155, 211, 212, 213, 214, 215, 221, 222

Offset: 1

Robert R. Forslund (forslund(AT)tbaytel.net), Jun 27 2003

			From _Hieronymus Fischer_, Jun 06 2012: (Start)
a(100)  = 345.
a(10^3) = 12445.
a(10^4) = 254445.
a(10^5) = 11144445.
a(10^6) = 223444445.
a(10^7) = 4524444445.
a(10^8) = 145544444445.
a(10^9) = 3521444444445. (End)

Hieronymus Fischer, Table of n, a(n) for n = 1..10000
EMIS, Mirror site for Southwest Journal of Pure and Applied Mathematics
R. R. Forslund, A logical alternative to the existing positional number system, Southwest Journal of Pure and Applied Mathematics, Vol. 1 1995 pp. 27-29.
R. R. Forslund, Positive Integer Pages [Wayback Machine link]
James E. Foster, A Number System without a Zero-Symbol, Mathematics Magazine, Vol. 21, No. 1. (1947), pp. 39-41.
Index entries for 10-automatic sequences.

Cf. A007931, A007932, A052382, A084544, A046034, A089581, A084984, A001742, A001743, A001744, A202267, A202268, A014261, A014263.

a(n) = my (w=5); while (n>w, n -= w; w *= 5); my (d=digits(w+n-1, 5)); d[1] = 0; fromdigits(d) + (10^(#d-1)-1)/9 \\ Rémy Sigrist, Dec 04 2019

From Hieronymus Fischer, Jun 06 and Jun 08 2012: (Start)
The formulas are designed to calculate base-10 numbers only using the digits 1..5.
a(n) = Sum_{j=0..m-1} (1 + b(j) mod 5)*10^j, where m = floor(log_5(4*n+1)), b(j) = floor((4*n+1-5^m)/(4*5^j)).
a(k*(5^n-1)/4) = k*(10^n-1)/9, for k = 1,2,3,4,5.
a((9*5^n-5)/4) = (14*10^n-5)/9 = 10^n + 5*(10^n-1)/9.
a((5^n-1)/4 - 1) = 5*(10^(n-1)-1)/9, n>1.
a(n) <= (10^log_5(4*n+1)-1)/9, equality holds for n=(5^k-1)/4, k>0.
a(n) > (5/10)*(10^log_5(4*n+1)-1)/9, n>0.
lim inf a(n)/10^log_5(4*n) = 1/18, for n --> infinity.
lim sup a(n)/10^log_5(4*n) = 1/9, for n --> infinity.
G.f.: g(x) = (x^(1/4)*(1-x))^(-1) sum_{j>=0} 10^j*z(j)^(5/4)*(1 - 6z(j)^5 + 5z(j)^6)/((1-z(j))(1-z(j)^5)), where z(j) = x^5^j.
Also: g(x) = (1/(1-x)) sum_{j>=0} (1-6(x^5^j)^5+5(x^5^j)^6)*x^5^j*f_j(x)/(1-x^5^j), where f_j(x) = 10^j*x^((5^j-1)/4)/(1-(x^5^j)^5). The f_j obey the recurrence f_0(x) = 1/(1-x^5), f_(j+1)(x) = 10x*f_j(x^5).
Also: g(x) = 1/(1-x))*(h_(5,0)(x) + h_(5,1)(x) + h_(5,2)(x) + h_(4,1)(x) + h_(5,4)(x) - 5*h_(5,5)(x)), where h_(5,k)(x) = sum_{j>=0} 10^j*x^((5^(j+1)-1)/4) * (x^5^j)^k/(1-(x^5^j)^5).
(End)

Offset set to 1 according to A007931, A007932 and more terms added by Hieronymus Fischer, Jun 06 2012

A169966 Numbers whose decimal expansion contains only 0's and 3's.

Original entry on oeis.org

0, 3, 30, 33, 300, 303, 330, 333, 3000, 3003, 3030, 3033, 3300, 3303, 3330, 3333, 30000, 30003, 30030, 30033, 30300, 30303, 30330, 30333, 33000, 33003, 33030, 33033, 33300, 33303, 33330, 33333, 300000, 300003, 300030, 300033, 300300, 300303, 300330, 300333

Offset: 1

N. J. A. Sloane, Aug 07 2010

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

Cf. A007088, A014263, A169964, A169965, A169967.

Cf. A078242, A204093, A204094, A204095, A097256.

a169966 n = a169966_list !! (n-1)
a169966_list = map (* 3) a007088_list
-- Reinhard Zumkeller, Jan 10 2012

Map[FromDigits,Tuples[{0,3},6]] (* Paolo Xausa, Oct 30 2023 *)

print1(0);for(d=1,5,for(n=2^(d-1),2^d-1,print1(", ");forstep(i=d-1,0,-1,print1((n>>i)%2*3)))) \\ Charles R Greathouse IV, Nov 16 2011

def a(n): return 3*int(bin(n)[2:])
print([a(n) for n in range(40)]) # Michael S. Branicky, Mar 30 2021

a(n+1) = Sum_{k>=0} A030308(n,k)*A093138(k+1). - Philippe Deléham, Oct 16 2011

a(n) = 3 * A007088(n-1).

A179082 Even numbers having an even sum of digits in their decimal representation.

Original entry on oeis.org

0, 2, 4, 6, 8, 20, 22, 24, 26, 28, 40, 42, 44, 46, 48, 60, 62, 64, 66, 68, 80, 82, 84, 86, 88, 110, 112, 114, 116, 118, 130, 132, 134, 136, 138, 150, 152, 154, 156, 158, 170, 172, 174, 176, 178, 190, 192, 194, 196, 198, 200, 202, 204, 206, 208, 220, 222, 224, 226

Offset: 1

Reinhard Zumkeller, Jun 28 2010

a(n) = A014263(n) for n <= 25;
intersection of A005843 and A054683: A059841(a(n))*(1-A179081(a(n)))=1;
complement of A179083 with respect to A005843;
complement of A179084 with respect to A054683;
a(n) mod 2 = 0 and A007953(a(n)) mod 2 = 0.

Select[Range[0,250,2],EvenQ[Total[IntegerDigits[#]]]&] (* Harvey P. Dale, Mar 19 2012 *)

A029581 Numbers in which all digits are composite.

Original entry on oeis.org

4, 6, 8, 9, 44, 46, 48, 49, 64, 66, 68, 69, 84, 86, 88, 89, 94, 96, 98, 99, 444, 446, 448, 449, 464, 466, 468, 469, 484, 486, 488, 489, 494, 496, 498, 499, 644, 646, 648, 649, 664, 666, 668, 669, 684, 686, 688, 689, 694, 696, 698, 699, 844, 846, 848

Offset: 1

N. J. A. Sloane

If n is represented as a zerofree base-4 number (see A084544) according to n=d(m)d(m-1)...d(3)d(2)d(1)d(0) then a(n) = Sum_{j=0..m} c(d(j))*10^j, where c(k)=4,6,8,9 for k=1..4. - Hieronymus Fischer, May 30 2012

			From _Hieronymus Fischer_, May 30 2012: (Start)
a(1000) = 88649.
a(10^4) = 6468989
a(10^5) = 449466489. (End)

Hieronymus Fischer, Table of n, a(n) for n = 1..10000
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
Index entries for 10-automatic sequences.

Cf. A002808, A001744, A046034, A084544, A084984, A017042, A001743, A014261, A014263, A202267, A202268.

[n: n in [1..1000] | Set(Intseq(n)) subset [4, 6, 8, 9]]; // Vincenzo Librandi, Dec 17 2018

Table[FromDigits/@Tuples[{4, 6, 8, 9}, n], {n, 3}] // Flatten (* Vincenzo Librandi, Dec 17 2018 *)

From Hieronymus Fischer, May 30 and Jun 25 2012: (Start)
a(n) = Sum_{j=0..m-1} (2*b(j) mod 8 + 4 + floor(b(j)/4) - floor((b(j)+1)/4))*10^j, where m = floor(log_4(3*n+1)), b(j) = floor((3*n+1-4^m)/(3*4^j)).
Also: a(n) = Sum_{j=0..m-1} (A010877(2*b(j)) + 4 + A002265(b(j)) - A002265(b(j)+1))*10^j.
Special values:
a(1*(4^n-1)/3) = 4*(10^n-1)/9.
a(2*(4^n-1)/3) = 2*(10^n-1)/3.
a(3*(4^n-1)/3) = 8*(10^n-1)/9.
a(4*(4^n-1)/3) = 10^n-1.
a(n) < 4*(10^log_4(3*n+1)-1)/9, equality holds for n=(4^k-1)/3, k > 0.
a(n) < 4*A084544(n), equality holds iff all digits of A084544(n) are 1.
a(n) > 2*A084544(n).
Lower and upper limits:
lim inf a(n)/10^log_4(n) = 1/10*10^log_4(3) = 0.62127870, for n --> inf.
lim sup a(n)/10^log_4(n) = 4/9*10^log_4(3) = 2.756123868970, for n --> inf.
where 10^log_4(n) = n^1.66096404744...
G.f.: g(x) = (x^(1/3)*(1-x))^(-1) Sum_{j>=0} 10^j*z(j)^(4/3)*(1-z(j))*(4 + 6z(j) + 8*z(j)^2 + 9*z(j)^3)/(1-z(j)^4), where z(j) = x^4^j.
Also: g(x) = (1/(1-x))*(4*h_(4,0)(x) + 2*h_(4,1)(x) + 2*h_(4,2)(x) + h_(4,3)(x) - 9*h_(4,4)(x)), where h_(4,k)(x) = Sum_{j>=0} 10^j*x^((4^(j+1)-1)/3)*(x^(k*4^j)/(1-x^4^(j+1)). (End)
Sum_{n>=1} 1/a(n) = 1.039691381254753739202528087006945643166147087095114911673083135126969046250... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 15 2024

Offset corrected by Arkadiusz Wesolowski, Oct 03 2011

A062293 Smallest multiple k*n of n which has even digits and is a palindrome or becomes a palindrome when 0's are added on the left (e.g., 10 becomes 010, which is a palindrome).

Original entry on oeis.org

0, 2, 2, 6, 4, 20, 6, 686, 8, 666, 20, 22, 60, 2002, 686, 60, 80, 646, 666, 646, 20, 6006, 22, 828, 600, 200, 2002, 8886888, 868, 464, 60, 868, 800, 66, 646, 6860, 828, 222, 646, 6006, 40, 22222, 6006, 68886, 44, 6660, 828, 282, 4224, 686, 200, 42024, 4004, 424, 8886888, 220, 8008, 68286, 464, 68086, 60

Offset: 0

Amarnath Murthy, Jun 18 2001

Every integer n has a multiple of the form 99...9900...00. To see that n has a multiple that's a palindrome (allowing 0's on the left) with even digits, let 9n divide 99...9900...00; then n divides 22...2200...00. - Dean Hickerson, Jun 29 2001

			a(7) = 686 as 686 = 98*7 is the smallest palindrome multiple of 7 with even digits.

Cf. A062279. Values of k are given in A061797.

Cf. A014263, A136522, A004151.

stop := 500000; for n := 0 to 60 do k := 1; test := true; while test and k < stop do m := omit_trailzeros(n*k); if test := not all_even(m) or m <> int_reverse(m) then inc(k); end; end; if k < stop then write(n*k," "); else write(-1," "); end; end;

a062293 0 = 0
a062293 n = head [x | x <- map (* n) [1..],
                 all (`elem` "02468") $ show x, a136522 (a004151 x) == 1]
-- Reinhard Zumkeller, Feb 01 2012

Corrected and extended by Klaus Brockhaus, Jun 21 2001

A338839 No odd digit is present in a(n) + a(n+1).

Original entry on oeis.org

0, 2, 4, 16, 6, 14, 8, 12, 10, 18, 22, 20, 24, 36, 26, 34, 28, 32, 30, 38, 42, 40, 44, 156, 46, 154, 48, 152, 50, 150, 52, 148, 54, 146, 56, 144, 58, 142, 60, 140, 62, 138, 64, 136, 66, 134, 68, 132, 70, 130, 72, 128, 74, 126, 76, 124, 78, 122, 80, 120, 82, 118, 84, 116, 86, 114, 88, 112, 90, 110, 92

Offset: 1

Eric Angelini and Carole Dubois, Nov 11 2020

This is the lexicographically earliest sequence of distinct nonnegative terms with this property.

			a(1) + a(2) = 0 + 2 = 2 (no odd digit is present);
a(2) + a(3) = 2 + 4 = 6 (no odd digit is present);
a(3) + a(4) = 4 + 16 = 20 (no odd digit is present); etc.

Carole Dubois, Table of n, a(n) for n = 1..9999

Cf. A014263 (no odd digit).

Cf. A338840, A338841, A338842, A338843, A338844, A338845, A338846 (variants on the same idea).

Block[{a = {0}}, Do[Block[{k = 2}, While[Nand[FreeQ[a, k], AllTrue[IntegerDigits@ Total[a[[-1]] + k], EvenQ]], k += 2]; AppendTo[a, k]], {i, 2, 71}]; a] (* Michael De Vlieger, Nov 12 2020 *)

A338841 No odd digit is present in a(n) * a(n+1).

Original entry on oeis.org

0, 1, 2, 3, 8, 5, 4, 6, 7, 12, 17, 24, 10, 20, 11, 22, 13, 16, 14, 19, 32, 9, 52, 39, 58, 36, 18, 26, 31, 28, 15, 40, 21, 42, 53, 54, 46, 44, 47, 60, 34, 59, 38, 69, 94, 30, 68, 33, 62, 43, 48, 50, 56, 73, 88, 23, 96, 25, 80, 35, 64, 41, 104, 27, 84, 51, 122, 71, 66, 37, 72, 29, 76, 79, 112, 74, 63, 102, 61

Offset: 1

Eric Angelini and Carole Dubois, Nov 11 2020

This is the lexicographically earliest sequence of distinct nonnegative terms with this property and also a permutation of the nonnegative integers.

			a(1) * a(2) = 0 * 1 = 0 (no odd digit is present);
a(2) * a(3) = 1 * 2 = 2 (no odd digit is present);
a(3) * a(4) = 2 * 3 = 6 (no odd digit is present);
a(4) * a(5) = 3 * 8 = 24 (no odd digit is present); etc.

Cf. A014263 (no odd digits).

Cf. A338839, A338840, A338842, A338843, A338844, A338845, A338846 (variants on the same idea).

Block[{a = {0}}, Do[Block[{k = 1}, While[Nand[FreeQ[a, k], NoneTrue[IntegerDigits@ Total[a[[-1]]*k], OddQ]], k++]; AppendTo[a, k]], {i, 2, 79}]; a] (* Michael De Vlieger, Nov 12 2020 *)

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

Original entry on oeis.org

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).

A045926 All digits even and nonzero.

Original entry on oeis.org

2, 4, 6, 8, 22, 24, 26, 28, 42, 44, 46, 48, 62, 64, 66, 68, 82, 84, 86, 88, 222, 224, 226, 228, 242, 244, 246, 248, 262, 264, 266, 268, 282, 284, 286, 288, 422, 424, 426, 428, 442, 444, 446, 448, 462, 464, 466, 468, 482, 484, 486, 488, 622, 624, 626, 628, 642

Offset: 1

Felice Russo

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.

Cf. A045927, A014261, A014263, A192370.

a045926 n = a045926_list !! (n-1)
a045926_list = filter (all (`elem` "2468") . show) [2, 4..]
-- Reinhard Zumkeller, Jan 01 2013

def A045926(n):
    m = (3*n+1).bit_length()-1>>1
    return int(''.join((str(((3*n+1-(1<<(m<<1)))//(3<<((m-1-j)<<1))&3)+1) for j in range(m))))<<1 # Chai Wah Wu, Feb 08 2023

a(n) = 2 * A084544(n). - Reinhard Zumkeller, Jan 01 2013

More terms from Patrick De Geest, Jun 15 1999

A085557 Numbers that have more prime digits than nonprime digits.

Original entry on oeis.org

2, 3, 5, 7, 22, 23, 25, 27, 32, 33, 35, 37, 52, 53, 55, 57, 72, 73, 75, 77, 122, 123, 125, 127, 132, 133, 135, 137, 152, 153, 155, 157, 172, 173, 175, 177, 202, 203, 205, 207, 212, 213, 215, 217, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232

Offset: 1

Jason Earls, Jul 04 2003

Begins to differ from A046034 at the 21st term (which is the first 3-digit term).

			133 is in the sequence as the prime digits are 3 and 3 (those are two digits; counted with multiplicity) and one nonprime digit 1 and so there are more prime digits than nonprime digits. - _David A. Corneth_, Sep 06 2020

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A193238, A046034, A046035, A118950, A019546, A203263, A035232, A039996, A085823, A052382, A084544, A084984, A017042, A001743, A001744, A014261, A014263, A202267, A202268, A211681.

is(n) = my(d = digits(n), c = 0); for(i = 1, #d, if(isprime(d[i]), c++)); c<<1 > #d \\ David A. Corneth, Sep 06 2020

from itertools import count, islice
def A085557_gen(startvalue=1): # generator of terms
    return filter(lambda n:len(s:=str(n))<(sum(1 for d in s if d in {'2','3','5','7'})<<1),count(max(startvalue,1)))
A085557_list = list(islice(A085557_gen(),20)) # Chai Wah Wu, Feb 08 2023

cp's OEIS Frontend

A084545 Alternate number system in base 5.

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A169966 Numbers whose decimal expansion contains only 0's and 3's.

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A179082 Even numbers having an even sum of digits in their decimal representation.

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A029581 Numbers in which all digits are composite.

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A062293 Smallest multiple k*n of n which has even digits and is a palindrome or becomes a palindrome when 0's are added on the left (e.g., 10 becomes 010, which is a palindrome).

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A338839 No odd digit is present in a(n) + a(n+1).

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A338841 No odd digit is present in a(n) * a(n+1).

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

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