A023729 -id:A023729 - cp's OEIS frontend

A023733 Numbers with no 3's in base-5 expansion.

0, 1, 2, 4, 5, 6, 7, 9, 10, 11, 12, 14, 20, 21, 22, 24, 25, 26, 27, 29, 30, 31, 32, 34, 35, 36, 37, 39, 45, 46, 47, 49, 50, 51, 52, 54, 55, 56, 57, 59, 60, 61, 62, 64, 70, 71, 72, 74, 100, 101, 102, 104, 105, 106, 107, 109, 110, 111, 112, 114

Offset: 1

Olivier Gérard

			14 in base 5 is 24, which contains no 3's, so 14 is in the sequence.
15 in base 5 is 30, so 15 is not in the sequence.

Nathaniel Johnston, 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 5-automatic sequences.

Cf. A020654, A023721, A023725, A023729.

seq(`if`(numboccur(3,convert(n,base,5))=0,n,NULL),n=0..127); # Nathaniel Johnston, Jun 27 2011

Select[Range[0, 124], Count[IntegerDigits[#, 5], 3] == 0 &]
Select[Range[0,200],DigitCount[#,5,3]==0&] (* Harvey P. Dale, Jul 29 2024 *)

is(n)=while(n>3, if(n%5==3, return(0)); n\=5); 1 \\ Charles R Greathouse IV, Feb 12 2017

(0 to 124).filter(Integer.toString(, 5).indexOf("3") == -1) // _Alonso del Arte, Feb 05 2019

Sum_{n>=2} 1/a(n) = 7.2918685472993284072384543509909968409572571215800451577936556651148540560895813691253670323741759722063... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Apr 14 2025

A023721 Numbers with no 0's in their base-5 expansion.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24, 31, 32, 33, 34, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 49, 56, 57, 58, 59, 61, 62, 63, 64, 66, 67, 68, 69, 71, 72, 73, 74, 81, 82, 83, 84, 86, 87, 88, 89, 91, 92, 93

Offset: 1

Olivier Gérard

Nathaniel Johnston, 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 5-automatic sequences.

Cf. A020654, A023725, A023729, A023733.

seq(`if`(numboccur(0,convert(n,base,5))=0,n,NULL),n=1..127); # Nathaniel Johnston, Jun 27 2011

Select[ Range[ 120 ], (Count[ IntegerDigits[ #, 5 ], 0 ]==0)& ]
Select[Range[120], DigitCount[#, 5, 0] == 0 &] (* Amiram Eldar, Apr 14 2025 *)

is(n)=while(n, if(n%5==0, return(0)); n\=5); 1 \\ Charles R Greathouse IV, Feb 12 2017

Sum_{n>=1} 1/a(n) = 8.1899922882413061715479525413921657841497267151276815624858907606158756278085270372763455153366655369098... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Apr 14 2025

A204061 G.f.: exp( Sum_{n>=1} A001333(n)^2 * x^n/n ) where A001333(n) = A002203(n)/2, one-half the companion Pell numbers.

Original entry on oeis.org

1, 1, 5, 21, 101, 501, 2561, 13345, 70561, 377281, 2035285, 11059205, 60454005, 332138405, 1832677185, 10150115201, 56398558081, 314273655745, 1755700634981, 9830544087221, 55155558312901, 310027473436821, 1745567243959105, 9843160519978401, 55582528404717601

Offset: 0

Paul D. Hanna, Jan 10 2012

nonn

a(n) == 1 (mod 5) iff n has no 2's in its base 5 expansion (A023729), otherwise a(n) == 0 (mod 5); this is a conjecture needing proof.

			G.f.: A(x) = 1 + x + 5*x^2 + 21*x^3 + 101*x^4 + 501*x^5 + 2561*x^6 +...
where log(A(x)) = x + 3^2*x^2/2 + 7^2*x^3/3 + 17^2*x^4/4 + 41^2*x^5/5 + 99^2*x^6/6 + 239^2*x^7/7 +...+ A001333(n)^2*x^n/n +...
The last digit of the terms in this sequence seems to be either a '1' or a '5':
by conjecture, a(n) == 0 (mod 5) whenever n has a 2 in its base 5 expansion;
if true, terms a(2*5^k) through a(3*5^k - 1) all end with digit '5' for k>=0.

Paul D. Hanna, Table of n, a(n) for n = 0..625

Cf. A204062, A026933, A001333, A023729.

{A001333(n)=polcoeff((1-x)/(1-2*x-x^2+x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(k=1, n, A001333(k)^2*x^k/k)+x*O(x^n)), n)}

{a(n)=polcoeff(1/(sqrt(1+x+x*O(x^n))*(1-6*x+x^2+x*O(x^n))^(1/4)),n)}

G.f.: 1 / ( sqrt(1+x) * (1-6*x+x^2)^(1/4) ).
Self-convolution yields A026933: Sum_{k=0..n} a(n-k)*a(k) = Sum_{k=0..n} D(n-k,k)^2 where D(n,k) = A008288(n,k) are the Delannoy numbers.
a(n) ~ 2^(1/8) * GAMMA(3/4) * (3+2*sqrt(2))^(n+1/2) / (4 * Pi * n^(3/4)). - Vaclav Kotesovec, Oct 30 2014

A023725 Numbers with no 1's in their base-5 expansion.

Original entry on oeis.org

0, 2, 3, 4, 10, 12, 13, 14, 15, 17, 18, 19, 20, 22, 23, 24, 50, 52, 53, 54, 60, 62, 63, 64, 65, 67, 68, 69, 70, 72, 73, 74, 75, 77, 78, 79, 85, 87, 88, 89, 90, 92, 93, 94, 95, 97, 98, 99, 100, 102, 103, 104, 110, 112, 113, 114, 115, 117, 118, 119, 120

Offset: 1

Olivier Gérard

Nathaniel Johnston, 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 5-automatic sequences.

Cf. A020654, A023721, A023729, A023733.

seq(`if`(numboccur(1,convert(n,base,5))=0,n,NULL),n=0..127); # Nathaniel Johnston, Jun 27 2011

Select[ Range[ 0, 125 ], (Count[ IntegerDigits[ #, 5 ], 1 ]==0)& ]
Select[Range[0, 120], DigitCount[#, 5, 1] == 0 &] (* Amiram Eldar, Apr 14 2025 *)

is(n)=while(n>1, if(n%5==1, return(0)); n\=5); 1 \\ Charles R Greathouse IV, Feb 12 2017

Sum_{n>=2} 1/a(n) = 4.7113203882192880160403245816366085015069192113921100121384809791433027046475062716543062654277431569224... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Apr 14 2025

A249102 Numbers with no 1's in base-7 expansion.

Original entry on oeis.org

0, 2, 3, 4, 5, 6, 14, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 98, 100, 101, 102, 103, 104, 112, 114, 115, 116, 117, 118, 119, 121, 122, 123, 124, 125, 126, 128, 129, 130, 131, 132, 133

Offset: 1

Zak Seidov, Oct 21 2014

			14_10 = 20_7, 16_10 = 22_10, 17_10 = 23_7.
14 in base 7 is 20, which contains no 1s, so 14 is in the sequence.
15 in base 7 is 21, which contains one 1, so 15 is not in the sequence.
16 in base 7 is 22, so 16 is in the sequence.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Subsequence of A047306. Cf. A023721, A023725, A023729, A023733, A005823. This sequence has no terms in common with A016993.

Select[Range[0, 200], FreeQ[IntegerDigits[#, 7], 1] &] (* Seidov *)
Select[Range[0, 139], DigitCount[#, 7, 1] == 0 &] (* Alonso del Arte, Oct 26 2014 *)

fromdigits(v, b=10)=subst(Pol(v), 'x, b) \\ needed for gp < 2.63 or so
a(n)=a(n)=fromdigits(apply(k->if(k, k+1, 0), digits(n, 6)),7) \\ Charles R Greathouse IV, Oct 30 2014

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A023733 Numbers with no 3's in base-5 expansion.

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A023721 Numbers with no 0's in their base-5 expansion.

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A204061 G.f.: exp( Sum_{n>=1} A001333(n)^2 * x^n/n ) where A001333(n) = A002203(n)/2, one-half the companion Pell numbers.

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A023725 Numbers with no 1's in their base-5 expansion.

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A249102 Numbers with no 1's in base-7 expansion.

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