A020654 -id:A020654 - cp's OEIS frontend

A240556 Earliest nonnegative increasing sequence with no 5-term subsequence of constant third differences.

0, 1, 2, 3, 5, 7, 11, 15, 16, 27, 47, 48, 64, 95, 175, 196, 211, 212, 214, 247, 249, 252, 398, 839, 1002, 1014, 1016, 1035, 1036, 1037, 1051, 1054, 1072, 1121, 1143, 1146, 1172, 1258, 4271, 4282, 4284, 4336, 4571, 4578, 4582, 4598, 4613, 4622, 4628, 4646

Offset: 1

T. D. Noe, Apr 09 2014

nonn

For the positive sequence, see A240557, which is this sequence plus 1. Is there a simple way of determining this sequence, as in the case of the no 3-term arithmetic progression?

			After (0, 1, 2, 3, 5, 7), the number 10 is excluded since else the subsequence (0, 2, 3, 5, 10) would have successive 1st, 2nd and 3rd differences (2, 1, 2, 5), (-1, 1, 3) and (2, 2), which is constant and thus excluded.

Cf. A240557 (starting with 1).
No 3-term AP: A005836 (>=0), A003278 (>0);
no 4-term AP: A240075 (>=0), A240555 (>0);
no 5-term AP: A020654 (>=0), A020655 (>0);
no 6-term AP: A020656 (>=0), A005838 (>0);
no 7-term AP: A020657 (>=0), A020658 (>0);
no 8-term AP: A020659 (>=0), A020660 (>0);
no 9-term AP: A020661 (>=0), A020662 (>0);
no 10-term AP: A020663 (>=0), A020664 (>0).
Cf. A240075 and A240555 for sequences avoiding 4-term subsequences with constant second differences.

t = {0, 1, 2, 3}; Do[s = Table[Append[i, n], {i, Subsets[t, {4}]}]; If[! MemberQ[Flatten[Table[Differences[i, 4], {i, s}]], 0], AppendTo[t, n]], {n, 4, 5000}]; t

A240556(n,show=0,L=5,o=3,v=[0],D=v->v[2..-1]-v[1..-2])={ my(d,m); while( #v1,);#Set(d)>1||next(2),2);break));v[#v]} \\ M. F. Hasler, Jan 12 2016

A240557 Earliest positive increasing sequence with no 5-term subsequence of constant third differences.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 16, 17, 28, 48, 49, 65, 96, 176, 197, 212, 213, 215, 248, 250, 253, 399, 840, 1003, 1015, 1017, 1036, 1037, 1038, 1052, 1055, 1073, 1122, 1144, 1147, 1173, 1259, 4272, 4283, 4285, 4337, 4572, 4579, 4583, 4599, 4614, 4623, 4629, 4647

Offset: 1

T. D. Noe, Apr 09 2014

nonn

For the nonnegative sequence, see A240556, which is this sequence minus 1. Is there a simple way of determining this sequence, as in the case of the no 3-term arithmetic progression?

See crossreferences for sequences avoiding arithmetic progressions. - M. F. Hasler, Jan 12 2016

Cf. A240556 (starting with 0).
No 3-term AP: A005836 (>=0), A003278 (>0);
no 4-term AP: A240075 (>=0), A240555 (>0);
no 5-term AP: A020654 (>=0), A020655 (>0);
no 6-term AP: A020656 (>=0), A005838 (>0);
no 7-term AP: A020657 (>=0), A020658 (>0);
no 8-term AP: A020659 (>=0), A020660 (>0);
no 9-term AP: A020661 (>=0), A020662 (>0);
no 10-term AP: A020663 (>=0), A020664 (>0).
Cf. A240075 and A240555 for sequences avoiding 4-term subsequences with constant second differences.

t = {1, 2, 3, 4}; Do[s = Table[Append[i, n], {i, Subsets[t, {4}]}]; If[! MemberQ[Flatten[Table[Differences[i, 4], {i, s}]], 0], AppendTo[t, n]], {n, 5, 5000}]; t

A240557(n,show=0,L=5,o=3,v=[1],D=v->v[2..-1]-v[1..-2])={ my(d,m); while( #v1,);#Set(d)>1||next(2),2);break));v[#v]} \\ M. F. Hasler, Jan 12 2016

Definition corrected by M. F. Hasler, Jan 12 2016

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

Original entry on oeis.org

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

A267300 Earliest positive increasing sequence having no 5-term subsequence with constant second differences.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 8, 11, 13, 16, 19, 20, 22, 24, 30, 31, 36, 45, 46, 52, 55, 60, 62, 63, 66, 69, 71, 75, 86, 89, 92, 103, 111, 115, 119, 134, 137, 145, 152, 163, 176, 178, 179, 196, 200, 220, 223, 275, 276, 278, 281, 282, 284, 286, 294, 304, 316, 319, 326, 339, 353, 360, 363, 376, 379, 384, 390, 402, 414, 423, 429, 442

Offset: 1

M. F. Hasler, Jan 12 2016

nonn

Cf. A267301 (positive variant: starting with 1).
No 3-term AP: A005836 (>=0), A003278 (>0);
no 4-term AP: A240075 (>=0), A240555 (>0);
no 5-term AP: A020654 (>=0), A020655 (>0);
no 6-term AP: A020656 (>=0), A005838 (>0);
no 7-term AP: A020657 (>=0), A020658 (>0);
no 8-term AP: A020659 (>=0), A020660 (>0);
no 9-term AP: A020661 (>=0), A020662 (>0);
no 10-term AP: A020663 (>=0), A020664 (>0).
Cf. A240075 and A240555 for sequences avoiding 4-term subsequences with constant second differences.
Cf. A240556 and A240557 for sequences avoiding 5-term subsequences with constant third differences.

A267300(n, show=0, L=5, o=2, v=[0], D=v->v[2..-1]-v[1..-2])={ my(d, m); while( #v1, ); #Set(d)>1||next(2), 2); break)); v[#v]} \\ M. F. Hasler, Jan 12 2016

A267301 Earliest positive increasing sequence having no 5-term subsequence with constant second differences.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 17, 20, 21, 23, 25, 31, 32, 37, 46, 47, 53, 56, 61, 63, 64, 67, 70, 72, 76, 87, 90, 93, 104, 112, 116, 120, 135, 138, 146, 153, 164, 177, 179, 180, 197, 201, 221, 224, 276, 277, 279, 282, 283, 285, 287, 295, 305, 317, 320, 327, 340, 354, 361, 364, 377, 380, 385, 391, 403, 415, 424, 430, 443

Offset: 1

M. F. Hasler, Jan 12 2016

nonn

Cf. A267300 (nonnegative variant: starting with 0).
No 3-term AP: A005836 (>=0), A003278 (>0);
no 4-term AP: A240075 (>=0), A240555 (>0);
no 5-term AP: A020654 (>=0), A020655 (>0);
no 6-term AP: A020656 (>=0), A005838 (>0);
no 7-term AP: A020657 (>=0), A020658 (>0);
no 8-term AP: A020659 (>=0), A020660 (>0);
no 9-term AP: A020661 (>=0), A020662 (>0);
no 10-term AP: A020663 (>=0), A020664 (>0).
Cf. A240075 and A240555 for sequences avoiding 4-term subsequences with constant second differences.
Cf. A240556 and A240557 for sequences avoiding 5-term subsequences with constant third differences.

A267301(n, show=0, L=5, o=2, v=[1], D=v->v[2..-1]-v[1..-2])={ my(d, m); while( #v1, ); #Set(d)>1||next(2), 2); break)); v[#v]} \\ M. F. Hasler, Jan 12 2016

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

A337536 Numbers k for which there are only 2 bases b (2 and k+1) where the digits of k contain the digit b-1.

Original entry on oeis.org

2, 3, 4, 10, 36, 40, 82, 256

Offset: 1

Michel Marcus, Aug 31 2020

These could be called "nine-free numbers".
From David A. Corneth, Aug 31 2020: (Start)
This sequence has density 0. Conjecture: this sequence is finite and full. a(9) > 10^100 if it exists.
Suppose we want to see if 22792 = 1011021011_3 is a term. Since it has a digit of 2 in base 3, we can see that it is not. The next number that does not have the digit 2 in base 3 is 1011100000_3 = 22842, so we can proceed from there. In a similar way we can skip numbers based on bases b > 3. (End)
All terms of this sequence increased by 1 (except a(2)=3) are prime. - François Marques, Aug 31 2020
From Devansh Singh, Sep 19 2020: (Start)
If n is one less than an odd prime and we are interested in bases 3 <= b <= n-1 such that n in base b contains the digit b-1, then divisor of b (except 1) -1 cannot be the last digit since divisor of b divides n+1, which is not possible as n+1 is an odd prime.
If the last digit is 1, then b is odd as 1 = 2-1 and 2 cannot divide b as n+1 is an odd prime.
If the last digit is 0, then b-1 is the last digit of n-1 in base b.
b <= n/2 for even n,b <= (n+1)/2 for odd n.
This sequence is equivalent to the existence of only one prime generating polynomial = F(x) (having positive integer coefficients >=0 and <=b-1 for F(b)) such that F(2) = p.
There is no other prime generating polynomial = G(x) (having positive integer coefficients >=0 and <= b-1 for G(b)) that generates p for 2 < x = b <= (p-1)/2.
(End)

			2 is a term because 2 = 10_2 = 2_3, so both have the digit b-1, and there are no other bases where this happens.
4 is a term because 4 = 100_2 = 4_5, so both have the digit b-1, and there are no other bases where this happens.

David A. Corneth, PARI program

Cf. A005836, A007095, A023717, A020654, A020657, A037465, A037474, A037477, A337496, A337535.

Subsequence of A005836 U {2} and of A068499.

isok(n, b) = vecmax(digits(n, b)) == b-1;
b(n) = if (n==1, return (1)); my(b=3); while(!isok(n, b), b++); b; \\ A337535
is(n) = b(n) == n+1;

\\ See Corneth link \\ David A. Corneth, Aug 31 2020

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

A023729 Numbers with no 2's in their base-5 expansion.

Original entry on oeis.org

0, 1, 3, 4, 5, 6, 8, 9, 15, 16, 18, 19, 20, 21, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 40, 41, 43, 44, 45, 46, 48, 49, 75, 76, 78, 79, 80, 81, 83, 84, 90, 91, 93, 94, 95, 96, 98, 99, 100, 101, 103, 104, 105, 106, 108, 109, 115, 116, 118, 119, 120, 121

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, A023725, A023733.

seq(`if`(numboccur(2,convert(n,base,5))=0,n,NULL),n=0..127); # Nathaniel Johnston, Jun 27 2011

Select[ Range[ 0, 125 ], (Count[ IntegerDigits[ #, 5 ], 2 ]==0)& ]
Select[Range[0, 120], DigitCount[#, 5, 2] == 0 &] (* Amiram Eldar, Apr 14 2025 *)

is(n)=while(n>2, if(n%5==2, return(0)); n\=5); 1 \\ Charles R Greathouse IV, Feb 12 2017

Sum_{n>=2} 1/a(n) = 6.4926328481629227744899858111920644967528391159751448517967690160220746453627777857879224296518328062481... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Apr 14 2025

A267302 Earliest nonnegative increasing sequence having no 6-term subsequence with constant second differences.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 17, 19, 20, 21, 24, 25, 27, 34, 35, 38, 40, 42, 45, 46, 48, 53, 54, 55, 63, 67, 73, 74, 80, 82, 83, 84, 86, 87, 89, 90, 92, 94, 102, 107, 108, 110, 117, 128, 133, 136, 139, 143, 144, 149, 150, 151, 152

Offset: 1

M. F. Hasler, Jan 12 2016

nonn

Cf. A267303 (positive variant: starting with 1).
No 3-term AP: A005836 (>=0), A003278 (>0);
no 4-term AP: A240075 (>=0), A240555 (>0);
no 5-term AP: A020654 (>=0), A020655 (>0);
no 6-term AP: A020656 (>=0), A005838 (>0);
no 7-term AP: A020657 (>=0), A020658 (>0);
no 8-term AP: A020659 (>=0), A020660 (>0);
no 9-term AP: A020661 (>=0), A020662 (>0);
no 10-term AP: A020663 (>=0), A020664 (>0).
Cf. A240075 and A240555 for sequences avoiding 4-term subsequences with constant second differences.
Cf. A267300 and A267301 for sequences avoiding 5-term subsequences with constant second differences.
Cf. A240556 and A240557 for sequences avoiding 5-term subsequences with constant third differences.

A267302(n, show=0, L=6, o=2, v=[0], D=v->v[2..-1]-v[1..-2])={ my(d, m); while( #v1, ); #Set(d)>1||next(2), 2); break)); v[#v]} \\ M. F. Hasler, Jan 12 2016

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A240556 Earliest nonnegative increasing sequence with no 5-term subsequence of constant third differences.

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A240557 Earliest positive increasing sequence with no 5-term subsequence of constant third differences.

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

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A267300 Earliest positive increasing sequence having no 5-term subsequence with constant second differences.

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A267301 Earliest positive increasing sequence having no 5-term subsequence with constant second differences.

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PARI

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

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A337536 Numbers k for which there are only 2 bases b (2 and k+1) where the digits of k contain the digit b-1.

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PARI

PARI

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

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PARI