cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 31-40 of 42 results. Next

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

Views

Author

T. D. Noe, Apr 09 2014

Keywords

Comments

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

Crossrefs

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.

Programs

  • Mathematica
    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
  • PARI
    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

Extensions

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

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

Views

Author

M. F. Hasler, Jan 12 2016

Keywords

Crossrefs

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.

Programs

  • PARI
    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

Views

Author

M. F. Hasler, Jan 12 2016

Keywords

Crossrefs

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.

Programs

  • PARI
    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

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

Views

Author

Michel Marcus, Aug 31 2020

Keywords

Comments

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)

Examples

			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.

Links

Crossrefs

Cf. A005836, A007095, A023717, A020654, A020657, A037465, A037474, A037477, A337496, A337535.
Subsequence of A005836 U {2} and of A068499.

Programs

  • PARI
    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;
  • PARI
    \\ See Corneth link \\ David A. Corneth, Aug 31 2020

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

Views

Author

M. F. Hasler, Jan 12 2016

Keywords

Crossrefs

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.

Programs

  • PARI
    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

A267303 Earliest positive increasing sequence having no 6-term subsequence with constant second differences.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 18, 20, 21, 22, 25, 26, 28, 35, 36, 39, 41, 43, 46, 47, 49, 54, 55, 56, 64, 68, 74, 75, 81, 83, 84, 85, 87, 88, 90, 91, 93, 95, 103, 108, 109, 111, 118, 129, 134, 137, 140, 144, 145, 150, 151, 152, 153
Offset: 1

Views

Author

M. F. Hasler, Jan 12 2016

Keywords

Crossrefs

Cf. A267302 (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. 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.

Programs

  • PARI
    A267303(n, show=0, L=6, 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

A267304 Earliest nonnegative increasing sequence having no 7-term subsequence with constant second differences.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 21, 24, 26, 28, 29, 31, 33, 35, 38, 40, 41, 43, 49, 50, 52, 53, 58, 59, 62, 63, 64, 69, 70, 72, 73, 77, 81, 82
Offset: 1

Views

Author

M. F. Hasler, Jan 12 2016

Keywords

Crossrefs

Cf. A267305 (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. A267302 and A267303 for sequences avoiding 6-term subsequences with constant second differences.
Cf. A240556 and A240557 for sequences avoiding 5-term subsequences with constant third differences.

Programs

  • PARI
    A267304(n, show=0, L=7, 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

A267305 Earliest positive increasing sequence having no 7-term subsequence with constant second differences.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 22, 25, 27, 29, 30, 32, 34, 36, 39, 41, 42, 44, 50, 51, 53, 54, 59, 60, 63, 64, 65, 70, 71, 73, 74, 78, 82, 83
Offset: 1

Views

Author

M. F. Hasler, Jan 12 2016

Keywords

Crossrefs

Cf. A267304 (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. A267300 and A267301 for sequences avoiding 5-term subsequences with constant second differences.
Cf. A267302 and A267303 for sequences avoiding 6-term subsequences with constant second differences.
Cf. A240556 and A240557 for sequences avoiding 5-term subsequences with constant third differences.

Programs

  • PARI
    A267305(n, show=0, L=7, 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

A037470 a(n)=Sum{d(i)*7^i: i=0,1,...,m}, where Sum{d(i)*6^i: i=0,1,...,m} is the base 6 representation of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 82, 84, 85, 86
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

Except for the offset, identical to A020657: a(n)=A020657(n+1). - M. F. Hasler, Oct 05 2014

Links

Crossrefs

Essentially a duplicate of A020657. - N. J. A. Sloane, Jan 04 2016

Programs

  • Mathematica
    Table[FromDigits[RealDigits[n, 6], 7], {n, 0, 100}] (* Clark Kimberling, Aug 14 2012 *)
  • PARI
    a(n)=vector(#n=digits(n, 6), i, 7^(#n-i))*n~ \\ M. F. Hasler, Oct 05 2014

Extensions

Offset changed to 0 by Clark Kimberling, Aug 14 2012

A267306 Earliest nonnegative increasing sequence having no 6-term subsequence with constant third differences.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 28, 29, 31, 32, 33, 34, 35, 37, 38, 40, 47, 79, 93, 94, 96, 97, 98, 99, 100, 102, 103, 105, 110, 116, 122, 128, 130, 140, 148, 266, 281, 296, 303, 304, 306, 308, 311, 313, 318, 324, 326, 327, 328, 330, 331, 332, 337
Offset: 1

Views

Author

M. F. Hasler, Jan 12 2016

Keywords

Crossrefs

Cf. A267307 (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. A267302 and A267303 for sequences avoiding 6-term subsequences with constant second differences.
Cf. A267304 and A267305 for sequences avoiding 7-term subsequences with constant second differences.
Cf. A240556 and A240557 for sequences avoiding 5-term subsequences with constant third differences.

Programs

  • PARI
    A267306(n, show=0, L=6, 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

Extensions

More terms from Jinyuan Wang, Jan 01 2021
Previous Showing 31-40 of 42 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.