A020654 -id:A020654 - cp's OEIS frontend

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

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

M. F. Hasler, Jan 12 2016

nonn

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.

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

M. F. Hasler, Jan 12 2016

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.

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

M. F. Hasler, Jan 12 2016

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.

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

A303787 a(n) = Sum_{i=0..m} d(i)4^i, where Sum_{i=0..m} d(i)5^i is the base-5 representation of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 4, 5, 6, 7, 8, 8, 9, 10, 11, 12, 12, 13, 14, 15, 16, 16, 17, 18, 19, 20, 16, 17, 18, 19, 20, 20, 21, 22, 23, 24, 24, 25, 26, 27, 28, 28, 29, 30, 31, 32, 32, 33, 34, 35, 36, 32, 33, 34, 35, 36, 36, 37, 38, 39, 40, 40, 41, 42, 43, 44, 44, 45, 46, 47, 48, 48, 49, 50, 51

Offset: 0

Seiichi Manyama, Apr 30 2018

			13 = 23_5, so a(13) = 2*4 + 3 = 11.
14 = 24_5, so a(14) = 2*4 + 4 = 12.
15 = 30_5, so a(15) = 3*4 + 0 = 12.
16 = 31_5, so a(16) = 3*4 + 1 = 13.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Sum_{i=0..m} d(i)*b^i, where Sum_{i=0..m} d(i)*(b+1)^i is the base (b+1) representation of n: A065361 (b=2), A215090 (b=3), this sequence (b=4), A303788 (b=5), A303789 (b=6).

Cf. A020654, A037459.

function a(n)
    m, r, b = n, 0, 1
    while m > 0
        m, q = divrem(m, 5)
        r += b * q
        b *= 4
    end
r end; [a(n) for n in 0:73] |> println # Peter Luschny, Jan 03 2021

a(n) = fromdigits(digits(n, 5), 4); \\ Michel Marcus, May 02 2018

def f(k, ary)
  (0..ary.size - 1).inject(0){|s, i| s + ary[i] * k ** i}
end
def A(k, n)
  (0..n).map{|i| f(k, i.to_s(k + 1).split('').map(&:to_i).reverse)}
end
p A(4, 100)

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

M. F. Hasler, Jan 12 2016

nonn

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.

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

More terms from Jinyuan Wang, Jan 01 2021

A267307 Earliest positive increasing sequence having no 6-term subsequence with constant third differences.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 29, 30, 32, 33, 34, 35, 36, 38, 39, 41, 48, 80, 94, 95, 97, 98, 99, 100, 101, 103, 104, 106, 111, 117, 123, 129, 131, 141, 149, 267, 282, 297, 304, 305, 307, 309, 312, 314, 319, 325, 327, 328, 329, 331, 332, 333, 338

Offset: 1

M. F. Hasler, Jan 12 2016

nonn

Cf. A267306 (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. 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.

A267307(n, show=0, L=6, 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

More terms from Jinyuan Wang, Jan 08 2021

A037459 Sum{d(i)5^i: i=0,1,...,m}, where Sum{d(i)4^i: i=0,1,...,m} is the base 4 representation of n.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 15, 16, 17, 18, 25, 26, 27, 28, 30, 31, 32, 33, 35, 36, 37, 38, 40, 41, 42, 43, 50, 51, 52, 53, 55, 56, 57, 58, 60, 61, 62, 63, 65, 66, 67, 68, 75, 76, 77, 78, 80, 81, 82, 83, 85, 86, 87, 88, 90, 91, 92, 93

Offset: 1

Clark Kimberling

A020654 \ {0}. [From R. J. Mathar, Oct 20 2008]

A337143 Numbers k for which there are only 3 bases b (2, k+1 and another one) in which the digits of k contain the digit b-1.

Original entry on oeis.org

5, 6, 8, 9, 12, 16, 18, 28, 37, 81, 85, 88, 130, 150, 262, 810, 1030, 1032, 4132, 9828, 9832, 10662, 10666, 562576, 562578

Offset: 1

François Marques, Sep 14 2020

This sequence is the list of indices k such that A337496(k)=3.
Conjecture: this sequence is finite and full. a(26) > 3.8*10^12 if it exists.
All terms of this sequence increased by 1 are either prime numbers, or prime numbers squared, or 2 times a prime number because if b is a strict divisor of k+1, the digit for the units in the expansion of k in base b is b-1 so it must be 2 or the third base. In fact k+1 could have been equal to 8=2*4 but 7 is not a term of the sequence (7 = 111_2 = 21_3 = 13_4 = 7_8).

			a(7)=18 because there are only 3 bases (2, 19 and 3) which satisfy the condition of the definition (18=200_3) and 18 is the seventh of these numbers.

Cf. Numbers with at least one digit b-1 in base b : A074940 (b=3), A337250 (b=4), A337572 (b=5), A011539 (b=10), A095778 (b=11).
Cf. Numbers with no digit b-1 in base b: A005836 (b=3), A023717 (b=4), A020654 (b=5), A037465 (b=6), A020657 (b=7), A037474 (b=8), A037477 (b=9), A007095 (b=10), A065039 (b=11).
Cf. A337496, A337535, A337536.

cp's OEIS Frontend

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

Views

Author

Keywords

Crossrefs

Programs

PARI

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

Views

Author

Keywords

Crossrefs

Programs

PARI

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

Views

Author

Keywords

Crossrefs

Programs

PARI

A303787 a(n) = Sum_{i=0..m} d(i)*4^i, where Sum_{i=0..m} d(i)*5^i is the base-5 representation of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Julia

PARI

Ruby

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

Views

Author

Keywords

Crossrefs

Programs

PARI

Extensions

A267307 Earliest positive increasing sequence having no 6-term subsequence with constant third differences.

Views

Author

Keywords

Crossrefs

Programs

PARI

Extensions

A037459 Sum{d(i)*5^i: i=0,1,...,m}, where Sum{d(i)*4^i: i=0,1,...,m} is the base 4 representation of n.

Views

Author

Keywords

Formula

A337143 Numbers k for which there are only 3 bases b (2, k+1 and another one) in which the digits of k contain the digit b-1.

Views

Author

Keywords

Comments

Examples

Crossrefs

A303787 a(n) = Sum_{i=0..m} d(i)4^i, where Sum_{i=0..m} d(i)5^i is the base-5 representation of n.

A037459 Sum{d(i)5^i: i=0,1,...,m}, where Sum{d(i)4^i: i=0,1,...,m} is the base 4 representation of n.