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.

Showing 1-5 of 5 results.

A065438 Complement of A065039.

Original entry on oeis.org

10, 21, 32, 43, 54, 65, 76, 87, 98, 109, 110, 121, 132, 143, 154, 165, 176, 187, 198, 209, 220, 221, 232, 243, 254, 265, 276, 287, 298, 309, 320, 331, 332, 343, 354, 365, 376, 387, 398, 409, 420, 431, 442, 443, 454, 465, 476, 487, 498, 509, 520, 531, 542
Offset: 1

Views

Author

Santi Spadaro, Nov 17 2001

Keywords

A067080 If n = ab...def in decimal notation then the left digitorial function Ld(n) = ab...def*ab...de*ab...d*...*ab*a.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 160, 164, 168, 172, 176, 180, 184, 188, 192, 196, 250, 255, 260, 265, 270, 275, 280, 285, 290, 295, 360, 366, 372
Offset: 1

Views

Author

Amarnath Murthy, Jan 05 2002

Keywords

Comments

This entry should probably start at n=0, just as A067079 does. But that would require a number of changes, so it can wait until the editors have more free time. - N. J. A. Sloane, Nov 29 2014

Examples

			Ld(256) = 256*25*2 =12800.
a(31)=floor(31/10^0)*floor(31/10^1)=31*3=93;
a(42)=168 since 42=42(base-10) and so a(42)=42*4(base-10)=42*4=168.
		

Crossrefs

For formulas regarding a general parameter p (i.e. terms floor(n/p^k)) see A132264.
For the product of terms floor(n/p^k) for p=2 to p=12 see A098844(p=2), A132027(p=3)-A132033(p=9), A132263(p=11), A132264(p=12).
For the products of terms 1+floor(n/p^k) see A132269-A132272, A132327, A132328.

Programs

  • Haskell
    a067080 n = if n <= 9 then n else n * a067080 (n `div` 10)
    -- Reinhard Zumkeller, Nov 29 2012
  • Mathematica
    Table[d = IntegerDigits[n]; rd = 1; While[ Length[d] > 0, rd = rd*FromDigits[d]; d = Drop[d, -1]]; rd, {n, 1, 75} ]
    Table[Times@@NestList[Quotient[#,10]&,n,IntegerLength[n]-1],{n,70}] (* Harvey P. Dale, Dec 16 2013 *)
  • PARI
    a(n)=my(t=n);while(n\=10,t*=n); t \\ Charles R Greathouse IV, Nov 20 2012
    

Formula

a(n) = Product_{k=1..length(n)} floor(n/10^(k-1)). - Vladeta Jovovic, Jan 08 2002
From Hieronymus Fischer, Aug 13 2007: (Start)
a(n) = product{0<=k<=floor(log_10(n)), floor(n/10^k)}, n>=1.
Recurrence:
a(n) = n*a(floor(n/10));
a(n*10^m) = n^m*10^(m(m+1)/2)*a(n).
a(k*10^m) = k^(m+1)*10^(m(m+1)/2), for 0
a(n) <= b(n), where b(n)=n^(1+floor(log_10(n)))/10^(1/2*(1+floor(log_10(n)))*floor(log_10(n))); equality holds for n=k*10^m, m>=0, 1<=k<10. Here b(n) can also be written n^(1+floor(log_10(n)))/10^A000217(floor(log_10(n))).
Also: a(n) <= 3^((1-log_10(3))/2)*n^((1+log_10(n))/2)=1.332718...*10^A000217(log_10(n)), equality for n=3*10^m, m>=0.
a(n) > c*b(n), where c=0.472362443816572... (see constant A132026).
Also: a(n) > c*2^((1-log_10(2))/2)*n^((1+log_10(n))/2) = 0.601839...*10^A000217(log_10(n)).
lim inf a(n)/b(n) = 0.472362443816572..., for n-->oo.
lim sup a(n)/b(n) = 1, for n-->oo.
lim inf a(n)/n^((1+log_10(n))/2) = 0.472362443816572...*sqrt(2)/2^log_10(sqrt(2)), for n-->oo.
lim sup a(n)/n^((1+log_10(n))/2) = sqrt(3)/3^log_10(sqrt(3)), for n-->oo.
lim inf a(n)/a(n+1) = 0.472362443816572... for n-->oo (see constant A132026).
a(n) = O(n^((1+log_10(n))/2)). (End)

Extensions

More terms from Robert G. Wilson v, Jan 07 2002

A171397 Write n in base 10, but then read it as if it were written in base 11: if n = Sum_{i >= 0} d_i*10^i, with 0 <= d_i <= 9, then a(n) = Sum_{i >= 0} d_i*11^i.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72
Offset: 0

Author

Paul Weisenhorn, Jul 11 2011

Keywords

Comments

This is the sequence of all decimal integers that are created when base 10 numbers are interpreted as base 11 numbers.
Numbers without digit A (=10) in their representation in base 11. Complement of A095778. - François Marques, Oct 20 2020
Original definition: Earliest sequence containing no 11-term arithmetic progression.
In general, if p is prime, the earliest sequence containing no p-term arithmetic progression is created when base (p-1) numbers are interpreted as base p numbers.

Examples

			a(53)=58 because 53_11 in base 11 equals 58. - _François Marques_, Oct 20 2020
		

References

  • D. E. Arganbright, Mathematical Modeling with Spreadsheets, ABACUS, Vol. 3, #4(1986), 19-31.

Crossrefs

Different from A065039. - Alois P. Heinz, Sep 07 2011
CNumbers with at least one digit b-1 in base b : A074940 (b=3), A337250 (b=4), A337572 (b=5), A333656 (b=6), A337141 (b=7), A337239 (b=8), A338090 (b=9), A011539 (b=10), A095778 (b=11).
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), this sequence (b=11).

Programs

  • Maple
    seq(`if`(numboccur (10, convert (n, base, 11))=0, n, NULL), n=0..122);
    # second Maple program:
    a:= n-> (l-> add(l[i]*11^(i-1), i=1..nops(l)))(convert(n, base, 10)):
    seq(a(n), n=0..66);  # Alois P. Heinz, Aug 30 2024
  • Mathematica
    Table[FromDigits[RealDigits[n, 10], 11], {n, 0, 100}] (* François Marques, Oct 20 2020 *)
  • PARI
    a(n) = fromdigits(digits(n), 11); \\ Michel Marcus, Oct 09 2020
    
  • Python
    def A171397(n): return int(str(n),11) # Chai Wah Wu, Aug 30 2024

Extensions

Edited by N. J. A. Sloane, Aug 31 2024

A067082 If n = abc...def in decimal notation then the right digit sum function = abc...def + bc...def + c...def + ... + def + ef + f.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 70, 72
Offset: 0

Author

Amarnath Murthy, Jan 05 2002

Keywords

Examples

			a(256) = 256 + 56 + 6 = 318.
		

Crossrefs

Programs

  • Mathematica
    Table[d = IntegerDigits[n]; rd = 0; While[ Length[d] > 0, rd = rd + FromDigits[d]; d = Drop[d, 1]]; rd, {n, 0, 75} ]

Formula

a(abcd) = abcd+(abcd-1000a)+(abcd-1000a-100b)+(abcd-1000a-100b-10c).
n*length(n)-Sum_{k=1..length(n)} 10^k*floor(n/10^k). - Vladeta Jovovic, Jan 08 2002

Extensions

More terms from Robert G. Wilson v, Jan 07 2002

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

Author

François Marques, Sep 14 2020

Keywords

Comments

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).

Examples

			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.
		

Crossrefs

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).
Showing 1-5 of 5 results.