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.

A040115 Concatenate absolute values of differences between adjacent digits of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1
Offset: 0

Views

Author

Felice Russo

Keywords

Comments

Let the decimal expansion of n be abcd...efg, say. Then a(n) has decimal expansion |a-b| |b-c| |c-d| ... |e-f| |f-g|. Leading zeros in a(n) are omitted.
From M. F. Hasler, Nov 09 2019: (Start)
This sequence coincides with A080465 up to a(109) but is thereafter completely different.
Eric Angelini calls a(n) the "ghost" of the number n and considers iterations of n -> n +- a(n) depending on parity of a(n), cf. A329200 and A329201. (End)

Examples

			a(371) = 46, for example.
a(110) = 01 = 1, while A080465(110) = 10 - 1 = 9. - _M. F. Hasler_, Nov 09 2019

Links

Crossrefs

Cf. A010785, A037904, A040114, A040163, A040997.
Cf. A329200, A329201: iterations of n +- a(n).

Programs

  • Mathematica
    Table[FromDigits[Abs[Differences[IntegerDigits[n]]]],{n,110}] (* Harvey P. Dale, Dec 16 2021 *)
  • PARI
    apply( A040115(n)=fromdigits(abs((n=digits(n+!n))[^-1]-n[^1])), [10..199]) \\ Works for all n >= 0. - M. F. Hasler, Nov 09 2019

Formula

a(n) = 0 iff n is a repdigit >= 11 (A010785). - Bernard Schott, May 09 2022

Extensions

Definition clarified by N. J. A. Sloane, Aug 19 2008
Name edited by M. F. Hasler, Nov 09 2019
Terms a(0) = a(1) = ... = a(9) = 0 prepended by Max Alekseyev, Jul 26 2024

A080464 Product of the two numbers formed by alternate digits of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 0, 8, 16, 24, 32, 40, 48, 56, 64, 72, 0
Offset: 10

Views

Author

Amarnath Murthy, Mar 02 2003

Keywords

Examples

			a(132546) = 124 * 356 = 44144.

Links

Crossrefs

Cf. A007954, A080463, A080465, A087471, A087472, A087473, A087474.

Programs

  • Mathematica
    nad[n_]:=Module[{idn=IntegerDigits[n]},FromDigits[Take[idn,{1,-1,2}]] FromDigits[ Take[idn,{2,-1,2}]]]; Array[nad,120,10] (* Harvey P. Dale, Aug 07 2019 *)
  • PARI
    A080464(n,d=digits(n))={n=d*matrix(#d,2,z,s,if((z-s)%2,10^((#d-z)\2)));n[1]*n[2]}

Formula

a(n) < n for all n. - M. F. Hasler, Jan 10 2016

Extensions

More terms from Ray Chandler, Oct 11 2003

A080463 Sum of the two numbers formed by alternate digits of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 8, 9, 10, 11, 12, 13, 14, 15, 16
Offset: 0

Views

Author

Amarnath Murthy, Mar 02 2003

Keywords

Comments

First 99 terms match with those of A007953.
They also match A209685. - M. F. Hasler, Jan 10 2016

Examples

			a(132546) = 124 + 356 = 480.

Links

Crossrefs

Cf. A007953, A080464, A080465.

Programs

  • Maple
    f:= proc(n) option remember; n mod 10 + (floor(n/10) mod 10) + 10*procname(floor(n/100)) end proc:
f(0):= 0:
seq(f(n),n=0..1000); # Robert Israel, Jan 10 2016
  • PARI
    A080463(n)=abs(vector(#n=digits(n),j,10^((#n-j)\2))*n~) \\ M. F. Hasler, Jan 10 2016

Formula

From Robert Israel, Jan 10 2016: (Start)
f(n) = n mod 10 + floor(n/10) mod 10 + 10*f(floor(n/100)).
G.f. G(x) satisfies G(x) = (x + 2x^2 + ... + 9x^9)/(1-x^10) + (x^10 + 2 x^20 + ... + 9 x^90)/((1-x)(1+x^10+...+x^90) + 10 (1 + x + ... + x^99) G(x^10).
(End)

Extensions

More terms from Ray Chandler, Oct 11 2003
Extended to offset 0 and b-file by M. F. Hasler, Jan 10 2016

A267086 Numbers such that the number formed by digits in even positions divides, or is divisible by, the number formed by the digits in odd positions; zero allowed.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 26, 28, 30, 31, 33, 36, 39, 40, 41, 42, 44, 48, 50, 51, 55, 60, 61, 62, 63, 66, 70, 71, 77, 80, 81, 82, 84, 88, 90, 91, 93, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 122, 124, 126, 128, 132, 135
Offset: 1

Views

Author

M. F. Hasler, Jan 10 2016

Keywords

Comments

The initial 0 is included by convention. The single-digit numbers are included with the reasoning that the number formed by digits in even positions is zero, and thus divisible by (= a multiple of) any other number, and here in particular the number formed by first digit.
By "digits in odd positions" we mean the first (most significant), third, fifth, etc. digits; e.g., for the numbers 12345 or 123456 this would be 135.
An extended version of Eric Angelini's "integears" A267085.
Sequence A263314 is a subsequence up to 120, but 121 is in A263314 and not in this sequence.

Examples

			12 is in the sequence because 1 divides 2.
213 is in the sequence because 1 divides 23.
1020 is in the sequence because 12 divides 00 = 0. (Any number divides 0 therefore any number which has every other digit equal to zero is in the sequence.)

Links

Crossrefs

Cf. A267085, A263314.
See also A080463, A080464 and A080465.

Programs

A267085 Numbers such that the number formed by digits in even position divides, or is divisible by, the number formed by the digits in odd position; both must be nonzero.

Original entry on oeis.org

11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 24, 26, 28, 31, 33, 36, 39, 41, 42, 44, 48, 51, 55, 61, 62, 63, 66, 71, 77, 81, 82, 84, 88, 91, 93, 99, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 122, 124, 126, 128, 132, 135, 138, 142, 146, 150, 155, 162, 168, 174, 186, 198
Offset: 1

Views

Author

Eric Angelini and M. F. Hasler, Jan 10 2016

Keywords

Comments

Termed "integears" by Eric Angelini. See A267086 for the "extended version" where zero is allowed.

Links

Crossrefs

See also A080463, A080464 and A080465.

Programs

  • Mathematica
    Select[Range[10, 200], Or[If[#2 == 0, False, Mod[#1, #2] == 0], If[#2 == 0, False, Mod[#2, #1] == 0]] & @@ {FromDigits@ Extract[#, Range[1, Length@ #, 2] /. x_Integer -> {x}], FromDigits@ Extract[#, Range[2, Length@#, 2] /. x_Integer -> {x}]} &@ IntegerDigits@ # &] (* Michael De Vlieger, Jan 21 2016 *)
  • PARI
    is(n,d=digits(n))={n=d*matrix(#d,2,z,s,if(z==Mod(s,2),10^((#d-z)\2))); n[2] && (n[1]%n[2]==0 || n[2]%n[1]==0)}
Showing 1-5 of 5 results.

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