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 1-7 of 7 results.

A169931 a(n) = 2*n in the arithmetic defined in A169918.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 32, 33, 34, 35, 36, 37, 38, 39, 30, 31, 42, 43, 44, 45, 46, 47, 48, 49, 40, 41, 52, 53, 54, 55, 56, 57, 58, 59, 50, 51, 62, 63, 64, 65, 66, 67, 68, 69, 60, 61, 72, 73, 74, 75, 76, 77, 78, 79, 70, 71, 82, 83, 84, 85, 86, 87, 88, 89, 80, 81, 92, 93, 94, 95, 96
Offset: 0

Views

Author

David Applegate, Marc LeBrun and N. J. A. Sloane, Jul 20 2010

Keywords

Comments

Equivalently, increase each (decimal) digit by 2 and take each result modulo 10. I.e., apply d -> (d+2 mod 10) to each digit of n. - M. F. Hasler, Mar 25 2015

Crossrefs

Cf. A048379, A169918.

Programs

  • Mathematica
    Table[FromDigits[Mod[IntegerDigits[n]+2,10]],{n,0,100}] (* Harvey P. Dale, Aug 17 2021 *)
  • PARI
    A169931(n)=2*!n+apply(d->(d+2)%10,n=digits(n))*vector(#n,i,10^(#n-i)) \\ M. F. Hasler, Mar 25 2015

Formula

A169931 = A048379 o A048379 (function A048379 applied twice). - M. F. Hasler, Mar 25 2015

A169933 a(n) = 2+n in the arithmetic defined in A169918.

Original entry on oeis.org

0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 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
Offset: 0

Views

Author

David Applegate, Marc LeBrun and N. J. A. Sloane, Jul 20 2010

Keywords

Comments

Equivalently, apply to the last (decimal) digit of n the operation d->((d*2) mod 10), i.e., multiply the last digit by 2 and take this modulo 10; keep all other digits the same. - M. F. Hasler, Mar 25 2015

Links

Crossrefs

Cf. A169931 - A169935.

Programs

Formula

For n > 9, a(n) = a(n-10) + 10. For n < 10, a(n) = (2*n mod 10). - M. F. Hasler, Mar 25 2015
From Chai Wah Wu, Feb 02 2023: (Start)
a(n) = a(n-1) + a(n-10) - a(n-11) for n > 10.
G.f.: 2*x*(x^9 + x^8 + x^7 + x^6 + x^5 - 4*x^4 + x^3 + x^2 + x + 1)/(x^11 - x^10 - x + 1). (End)

A169935 Numbers n such that for some k>0, n*k=1 in the arithmetic defined in A169918.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 16, 17, 22, 23, 27, 28, 33, 34, 38, 39, 40, 44, 45, 49, 50, 51, 55, 56, 61, 62, 66, 67, 72, 73, 77, 78, 83, 84, 88, 89, 90, 94, 95, 99, 111, 112, 116, 117, 222, 223, 227, 228, 333, 334, 338, 339, 440, 444, 445, 449
Offset: 1

Views

Author

David Applegate, Marc LeBrun and N. J. A. Sloane, Jul 21 2010

Keywords

Comments

These would be the units if the structure formed a ring (which it does not).

Examples

			22 is a member since 22*88 = 1. So is 23 since 23*8 = 1. 10 is not a member, even though 10 divides 9, since 10*9 = 0.

A169932 a(n) = 0+n in the arithmetic defined in A169918.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 70, 70, 70, 70, 70
Offset: 0

Views

Author

David Applegate, Marc LeBrun and N. J. A. Sloane, Jul 20 2010

Keywords

Comments

Equivalently, set to zero the last (decimal) digit of n, i.e., subtract (n mod 10). The digit-wise addition defined in A169918 consists of multiplying the digits and taking this product modulo 10 for each digit, and "Blanks are ignored". Since 0 has only one digit, only the last digit of n is set to zero in that way. - M. F. Hasler, Mar 25 2015

Crossrefs

Cf. A169918.

Programs

Formula

a(n) = [n/10]*10. - M. F. Hasler, Mar 25 2015

A048379 Apply the transformation 0->1->2->3->4->5->6->7->8->9->0 to digits of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 21, 22, 23, 24, 25, 26, 27, 28, 29, 20, 31, 32, 33, 34, 35, 36, 37, 38, 39, 30, 41, 42, 43, 44, 45, 46, 47, 48, 49, 40, 51, 52, 53, 54, 55, 56, 57, 58, 59, 50, 61, 62, 63, 64, 65, 66, 67, 68, 69, 60, 71, 72, 73, 74, 75, 76, 77, 78, 79, 70, 81, 82
Offset: 0

Views

Author

Patrick De Geest, Mar 15 1999

Keywords

Comments

This is the same as a(n) = 1*n in the arithmetic defined in A169918 (cf. A169930). - M. F. Hasler, Mar 25 2015

Examples

			a(8) = 9.
a(9) = 0.
a(10) = 21 because the original 1 is changed to a 2 and the 0 is changed to a 1.

Links

Programs

  • Haskell
    a048379 n = if n == 0 then 1 else x n where
   x m = if m == 0 then 0 else 10 * x m' + (d + 1) `mod` 10
         where (m',d) = divMod m 10
-- Reinhard Zumkeller, Feb 21 2014
  • Mathematica
    Table[FromDigits[ReplaceAll[IntegerDigits[n] + 1, 10 -> 0]], {n, 0, 79}] (* Alonso del Arte, Feb 27 2014 *)
  • PARI
    A048379(n)=n+sum(i=1, #n=digits(n), if(n[i]<9, 10^(i-1), -9*10^(i-1))) \\ M. F. Hasler, Mar 21 2015
  • PARI
    A048379(n)=!n+apply(t->(t+1)%10, n=digits(n))*vector(#n, i, 10^(#n-i))~ \\ M. F. Hasler, Mar 21 2015
  • Python
    d = {ord(str(i)):ord(str((i+1)%10)) for i in range(10)}
def a(n): return int(str(n).translate(d))
print([a(n) for n in range(72)]) # Michael S. Branicky, Dec 20 2022

Formula

a(A002283(n)) = 0. - Reinhard Zumkeller, Feb 21 2014

A169916 Squares in carryless arithmetic mod 10 with addition and multiplication of digits both defined to be addition mod 10.

Original entry on oeis.org

0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 220, 242, 264, 286, 208, 220, 242, 264, 286, 208, 440, 462, 484, 406, 428, 440, 462, 484, 406, 428, 660, 682, 604, 626, 648, 660, 682, 604, 626, 648, 880, 802, 824, 846, 868, 880, 802, 824, 846, 868, 0, 22, 44, 66, 88, 0, 22, 44, 66, 88, 220, 242
Offset: 0

Views

Author

David Applegate, Marc LeBrun and N. J. A. Sloane, Jul 20 2010

Keywords

Comments

The rules of arithmetic used in A169916, A169917, A169918 have very strange consequences. Many of the familiar laws fail. For instance, the arithmetic in A169916 is not associative: 10*(9*2) = 10*1 = 21 != (10*9)*2 = 9*2 = 1.

Examples

			a(16) = 16*16 = 242:
....16
....16
------
....72 (6*6 = 6+6 mod 10 = 2, 6*1 = 6+1 mod 10 = 7)
...27.
------
...242
------

Links

Crossrefs

The four versions are A059729, A169916, A169917, A169918.

Programs

  • PARI
    A169916(n)={u=vector(#n=digits(n),i,1);n=apply(d->n+d*u,n)%10;sum(i=0,2*#n-2,sum(j=max(1,#n-i),min(2*#n-1-i,#n),n[2*#n-i-j][j])%10*10^i)} \\ M. F. Hasler, Mar 26 2015

Formula

a(n)=a(n') if respective digits of n and n' differ by 0 or 5. In particular, a(10k+m) = a(10k+m+5) if 0 <= m <= 4.

A169917 Squares in carryless arithmetic mod 10 with addition and multiplication of digits both defined to be multiplication mod 10.

Original entry on oeis.org

0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 100, 111, 144, 199, 166, 155, 166, 199, 144, 111, 400, 441, 464, 469, 446, 405, 446, 469, 464, 441, 900, 991, 964, 919, 946, 955, 946, 919, 964, 991, 600, 661, 644, 649, 666, 605, 666, 649, 644, 661, 500, 551, 504, 559, 506, 555, 506, 559, 504
Offset: 0

Views

Author

David Applegate, Marc LeBrun and N. J. A. Sloane, Jul 20 2010

Keywords

Comments

The rules of arithmetic used in A169916, A169917, A169918 have very strange consequences. Many of the familiar laws fail. For instance, the arithmetic in A169916 is not associative: 10*(9*2) = 10*1 = 21 != (10*9)*2 = 9*2 = 1.

Examples

			a(24) = 24*24 = 446:
...24
...24
-----
...86
..48.
-----
..446
(The rule for "adding" the columns is to multiply mod 10: 8+8 = 8 * 8 mod 10 = 4.)

Links

Crossrefs

The four versions are A059729, A169916, A169917, A169918.

Programs

  • PARI
    A169917(n)={#n=digits(n);n=apply(d->n*d,n)%10;sum(i=0,2*#n-2,prod(j=max(1,#n-i),min(2*#n-1-i,#n),n[2*#n-i-j][j])%10*10^i)} \\ M. F. Hasler, Mar 26 2015

Formula

a(n) = a(n') if the i-th digit of n' either equals the i-th digit of n or (10 - the i-th digit of n): e.g., a(12345) = a(18365), because the 2nd and 4th digit of 12345 equal 10-(the 2nd resp. 4th digit of 18365), and the other digits are the same. In particular, a(10k+5+m) = a(10k+5-m), for m=0,...,4. - M. F. Hasler, Mar 26 2015
Showing 1-7 of 7 results.

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

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