A055120 -id:A055120 - cp's OEIS frontend

A089740 Primes which are digital complements (A055120) of primes.

3, 5, 7, 13, 31, 37, 43, 67, 73, 79, 97, 103, 113, 127, 139, 157, 163, 173, 181, 191, 199, 223, 227, 229, 233, 251, 257, 271, 281, 283, 313, 337, 349, 353, 359, 367, 383, 409, 419, 433, 449, 457, 463, 467, 479, 491, 523, 541, 547, 563, 569, 587, 601, 619, 631

Offset: 1

Roger L. Bagula, Jan 08 2004

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A055120.

A055120:= proc(n) local L;
  L:= map(t -> -t mod 10, convert(n,base,10));
  add(L[i]*10^(i-1),i=1..nops(L))
end proc:
filter:= t -> isprime(t) and isprime(A055120(t)):
select(filter, [seq(i,i=3..1000,2)]); # Robert Israel, Jun 29 2018

b=Delete[Union[ Table[If[PrimeQ[a[[n]]]==True,a[[n]],0],{n,1,Dimensions[a][[1]]}]],1]

Edited by Charles R Greathouse IV, Oct 28 2009

Name clarified by Robert Israel, Jun 29 2018

A004488 Tersum n + n.

Original entry on oeis.org

0, 2, 1, 6, 8, 7, 3, 5, 4, 18, 20, 19, 24, 26, 25, 21, 23, 22, 9, 11, 10, 15, 17, 16, 12, 14, 13, 54, 56, 55, 60, 62, 61, 57, 59, 58, 72, 74, 73, 78, 80, 79, 75, 77, 76, 63, 65, 64, 69, 71, 70, 66, 68, 67, 27, 29, 28, 33, 35, 34, 30, 32, 31, 45, 47, 46, 51

Offset: 0

N. J. A. Sloane

Could also be described as "Write n in base 3, then replace each digit with its base-3 negative" as with A048647 for base 4. - Henry Bottomley, Apr 19 2000
a(a(n)) = n, a self-inverse permutation of the nonnegative integers. - Reinhard Zumkeller, Dec 19 2003
First 3^n terms of the sequence form a permutation s(n) of 0..3^n-1, n>=1; the number of inversions of s(n) is A016142(n-1). - Gheorghe Coserea, Apr 23 2018

Gheorghe Coserea, Table of n, a(n) for n = 0..59048 (first 6561 terms from Alois P. Heinz)
Index entries for sequences related to carryless arithmetic
Index entries for sequences that are permutations of the natural numbers

Column k=0 of A253586, A253587.
Column k=3 of A248813.
Row / column 2 of A325820.
Main diagonal of A004489.
Cf. A048647, A055115, A055116, A055120, A059249, A117966, A117967, A117968, A225901, A242399, A244042, A263273, A289813, A289814, A289815, A289816, A289831, A289838, A300222, A321464.

a004488 0 = 0
a004488 n = if d == 0 then 3 * a004488 n' else 3 * a004488 n' + 3 - d
            where (n', d) = divMod n 3
-- Reinhard Zumkeller, Mar 12 2014

a:= proc(n) local t, r, i;
      t, r:= n, 0;
      for i from 0 while t>0 do
        r:= r+3^i *irem(2*irem(t, 3, 't'), 3)
      od; r
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Sep 07 2011

a[n_] := FromDigits[Mod[3-IntegerDigits[n, 3], 3], 3]; Table[a[n], {n, 0, 66}] (* Jean-François Alcover, Mar 03 2014 *)

a(n) = my(b=3); fromdigits(apply(d->(b-d)%b, digits(n, b)), b);
vector(67, i, a(i-1))  \\ Gheorghe Coserea, Apr 23 2018

from sympy.ntheory.factor_ import digits
def a(n): return int("".join([str((3 - i)%3) for i in digits(n, 3)[1:]]), 3) # Indranil Ghosh, Jun 06 2017

Tersum m + n: write m and n in base 3 and add mod 3 with no carries, e.g., 5 + 8 = "21" + "22" = "10" = 1.
a(n) = Sum(3-d(i)-3*0^d(i): n=Sum(d(i)*3^d(i): 0<=d(i)<3)). - Reinhard Zumkeller, Dec 19 2003
a(3*n) = 3*a(n), a(3*n+1) = 3*a(n)+2, a(3*n+2) = 3*a(n)+1. - Robert Israel, May 09 2014

A061601 9's complement of n: a(n) = 10^d - 1 - n where d is the number of digits in n. If a is a digit in n replace it with 9 - a.

Original entry on oeis.org

9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 89, 88, 87, 86, 85, 84, 83, 82, 81, 80, 79, 78, 77, 76, 75, 74, 73, 72, 71, 70, 69, 68, 67, 66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28

Offset: 0

Amarnath Murthy, May 19 2001

A109002 and A178500 give record values and where they occur: A109002(n+1)=a(A178500(n)) and a(m)<A109002(n+1) for m<A178500(n). - Reinhard Zumkeller, May 28 2010
If n is divisible by 3, so is a(n). The same goes for 9. - Alonso del Arte, Dec 01 2011
For n > 0, a(n-1) consists of the A055642(n) least significant digits of the 10-adic integer -n. - Stefano Spezia, Jan 21 2021

			a(7) = 2 = 10 - 1 -7. a(123) = 1000 -1 -123 = 876.

Kjartan Poskitt, Murderous Maths: Numbers, The Key to the Universe, Scholastic Ltd, 2002. See p 159.

Indranil Ghosh, Table of n, a(n) for n = 0..25000 (terms 0..1000 from Harry J. Smith)

Cf. A035327, A171960.
Cf. A055120.
See A267193 for complement obverse of n.

a061601 n = if n <= 9 then 9 - n else 10 * ad n' + 9 - d
            where (n',d) = divMod n 10
-- Reinhard Zumkeller, Feb 21 2014, Oct 04 2011

A061601 := proc(n)
        10^A055642(n)-1-n ;
end proc: # R. J. Mathar, Nov 30 2011

nineComplement[n_] := FromDigits[Table[9, {Length[IntegerDigits[n]]}] - IntegerDigits[n]]; Table[nineComplement[n], {n, 0, 71}] (* Alonso del Arte, Nov 30 2011 *)

A061601(n)=my(e=length(Str(n)));10^e-1 - n; \\ Joerg Arndt, Aug 28 2013

def A061601(n):
    return 10**len(str(n))-1-n # Indranil Ghosh, Jan 30 2017

a(n) = if n<10 then 9 - n else 10*a([n/10]) + 9 - n mod 10. - Reinhard Zumkeller, Jan 20 2010

a(n) <= 9n - 1. - Charles R Greathouse IV, Nov 15 2022

Corrected and extended by Matthew Conroy, Jan 19 2002

A248813 A(n,k) is the base-k complement of n; square array A(n,k), n>=0, k>=2, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 0, 3, 1, 3, 0, 4, 2, 6, 4, 0, 5, 3, 1, 8, 5, 0, 6, 4, 2, 12, 7, 6, 0, 7, 5, 3, 1, 15, 3, 7, 0, 8, 6, 4, 2, 20, 14, 5, 8, 0, 9, 7, 5, 3, 1, 24, 13, 4, 9, 0, 10, 8, 6, 4, 2, 30, 23, 8, 18, 10, 0, 11, 9, 7, 5, 3, 1, 35, 22, 11, 20, 11, 0, 12, 10, 8, 6, 4, 2, 42, 34, 21, 10, 19, 12

Offset: 0

Alois P. Heinz, Mar 03 2015

Every column is a permutation of the nonnegative integers.

			Square array A(n,k) begins:
   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, ...
   1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, ...
   2,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, ...
   3,  6,  1,  2,  3,  4,  5,  6,  7,  8,  9, ...
   4,  8, 12,  1,  2,  3,  4,  5,  6,  7,  8, ...
   5,  7, 15, 20,  1,  2,  3,  4,  5,  6,  7, ...
   6,  3, 14, 24, 30,  1,  2,  3,  4,  5,  6, ...
   7,  5, 13, 23, 35, 42,  1,  2,  3,  4,  5, ...
   8,  4,  8, 22, 34, 48, 56,  1,  2,  3,  4, ...
   9, 18, 11, 21, 33, 47, 63, 72,  1,  2,  3, ...
  10, 20, 10, 15, 32, 46, 62, 80, 90,  1,  2, ...

Columns k=2-16 give: A001477, A004488, A048647, A055115, A055116, A055117, A055118, A055119, A055120, A055121, A055122, A055123, A055124, A055125, A055126.

A:= proc(n, k) local t, r, i; t, r:= n, 0;
      for i from 0 while t>0 do
        r:= r+k^i *irem(k-irem(t, k, 't'), k)
      od; r
    end:
seq(seq(A(n, 2+d-n), n=0..d), d=0..14);

A(n,k)=fromdigits(apply(d->(k-d)%k, digits(n, k)), k); \\ Gheorghe Coserea, Apr 23 2018

A059692 Table of carryless products i * j, i>=0, j>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 4, 3, 0, 0, 4, 6, 6, 4, 0, 0, 5, 8, 9, 8, 5, 0, 0, 6, 0, 2, 2, 0, 6, 0, 0, 7, 2, 5, 6, 5, 2, 7, 0, 0, 8, 4, 8, 0, 0, 8, 4, 8, 0, 0, 9, 6, 1, 4, 5, 4, 1, 6, 9, 0, 0, 10, 8, 4, 8, 0, 0, 8, 4, 8, 10, 0, 0, 11, 20, 7, 2, 5, 6, 5, 2, 7, 20, 11, 0

Offset: 0

Henry Bottomley, Feb 19 2001

			Table begins:
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0,  0,  0,  0,  0,  0,  0 ...
  0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 ...
  0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 20, 22, 24, 26, 28, 20 ...
  0, 3, 6, 9, 2, 5, 8, 1, 4, 7, 30, 33, 36, 39, 32, 35 ...
  0, 4, 8, 2, 6, 0, 4, 8, 2, 6, 40, 44, 48, 42, 46, 40 ...
  ...
T(12, 97) = 954 since we have 12 X 97 = carryless sum of 900, (180 mod 100=)80, 70 and (14 mod 10=)4 = 954.

Stefano Spezia, First 140 antidiagonals of the table, flattened
David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version.
Index entries for sequences related to carryless arithmetic

Cf. A001477 for carryless 1 X n, A004520 for carryless 2 X 10 base 10, A055120 for carryless 9 X n, A008592 for carryless 10 X n.

Cf. A048720 (binary), A325820 (ternary).

len[num_]:=Length[IntegerDigits[num]]; digit[num_,d_]:=Part[IntegerDigits[num],d]; T[i_, j_] := FromDigits[Reverse[CoefficientList[PolynomialMod[Sum[digit[i,c]*x^(len[i]-c), {c, len[i]}]*Sum[digit[j,r]*x^(len[j]-r), {r, len[j]}], 10], x]]]; Flatten[Table[T[i - j, j], {i, 0, 12}, {j, 0, i}]] (* Stefano Spezia, Sep 26 2022 *)

T(n,k) = fromdigits(lift(Vec( Mod(Pol(digits(n)),10) * Pol(digits(k))))); \\ Kevin Ryde, Sep 27 2022

Minor edits by N. J. A. Sloane, Aug 24 2010

A059632 Carryless product 11 X n base 10.

Original entry on oeis.org

0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132, 143, 154, 165, 176, 187, 198, 109, 220, 231, 242, 253, 264, 275, 286, 297, 208, 219, 330, 341, 352, 363, 374, 385, 396, 307, 318, 329, 440, 451, 462, 473, 484, 495, 406, 417, 428, 439, 550, 561, 572, 583

Offset: 0

Henry Bottomley, Feb 19 2001

a(n) <= 11*n; a(m) = 11*m iff m is a term of A039691. - Reinhard Zumkeller, Jul 05 2014

			a(19)=109 since we have 11 X 19 = carryless sum of 100, 90, 10 and 9 =109

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version.

Cf. A001477 for carryless 1 X n, A004520 for carryless 2 X 10 base 10, A055120 for carryless 9 X n, A008592 for carryless 10 X n.
Cf. A048724 carryless 3Xn in base 2, A242399 carryless 4Xn in base 3.
Cf. A008593.

a059632 n = foldl (\v d -> 10 * v + d) 0 $
                  map (flip mod 10) $ zipWith (+) ([0] ++ ds) (ds ++ [0])
            where ds = map (read . return) $ show n
-- Reinhard Zumkeller, Jul 05 2014

A059691 Carryless product 12 X n base 10.

Original entry on oeis.org

0, 12, 24, 36, 48, 50, 62, 74, 86, 98, 120, 132, 144, 156, 168, 170, 182, 194, 106, 118, 240, 252, 264, 276, 288, 290, 202, 214, 226, 238, 360, 372, 384, 396, 308, 310, 322, 334, 346, 358, 480, 492, 404, 416, 428, 430, 442, 454, 466, 478, 500, 512, 524, 536

Offset: 0

Henry Bottomley, Feb 19 2001

			a(97)=954 since we have 12 X 97 = carryless sum of 900, 80, 70 and 4 = 954

Seiichi Manyama, Table of n, a(n) for n = 0..9999
David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version.
Index entries for sequences related to carryless arithmetic

Cf. A001477 for carryless 1 X n, A004520 for carryless 2 X n base 10, A055120 for carryless 9 X n, A008592 for carryless 10 X n.

A083991 Members of A083989 whose 10's complement is also a member of A083989.

Original entry on oeis.org

3, 7, 17, 29, 71, 83, 281, 719, 983, 997, 1637, 2309, 3701, 4493, 5507, 6299, 7691, 8363, 9161, 9803, 11003, 13163, 17117, 18371, 20807, 31181, 31793, 32693, 32843, 33617, 33893, 34211, 34673, 37277, 38453, 49409, 50591, 61547, 62723, 65327

Offset: 1

Amarnath Murthy, May 23 2003

			Leading zeros are removed before concatenation: 997 is in here because 997 is in A083989 (9973 is prime) and its 10-complement 3 is also in A083989 (37 is prime). Unlike the example of 17 and 83, the 10-complement is not a 1-to-1 relation in cases where 9 shows up as a most significant digit.
17 and 83 both are members and are each other's 10's complement.

Diana Mecum, Table of n, a(n) for n = 1..341

Cf. A083989, A083990.

A055120 := proc(n) local digs ; digs := ilog10(n)+1 ; 10^digs-n ; end: isA083989 := proc(n) local comp,ccat ; if isprime(n) then comp := A055120(n) ; ccat := n*10^(ilog10(comp)+1)+comp ; RETURN( isprime(ccat)) ; else false ; fi ; end: isA083991 := proc(n) local comp; if isA083989(n) then comp := A055120(n) ; RETURN( isA083989(comp) ) ; else false ; fi ; end: for n from 1 to 80000 do if isA083991(n) then printf("%d,",n) ; fi ; od ; # R. J. Mathar, Jul 18 2007

More terms from Diana L. Mecum and R. J. Mathar, Jul 18 2007

A300447 Numbers x such that sigma(x) = sigma(y), with x<>y, where y is the 10's complement mod 10 of the digits of x.

Original entry on oeis.org

54, 56, 513, 520, 546, 564, 580, 597, 4845, 5130, 5223, 5454, 5656, 5887, 5970, 6265, 44226, 46365, 48450, 50260, 50840, 51300, 52230, 52520, 53768, 57342, 58580, 58870, 59700, 62650, 64745, 66884, 463650, 477972, 484500, 489132, 489632, 493752, 501536, 503274

Offset: 1

Paolo P. Lava, Mar 06 2018

			sigma(54) = sigma(56) = 120;
sigma(513) = sigma(597) = 800;
sigma(477972) = sigma(633138) = 1415232.

Paolo P. Lava, Table of n, a(n) for n = 1..150

Cf. A000203, A055120.

with(numtheory): P:=proc(q) local a,b,k,n;
for n from 1 to q do a:=convert(n,base,10);
for k from 1 to nops(a) do a[k]:=(10-a[k]) mod 10; od; b:=0;
for k from 1 to nops(a) do b:=b*10+a[nops(a)-k+1]; od;
if b<>n and sigma(b)=sigma(n) then print(n); fi; od; end: P(10^6);

Select[Range[10^6], Apply[And[#1 != #2, DivisorSigma[1, #1] == DivisorSigma[1, #2]] &, {#, FromDigits[IntegerDigits[#] /. d_?Positive :> 10 - d]}] &] (* Michael De Vlieger, Mar 09 2018 *)

isok(x) = {my(dx = digits(x), dy = vector(#dx, k, (10-dx[k]) % 10), y = fromdigits(dy)); (x != y) && (sigma(x) == sigma(y));} \\ Michel Marcus, Mar 07 2018

A359697 Triangle T(n,k), n >= 1, 1 <= k <= n, read by rows, where T(n,k) is carryless product n X k base 10.

Original entry on oeis.org

1, 2, 4, 3, 6, 9, 4, 8, 2, 6, 5, 0, 5, 0, 5, 6, 2, 8, 4, 0, 6, 7, 4, 1, 8, 5, 2, 9, 8, 6, 4, 2, 0, 8, 6, 4, 9, 8, 7, 6, 5, 4, 3, 2, 1, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 12, 24, 36, 48, 50, 62, 74, 86, 98, 120, 132, 144

Offset: 1

Seiichi Manyama, Mar 08 2023

			Triangle begins:
   1;
   2,  4;
   3,  6,  9;
   4,  8,  2,  6;
   5,  0,  5,  0,  5;
   6,  2,  8,  4,  0,  6;
   7,  4,  1,  8,  5,  2,  9;
   8,  6,  4,  2,  0,  8,  6,  4;
   9,  8,  7,  6,  5,  4,  3,  2,  1;
  10, 20, 30, 40, 50, 60, 70, 80, 90, 100;
  11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121;
  12, 24, 36, 48, 50, 62, 74, 86, 98, 120, 132, 144;

Seiichi Manyama, Rows n = 1..99, flattened
David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version.
Index entries for sequences related to carryless arithmetic

T(n,n) gives A059729.

Cf. A001477 for carryless 1 X n, A004520 for carryless 2 X n base 10, A055120 for carryless 9 X n, A008592 for carryless 10 X n, A059691 for carryless 12 X n.

T(n, k) = fromdigits(Vec(Pol(digits(n))*Pol(digits(k)))%10);

cp's OEIS Frontend

A089740 Primes which are digital complements (A055120) of primes.

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A004488 Tersum n + n.

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A061601 9's complement of n: a(n) = 10^d - 1 - n where d is the number of digits in n. If a is a digit in n replace it with 9 - a.

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A248813 A(n,k) is the base-k complement of n; square array A(n,k), n>=0, k>=2, read by antidiagonals.

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A059692 Table of carryless products i * j, i>=0, j>=0, read by antidiagonals.

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A059632 Carryless product 11 X n base 10.

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A059691 Carryless product 12 X n base 10.

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