A061601 -id:A061601 - cp's OEIS frontend

A318785 Numbers which are prime if each digit is replaced by its 9's complement.

2, 4, 6, 7, 10, 16, 20, 26, 28, 32, 38, 40, 46, 52, 56, 58, 62, 68, 70, 76, 80, 82, 86, 88, 92, 94, 96, 97, 112, 116, 118, 122, 136, 140, 142, 146, 160, 170, 172, 176, 178, 188, 190, 202, 212, 226, 230, 238, 242, 248, 256, 260, 266, 272, 280, 290, 298, 308, 316, 322, 326, 338, 340, 346, 352, 356, 358

Offset: 1

Pierandrea Formusa, Sep 03 2018

			32 belongs to this sequence as its 9's complement is 67, which is prime.

Cf. A061601 (9's complement of n).

complement(n) = my(d=digits(n)); for(k=1, #d, d[k]=9-d[k]); subst(Pol(d), x, 10)
is(n) = ispseudoprime(complement(n)) \\ Felix Fröhlich, Sep 03 2018

nmax=500
def is_prime(num):
    if num == 0 or num == 1: return(0)
    for k in range(2, num):
       if (num % k) == 0:
           return(0)
    return(1)
def c9(num):
    s=str(num)
    l=len(str(num))
    n=""
    for k in range(l):
        n = n+str(9-int(s[k]))
    return(int(n))
ris = ""
for i in range(2,nmax):
    if is_prime(c9(i)):
       ris = ris+str(i)+","
print(ris)

A383787 Largest number obtainable by either keeping each decimal digit d in n or replacing it with 9-d.

Original entry on oeis.org

8, 7, 6, 5, 5, 6, 7, 8, 9, 89, 88, 87, 86, 85, 85, 86, 87, 88, 89, 79, 78, 77, 76, 75, 75, 76, 77, 78, 79, 69, 68, 67, 66, 65, 65, 66, 67, 68, 69, 59, 58, 57, 56, 55, 55, 56, 57, 58, 59, 59, 58, 57, 56, 55, 55, 56, 57, 58, 59, 69, 68, 67, 66, 65, 65, 66, 67, 68, 69, 79, 78, 77, 76, 75, 75, 76, 77, 78

Offset: 1

Ali Sada, May 09 2025

			To find a(129) we replace 1 with 8 and 2 with 7. So, a(129) = 879.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..10^6.

Cf. A061601, A383788.

a[n_] := FromDigits[IntegerDigits[n] /. d_?(# < 5 &) -> 9 - d]; Array[a, 100] (* Amiram Eldar, May 10 2025 *)

a(n) = fromdigits(apply(x->(if (x<5, 9-x, x)), digits(n))); \\ Michel Marcus, May 12 2025

def a(n): return int("".join(d if d>"4" else str(9-int(d)) for d in str(n)))
print([a(n) for n in range(1, 79)]) # Michael S. Branicky, May 10 2025

A069000 Numbers k such that k * (digit complement of k) is a square.

Original entry on oeis.org

0, 9, 99, 972, 999, 9900, 9999, 39024, 60975, 99999, 168399, 307692, 467775, 532224, 692307, 831600, 972972, 999999, 9946224, 9999999, 11678832, 12328767, 18797427, 19584972, 32618124, 42245775, 47819475, 52180524, 57754224, 67381875, 80415027, 81202572

Offset: 1

Joseph L. Pe, Mar 20 2002

The digit complement of a digit d is 9 - d; e.g., 8 and 3 have complements 1, 6, respectively. The digit complement of a number k is the number formed by replacing each digit of k by its complement; e.g., 83 has complement 16.

			972972 has complement 27027 (the leading 0 is ignored). 972972 * 27027 = 162162^2, so 972972 is a term of the sequence.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A061601 (9's complement).

A002283 is a subsequence.

j[n_] := 9 - n; Do[If[IntegerQ[Sqrt[n*FromDigits[Map[j, IntegerDigits[n]]]]], Print[n]], {n, 1, 10^6}]
Select[Range[0,81203000],IntegerQ[Sqrt[# FromDigits[9-IntegerDigits[ #]]]]&] (* Harvey P. Dale, Jun 26 2021 *)

isok(n) = {d = digits(n); nd = vector(#d, k, 9-d[k]); issquare(n*fromdigits(nd));}

More terms from Robert G. Wilson v, Apr 08 2002

a(1), a(21)-a(32) from Giovanni Resta, Apr 14 2017

A085927 a(n) is the digitwise absolute difference between the n-th palindrome and its 9's complement.

Original entry on oeis.org

7, 5, 3, 1, 1, 3, 5, 7, 9, 77, 55, 33, 11, 11, 33, 55, 77, 99, 797, 777, 757, 737, 717, 717, 737, 757, 777, 797, 595, 575, 555, 535, 515, 515, 535, 555, 575, 595, 393, 373, 353, 333, 313, 313, 333, 353, 373, 393, 191, 171, 151, 131, 111, 111, 131, 151, 171, 191

Offset: 1

Amarnath Murthy and Jason Earls, Jul 13 2003

			a(24) = 717 because A002113(24) = 151 and A061601(151) = 848. 8-1 = 7 and 5-4 = 1, thus 717.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A002113, A061601.

from sympy import isprime
from itertools import count, product
def f(s): return int("".join(str(abs(9 - 2*int(c))) for c in s))
def pals(base=10): # all (nonzero) palindromes as strings
    digits = "".join(str(i) for i in range(base))
    for d in count(1):
        for p in product(digits, repeat=d//2):
            if d > 1 and p[0] == "0": continue
            left = "".join(p); right = left[::-1]
            for mid in [[""], digits][d%2]:
                t = left + mid + right
                if t != '0': yield t
def aupton(nn): p = pals(); return [f(next(p)) for i in range(nn)]
print(aupton(58)) # Michael S. Branicky, Jul 05 2021

def A085927(n):
    y = 10*(x:=10**(len(str(n+1>>1))-1))
    m = str((c:=n+1-x)*x+int(str(c)[-2::-1] or 0) if n+1Chai Wah Wu, Jul 24 2024

Edited and extended by David Wasserman, Feb 11 2005

A296130 Replace each digit of n with its complement to 9; this will reproduce all digits of n in a different order.

Original entry on oeis.org

18, 27, 36, 45, 54, 63, 72, 81, 90, 1089, 1098, 1188, 1278, 1287, 1368, 1386, 1458, 1485, 1548, 1584, 1638, 1683, 1728, 1782, 1809, 1818, 1827, 1836, 1845, 1854, 1863, 1872, 1881, 1890, 1908, 1980, 2079, 2097, 2178, 2187, 2277, 2367, 2376, 2457, 2475, 2547, 2574, 2637, 2673, 2709, 2718

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Feb 14 2018

The complement of 0 is 9, of 1 is 8, of 2 is 7, etc. All terms of the sequence have an even number of digits, and all terms are divisible by 9.

			18 produces 81 when 1 is replaced by 8 and 8 is replaced by 1; 18 and 81 use the same set of digits, in a different order.
1089 produces 8910 when 1 is replaced by 8, 0 by 9, 8 by 1 and 9 by 0; 1089 and 8910 use the same set of digits, in a different order.

Jean-Marc Falcoz, Table of n, a(n) for n = 1..3000

Cf. A061601.

replace_digits(n) = my(d=digits(n)); for(k=1, #d, d[k]=abs(d[k]-9)); d
is(n) = vecsort(digits(n))==vecsort(replace_digits(n)) \\ Felix Fröhlich, Feb 14 2018

A377471 Smallest prime such that the number of distinct prime factors of its 9's complement is equal to n. If no such number exists return -1.

Original entry on oeis.org

2, 3, 29, 229, 10273, 1000099, 1061069, 101872769, 1569793669, 20207116829, 1069666778189, 102533896856389, 10003581910211789, 1000003754654504609, 100003356331318330889

Offset: 1

Jean-Marc Rebert, Jan 08 2025

			2 is prime, 9-2 = 7 and omega(7) = 1;

Cf. A000040, A001221, A061601, A378139.

nc(n) = my(e=length(Str(n))); 10^e-1 - n; \\ A061601
a(n) = my(p=2); while (omega(nc(p)) != n, p = nextprime(p+1)); p; \\ Michel Marcus, Jan 08 2025

a(12)-a(15) from Michael S. Branicky, Jan 12 2025

A383788 Smallest number obtainable by either keeping each decimal digit d in n or replacing it with 9-d.

Original entry on oeis.org

1, 2, 3, 4, 4, 3, 2, 1, 0, 10, 11, 12, 13, 14, 14, 13, 12, 11, 10, 20, 21, 22, 23, 24, 24, 23, 22, 21, 20, 30, 31, 32, 33, 34, 34, 33, 32, 31, 30, 40, 41, 42, 43, 44, 44, 43, 42, 41, 40, 40, 41, 42, 43, 44, 44, 43, 42, 41, 40, 30, 31, 32, 33, 34, 34, 33, 32, 31, 30, 20, 21, 22, 23, 24, 24, 23, 22

Offset: 1

Ali Sada, May 09 2025

			To find a(346), we replace 6 with 3. So, a(346) = 343.

Cf. A061601, A383787.

a[n_] := FromDigits[IntegerDigits[n] /. d_?(# > 4 &) -> 9 - d]; Array[a, 100] (* Amiram Eldar, May 10 2025 *)

a(n) = fromdigits(apply(x->(if (x>4, 9-x, x)), digits(n))); \\ Michel Marcus, May 12 2025

def a(n): return int("".join(d if d<"5" else str(9-int(d)) for d in str(n)))
print([a(n) for n in range(1, 78)]) # Michael S. Branicky, May 10 2025

A226587 Numbers n having at least two complementary pairs of divisors (q, p) and (p', q') such that n = pq = p'q' where the decimal digits of p' are the 9's complement of the decimal digits of p and the decimal digits of q' are the 9's complement of the decimal digits of q.

Original entry on oeis.org

88, 154, 198, 220, 888, 1554, 1998, 2220, 8888, 9768, 15554, 17094, 19998, 21978, 22220, 24420, 88888, 89890, 97768, 105444, 112918, 120190, 127260, 134128, 140794, 147258, 153520, 155554, 159580, 165438, 171094, 176548, 181800, 186850, 191698, 196344, 199998, 200788, 205030

Offset: 1

Michel Lagneau, Sep 02 2013

The 9's complement of a number m equals 10^d - 1 - m where d is the number of digits in m. If u is a digit in m replace it with 9 - u.
A pair of integer (p, q) is complementary for multiplication when the product p*q is the same as the product p'*q' where the decimal digits of p' are the 9's complement of the decimal digits of p and the decimal digits of q' are the 9's complement of the decimal digits of q.
A double pair shows a complementary structure, for example: 77*2 = 7*22; 888*11 = 88*111; 8989*10 = 89*1010.
The sequence is infinite: let two integers x and y with the decimal representation x = ppp...p (i times) and y = (9-p)(9-p)...(9-p) (j times). The product x*y = p*(9-p)*R_i*R_j where R_k is a string of k 1's (or a Repunit number of the form (10^k - 1)/9). But x’ = (9-p)*R_i and y' = p*R_j => x*y => x'*y'.

			198 is in the sequence because 66*3 = 6*33 = 198.

Cf. A002275, A061601.

with(numtheory):for n from 1 to 210000 do:x:=divisors(n):n1:=nops(x):ii:=0:for a from 2 to n1-1 while(ii=0) do:m:=n/x[a]:m1:=convert(m, base, 10):nn1:=nops(m1): m2:=convert(x[a], base, 10):nn2:=nops(m2): s1:=sum('(9-m1[i])*10^(i-1)', 'i'=1..nn1): s2:=sum('(9-m2[i])*10^(i-1)', 'i'=1..nn2):for b from a+1 to n1-1 while(ii=0) do:q:=n/x[b]:if s1=q and s2=x[b] and m<>x[b] then ii:=1:printf(`%d, `, n):else fi:od:od:od:
# warning: there were missing terms, so the above Maple program may be wrong. - N. J. A. Sloane, Sep 17 2017

compl(n) = my(dn = digits(n)); fromdigits(vector(#dn, k, 9 - dn[k]));
isok(n) = sumdiv(n, d, if ((d^2= 2; \\ Michel Marcus, Sep 16 2017

Missing terms 88, 888, 8888, 88888 added by Michel Marcus, Sep 16 2017

A240696 Prime numbers n such that replacing each digit d in the decimal expansion of n with its 9's complement produces a prime.

Original entry on oeis.org

2, 7, 97, 997, 99999999999999997

Offset: 1

Michel Lagneau, Apr 10 2014

a(n) = {2} union {primes of the form 10^n - 3} = {2} union {A093172}.
Primes p such that A061601(p) is also prime.
The next term has 140 digits.

			997 is in the sequence because 997 becomes (002) = 2, which is prime.

Cf. A093172, A173833.

lst={};f[n_]:=Block[{a=IntegerDigits[Prime[n]],b="",k=1,l},l=Length[a];While[k

A360324 Numbers k such that k divides Sum_{i=1..k} 10^(1 + floor(log_10(p(i)))) - 1 - p(i), where p(i) is the i-th prime number.

Original entry on oeis.org

1, 13, 313, 1359, 245895, 131186351, 468729047, 1830140937

Offset: 1

Ctibor O. Zizka, Feb 04 2023

Alternative Name : The arithmetic mean of the first k 9's complements of primes is an integer.

			k = 13: first 13 prime numbers are {2,3,5,7,11,13,17,19,23,29,31,37,41}, their 9's complements are {7,6,4,2,88,86,82,80,76,70,68,62,58} and (7 + 6 + ... + 62 + 58) / 13 = 53, thus 13 is a term.

Cf. A000040, A045345, A055642, A061601.

s = 0; p = 2; pow = 10; seq = {}; Do[s += pow - 1 - p; If[Divisible[s, k], AppendTo[seq, k]]; p = NextPrime[p]; If[p > pow, pow *= 10], {k, 1, 250000}]; seq (* Amiram Eldar, Feb 04 2023 *)

k: (Sum_{i=1..k} 10^(1 + floor(log_10(p(i)))) - 1 - p(i)) / k = c, c an integer.

a(5)-a(8) from Amiram Eldar, Feb 04 2023

cp's OEIS Frontend

A318785 Numbers which are prime if each digit is replaced by its 9's complement.

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A383787 Largest number obtainable by either keeping each decimal digit d in n or replacing it with 9-d.

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A069000 Numbers k such that k * (digit complement of k) is a square.

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A085927 a(n) is the digitwise absolute difference between the n-th palindrome and its 9's complement.

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A296130 Replace each digit of n with its complement to 9; this will reproduce all digits of n in a different order.

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A377471 Smallest prime such that the number of distinct prime factors of its 9's complement is equal to n. If no such number exists return -1.

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A383788 Smallest number obtainable by either keeping each decimal digit d in n or replacing it with 9-d.

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Mathematica

PARI

Python

A226587 Numbers n having at least two complementary pairs of divisors (q, p) and (p', q') such that n = p*q = p'*q' where the decimal digits of p' are the 9's complement of the decimal digits of p and the decimal digits of q' are the 9's complement of the decimal digits of q.

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A226587 Numbers n having at least two complementary pairs of divisors (q, p) and (p', q') such that n = pq = p'q' where the decimal digits of p' are the 9's complement of the decimal digits of p and the decimal digits of q' are the 9's complement of the decimal digits of q.