A076490 -id:A076490 - cp's OEIS frontend

A076489 Number of common (distinct) digits of consecutive natural numbers.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 2, 2, 2, 2, 2

Offset: 0

Labos Elemer, Oct 21 2002

a(A226637(n)) = 0. Reinhard Zumkeller, Sep 01 2013

This is the prefix overlap between the decimal expansions of n and n+1 (cf. A238845). - N. J. A. Sloane, Mar 22 2014

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A001477, A031298, A076490, A238845, A239092 (partial sums).

import Data.List (intersect, nub)
a076489 n = a076489_list !! n
a076489_list = map (length . nub) $
               zipWith intersect (tail a031298_tabf) a031298_tabf
-- Reinhard Zumkeller, Sep 01 2013

Table[Length[Intersection[IntegerDigits[w], IntegerDigits[w+1]]], {w, 0, 200}]

Initial zero prepended and offset adjusted by Reinhard Zumkeller, Sep 01 2013

A068803 Smaller of two consecutive primes which have no common digits.

Original entry on oeis.org

2, 3, 5, 7, 19, 29, 37, 47, 59, 79, 97, 397, 499, 599, 1999, 2999, 3989, 4999, 29989, 49999, 59999, 79999, 199999, 599999, 799999, 2999999, 4999999, 5999993, 19999999, 29999999, 59999999, 69999989, 99999989, 199999991, 699999953, 799999999, 5999999989, 6999999989

Offset: 1

Amarnath Murthy, Mar 06 2002

Is the sequence finite or infinite?

Except for 2, 3, 5, and 7, all such primes are of the form a*10^n-b with 1 <= a <= 8 and b mod 10 = 1, 3, 7 or 9. An example of a large pair is 10^101-203 and 10^101+3. The largest known pair of probable primes is 8*10^5002-6243 and 8*10^5002+14481. - Lewis Baxter, Mar 06 2023

			397 is a term as 397 and 401 are two consecutive primes with no common digits.

Michael S. Branicky, Table of n, a(n) for n = 1..804

Cf. A076490.

First /@ Select[Partition[Prime[Range[10^6]], 2, 1], Intersection @@ IntegerDigits /@ # == {} &] (* Jayanta Basu, Aug 06 2013 *)

isok(p) = isprime(p) && (#setintersect(Set(digits(p)), Set(digits(nextprime(p+1)))) == 0); \\ Michel Marcus, Mar 27 2023

from itertools import count, islice
from sympy import nextprime, prevprime
def agen(): # generator of terms
    yield from [2, 3, 5]
    for d in count(2):
        for b in range(10**(d-1), 10**d, 10**(d-1)):
            p, q = prevprime(b), nextprime(b)
            if set(str(p)) & set(str(q)) == set():
                yield p
print(list(islice(agen(), 40))) # Michael S. Branicky, May 09 2023

More terms from Larry Soule (lsoule(AT)gmail.com), Jun 21 2006

a(36) and beyond from Michael S. Branicky, May 09 2023

A076491 a(2n), a(2n+1) is the smallest consecutive prime pairs with at least n distinct common decimal digits.

Original entry on oeis.org

2, 3, 11, 13, 101, 103, 1031, 1033, 10223, 10243, 18379, 18397, 126079, 126097, 1206479, 1206497, 10258379, 10258397, 102346879, 102346897, 10127685439, 10127685493

Offset: 0

Labos Elemer, Oct 21 2002

If the common digits were not required to be distinct, the resulting sequence would be 2, 3, 11, 13, 101, 103, 1013, 1019, 1913, 1931, 18379, 18397, 109279, 109297, 1000213, 1000231, ... - Giovanni Resta, Oct 29 2019

Cf. A006880, A076489, A076490.

aa[n_] := Block[{p,q,cp,cq}, p = NextPrime[10^(n - 1)]; cp = IntegerDigits@ p; While[True, q = NextPrime[p]; cq = IntegerDigits[q]; If[ Length[ Intersection[cp, cq]] >= n, Break[]]; p=q; cp=cq]; {p, q}]; Flatten[aa /@ Range[0, 9]] (* Giovanni Resta, Oct 29 2019 *)

Corrected and extended by Giovanni Resta, Oct 29 2019

A076492 Number of common decimal digits of n! and (n+1)!.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Oct 21 2002

Diana Mecum, Table of n, a(n) for n = 1..1000

Cf. A006880, A076489, A076490, A076491.

Table[Length[Intersection[IntegerDigits[(n+1)!], IntegerDigits[n!]]], {n, 1, 100}]
Length[Intersection[#[[1]],#[[2]]]]&/@Partition[IntegerDigits[ Range[ 90]!],2,1] (* Harvey P. Dale, Jun 26 2021 *)

More terms from Diana L. Mecum, Jun 17 2007

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A076489 Number of common (distinct) digits of consecutive natural numbers.

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A068803 Smaller of two consecutive primes which have no common digits.

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A076491 a(2*n), a(2*n+1) is the smallest consecutive prime pairs with at least n distinct common decimal digits.

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A076492 Number of common decimal digits of n! and (n+1)!.

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A076491 a(2n), a(2n+1) is the smallest consecutive prime pairs with at least n distinct common decimal digits.