A076489 -id:A076489 - cp's OEIS frontend

A076493 Number of common (distinct) decimal digits of n and n^2.

1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 2, 1, 2, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 1, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 2, 2, 1, 1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 2, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 3, 2, 1, 1, 1

Offset: 0

Labos Elemer, Oct 21 2002

a(A029783(n)) = 0, a(A189056(n)) > 0; 0 <= a(n) <= 10, see example for first occurrences. [Reinhard Zumkeller, Apr 16 2011]

			a(2) = #{} = 0: 2^2 = 4;
a(0) = #{0} = 1: 0^2 = 0;
a(10) = #{0,1} = 2: 10^2 = 100;
a(105) = #{0,1,5} = 3: 105^2 = 11025;
a(1025) = #{0,1,2,5} = 4: 1025^2 = 1050625;
a(10245) = #{0,1,2,4,5} = 5: 10245^2 = 104960025;
a(102384) = #{0,1,2,3,4,8} = 6: 102384^2 = 10482483456;
a(1023456789) = #{0 .. 9} = 10: 1023456789^2 = 1047463798950190521.

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

Cf. A006880, A076489-A076491.

Cf. A000290, A043537.

import Data.List (intersect, nub)
import Data.Function (on)
a076493 n = length $ (intersect `on` nub . show) n (n^2)
-- Reinhard Zumkeller, Apr 16 2011

Table[Length[Intersection[IntegerDigits[n], IntegerDigits[n^2]]], {n, 1, 100}]

Initial 1 added and offset adjusted by Reinhard Zumkeller, Apr 16 2011

A076490 Number of common (distinct) digits of consecutive prime numbers.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Oct 21 2002

			1 to 9 common digits for {11 and 13},{101,103},{1031,1033}, {10223,10243},{97213,97231},{126079,126097},{1206479,1206497}, {10186237,10186273},{100438279,100438297} respectively.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A068803, A076489.

import Data.List (intersect, nub); import Data.Function (on)
a076490 n = a076490_list !! n
a076490_list = map (length . nub) $
   zipWith (intersect `on` show) (tail a000040_list) a000040_list
-- Reinhard Zumkeller, Sep 01 2013

Table[Length[Intersection[IntegerDigits[Prime[w]], IntegerDigits[ Prime[ w+1]]]], {w, 1, 200}] (* corrected by Harvey P. Dale, May 14 2014 *)
Length[Intersection@@IntegerDigits[#]]&/@Partition[Prime[Range[110]],2,1] (* Harvey P. Dale, May 14 2014 *)

a(n) = my(p=prime(n)); #setintersect(Set(digits(p)), Set(digits(nextprime(p+1)))); \\ Michel Marcus, Mar 27 2023

A238845 Prefix overlap between binary expansions of n and n+1.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 1, 3, 2, 3, 1, 3, 2, 3, 1, 4, 3, 4, 2, 4, 3, 4, 1, 4, 3, 4, 2, 4, 3, 4, 1, 5, 4, 5, 3, 5, 4, 5, 2, 5, 4, 5, 3, 5, 4, 5, 1, 5, 4, 5, 3, 5, 4, 5, 2, 5, 4, 5, 3, 5, 4, 5, 1, 6, 5, 6, 4, 6, 5, 6, 3, 6, 5, 6, 4, 6, 5, 6, 2, 6, 5, 6, 4, 6, 5, 6, 3, 6, 5, 6

Offset: 0

N. J. A. Sloane, Mar 22 2014

The prefix overlap between two words is the length of their longest common prefix.

			8 = 1000 and 9 = 1001 have prefix overlap of 3, so a(8)=3.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Luc Rousseau, Proof of the formula
Rémy Sigrist, Colored scatterplot of the ordinal transform of the first 10000 terms
Rodica Simion and Herbert S. Wilf, The distribution of prefix overlap in consecutive dictionary entries, SIAM J. Algebraic Discrete Methods, 7(1986), no. 3, 470--475. MR0844051.

Cf. A076489, A239091.

import Data.List (unfoldr); import Data.Tuple (swap)
a238845 n = length $ takeWhile (== 0) $ zipWith (-) (bin n) (bin (n+1))
where bin = reverse . unfoldr
(\x -> if x == 0 then Nothing else Just $ swap $ divMod x 2)
-- Reinhard Zumkeller, Mar 22 2014

# prefix overlap between n and n+1 in base b:
po:=proc(n,b) local t1,t2,l1,l2,c,L,i;
t1:=convert(n,base,b); l1:=nops(t1);
t2:=convert(n+1,base,b); l2:=nops(t2);
c:=0; L:=min(l1,l2);
for i from 1 to L do
if t1[l1+1-i] = t2[l2+1-i] then c:=c+1; else break; fi; od:
c;
end;
[seq(po(n,2),n=0..120)];

a[n_] := With[{v = IntegerExponent[n+1, 2]}, Floor[Log[2, n+1]] - v + Boole[n+1 == 2^v] - Boole[n == 0]]; Table[a[n], {n, 0, 90}] (* Jean-François Alcover, Feb 03 2018, after Charles R Greathouse IV *)
pol[n_]:=Module[{c1=IntegerDigits[n,2],c2=IntegerDigits[n+1,2]},Total[ Split[ If[#[[1]]==#[[2]],1,0]&/@Thread[{c1,Take[c2,Length[c1]]}]][[1]]]];Array[pol,100,0] (* Harvey P. Dale, Jun 12 2020 *)

a(n)=my(v=valuation(n+1,2)); logint(n+1,2) - v + (n+1==1<Charles R Greathouse IV, Dec 29 2017

For all n > 0, a(n-1) = A000523(n) - A007814(n) + A209229(n) - A063524(n) = floor(log_2(n)) - v_2(n) + [exists(k,n==2^k)] - [n==1]. (see link) - Luc Rousseau, Dec 29 2017

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

A226637 Numbers m having with m+1 no common digit in decimal representations.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 19, 29, 39, 49, 59, 69, 79, 99, 199, 299, 399, 499, 599, 699, 799, 999, 1999, 2999, 3999, 4999, 5999, 6999, 7999, 9999, 19999, 29999, 39999, 49999, 59999, 69999, 79999, 99999, 199999, 299999, 399999

Offset: 1

Reinhard Zumkeller, Sep 01 2013

Cf. A000030, A076489.

a226637 n = a226637_list !! (n-1)
a226637_list = filter ((== 0) . a076489) [0..]

isok(k) = #setintersect(Set(digits(k)), Set(digits(k+1))) == 0; \\ Michel Marcus, Feb 07 2025

A076489(a(n)) = 0;

A000030(a(n)) <> 8 for n > 9.

A239092 Prefix overlap of dictionary consisting of decimal expansions of 0 through n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 100, 102, 104, 106, 108, 110

Offset: 1

N. J. A. Sloane, Mar 22 2014

The prefix overlap between two words is the length of their longest common prefix.
The prefix overlap of a dictionary is the sum of the prefix overlaps between successive words.
Partial sums of A076489.
More than the usual number of terms are displayed in order to distinguish this from some closely related sequences.

Rodica Simion and Herbert S. Wilf, The distribution of prefix overlap in consecutive dictionary entries, SIAM J. Algebraic Discrete Methods, 7(1986), no. 3, 470--475. MR0844051.

Cf. A238845, A239091, A076489.

Different from A081600 and A028904.

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

A076494 Number of common decimal digits of 2^n and 2^(1+n).

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Oct 21 2002

Cf. A006880, A076489-A076493.

Table[Length[Intersection[IntegerDigits[2^n], IntegerDigits[2^(n+1)]]], {n, 1, 100}]

A164832 Least nonnegative integer k such that the decimal representations of k and k+1 have n distinct digits in common.

Original entry on oeis.org

0, 10, 100, 1020, 10230, 102340, 1023450, 10234560, 102345670, 1023456780, 10234567889

Offset: 0

Rick L. Shepherd, Aug 27 2009

Finding a(10), the final term, could be a simple but instructive puzzle.

a(1) through a(9) is a subsequence of A121030. a(0) through a(9) is a subsequence of A107411.

			a(10) = 10234567889 because 10234567889 and 10234567890 have all 10 decimal digits in common and this property does not hold for any smaller positive integer.

Cf. A076489, A121030, A107411.

For 1 <= n <= 10, a(n) is the least k such that A076489(k) = n. (This would be true for n = 0 also if A076489 considered nonnegative integers, having another initial 0 term and offset 0.).

cp's OEIS Frontend

A076493 Number of common (distinct) decimal digits of n and n^2.

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A076490 Number of common (distinct) digits of consecutive prime numbers.

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A238845 Prefix overlap between binary expansions of n and n+1.

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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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A226637 Numbers m having with m+1 no common digit in decimal representations.

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A239092 Prefix overlap of dictionary consisting of decimal expansions of 0 through n.

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

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Keywords

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Mathematica

Extensions

A076494 Number of common decimal digits of 2^n and 2^(1+n).

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Mathematica

A164832 Least nonnegative integer k such that the decimal representations of k and k+1 have n distinct digits in common.

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