A080719 -id:A080719

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A255734

Numbers n such that A080719(n) divides n.

+20
0

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 20, 30, 40, 50, 60, 70, 80, 90, 100, 105, 112, 120, 200, 210, 220, 240, 300, 330, 342, 360, 400, 408, 453, 462, 500, 504, 573, 600, 700, 720, 800, 900, 924, 1000, 1008, 1010, 1012, 1020, 1040, 1050, 1116, 1120, 1145, 1200, 1320, 1750, 1900, 2000

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..59.

EXAMPLE

Let us consider n = 342: 3 converted to base 2 is 11, 4 is 100, 2 is 10. Their concatenation is 1110010 that converted to base 10 is 114. So A080719(342) = 114. Finally 114 divides 342, so 342 is a member.

MAPLE

P:=proc(q) local a, b, k, n, x; for n from 1 to q do x:=[]; a:=n;

for k from 1 to length(n) do b:=convert((a mod 10), base, 2);

if b=[] then x:=[op(x), 0]; else x:=[op(x), op(b)]; fi; a:=trunc(a/10); od; a:=0;

for k from 1 to nops(x) do a:=2*a+x[-k]; od; if frac(n/a)=0 then print(n);

fi; od; end: P(2000);

MATHEMATICA

Select[Range[2000], Divisible[#, FromDigits[Flatten[IntegerDigits[#, 2]&/@ IntegerDigits[ #]], 2]]&] (* Harvey P. Dale, Mar 13 2019 *)

CROSSREFS

Cf. A007088, A080719.

KEYWORD

nonn,base

AUTHOR

Paolo P. Lava, Mar 05 2015

EXTENSIONS

Corrected and extended by Harvey P. Dale, Mar 13 2019

Corrected Maple code by Paolo P. Lava, Apr 01 2019

STATUS

approved

A257831

In decimal representation of n: replace each digit with its binary representation.

+10
3

0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 10, 11, 110, 111, 1100, 1101, 1110, 1111, 11000, 11001, 100, 101, 1010, 1011, 10100, 10101, 10110, 10111, 101000, 101001, 110, 111, 1110, 1111, 11100, 11101, 11110, 11111, 111000, 111001, 1000, 1001, 10010, 10011

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

a(n) = n iff (largest digit of n) = 1: a(A007088(n)) = A007088(n);

a(n) = A007088(A080719(n)).

LINKS

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

EXAMPLE

. n | dec --> bin | a(n) | A080719(n)

. -----+--------------------------+----------+------------

. 100 | [1,0,0] | [1,0,0] | 100 | 4

. 101 | [1,0,1] | [1,0,1] | 101 | 5

. 102 | [1,0,2] | [1,0,10] | 1010 | 10

. 103 | [1,0,3] | [1,0,11] | 1011 | 11

. 104 | [1,0,4] | [1,0,100] | 10100 | 20

. 105 | [1,0,5] | [1,0,101] | 10101 | 21

. 106 | [1,0,6] | [1,0,110] | 10110 | 22

. 107 | [1,0,7] | [1,0,111] | 10111 | 23

. 108 | [1,0,8] | [1,0,1000] | 101000 | 40

. 109 | [1,0,9] | [1,0,1001] | 101001 | 41

. 110 | [1,1,0] | [1,1,0] | 110 | 6

. 111 | [1,1,1] | [1,1,1] | 111 | 7

. 112 | [1,1,2] | [1,1,10] | 1110 | 14

. 113 | [1,1,3] | [1,1,11] | 1111 | 15

. 114 | [1,1,4] | [1,1,100] | 11100 | 28

. 115 | [1,1,5] | [1,1,101] | 11101 | 29

. 116 | [1,1,6] | [1,1,110] | 11110 | 30

. 117 | [1,1,7] | [1,1,111] | 11111 | 31

. 118 | [1,1,8] | [1,1,1000] | 111000 | 56

. 119 | [1,1,9] | [1,1,1001] | 111001 | 57

. 120 | [1,2,0] | [1,10,0] | 1100 | 12

. 121 | [1,2,1] | [1,10,1] | 1101 | 13

. 122 | [1,2,2] | [1,10,10] | 11010 | 26

. 123 | [1,2,3] | [1,10,11] | 11011 | 27

. 124 | [1,2,4] | [1,10,100] | 110100 | 52

. 125 | [1,2,5] | [1,10,101] | 110101 | 53 .

MATHEMATICA

Table[FromDigits[Flatten[IntegerDigits[#, 2]&/@IntegerDigits[n]]], {n, 0, 50}] (* Harvey P. Dale, Jun 06 2020 *)

PROG

(Haskell)

import Data.Maybe (mapMaybe)

a257831 = foldr (\b v -> 10 * v + b) 0 .

concat . mapMaybe (flip lookup bin) . a031298_row

where bin = zip [0..9] a030308_tabf

(Python)

def A257831(n):

....return int(''.join((format(int(d), 'b') for d in str(n))))

# Chai Wah Wu, May 10 2015

CROSSREFS

Cf. A007088, A030308, A031298, A080719.

KEYWORD

nonn,base

AUTHOR

Reinhard Zumkeller, May 10 2015

STATUS

approved

A322674

Square array read by antidiagonals: T(n, k) = 1 if the digits of p = n*k in base 2 are exactly the same as the digits of p when considering the base-2 representations of n, k and p as base-10 numbers, otherwise T(n, k) = 0.

+10
1

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 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, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

COMMENTS

As n * k = k * n, the array is symmetric.

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..10584; the first 145 antidiagonals of array

Jan Koornstra, Graph of all pairs up to (1024, 1024)

Index entries for characteristic functions

EXAMPLE

In base 2, 1001 * 10100 = 10110100. In base 10, 1001 * 10100 = 10110100. These digits match and therefore the pairs T(9, 20) and T(20, 9) are a 1 in the sequence (at a(444) and a(455)).

In base 2, the product of 11 * 11 = 1001, whereas 11 * 11 in base 10 yields 121. T(3, 3) is the 24th pair in the sequence and the first to fail. a(24) is thus a 0.

The array begins:

1, 1, 1, 1, 1, 1, 1, 1, 1, ...

1, 1, 1, 0, 1, 1, 0, 0, 1, ...

1, 1, 1, 1, 1, 1, 1, 1, 1, ...

1, 1, 1, 1, 1, 0, 1, 0, 1, ...

1, 1, 1, 0, 1, 1, 0, 0, 1, ...

1, 1, 1, 0, 1, 0, 0, 0, 1, ...

1, 1, 1, 1, 1, 1, 1, 1, 1, ...

PROG

(Python)

def a322674(k):

seq = []

i = 0

while len(seq) <= k:

j = 0

while len(seq) <= k and j < i + 1:

n = i - j

m = j

decn = int(bin(n).replace('0b', ''))

decm = int(bin(m).replace('0b', ''))

binProd = bin(n * m).replace('0b', '')

decProd = str(decn * decm)

seq.append(int(binProd == decProd))

j += 1

i += 1

print(seq)

a322674(100)

(PARI) T(n, k) = fromdigits(binary(n))*fromdigits(binary(k)) == fromdigits(binary(n*k)); \\ Michel Marcus, Apr 03 2019

CROSSREFS

Cf. A071998, A007088, A257831, A080719.

KEYWORD

nonn,easy,base,tabl

AUTHOR

Jan Koornstra, Jan 22 2019

STATUS

approved

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