A257831 -id:A257831

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A080719

Replace decimal digits with their binary values and convert back to decimal representation.

+10
4

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 3, 6, 7, 12, 13, 14, 15, 24, 25, 4, 5, 10, 11, 20, 21, 22, 23, 40, 41, 6, 7, 14, 15, 28, 29, 30, 31, 56, 57, 8, 9, 18, 19, 36, 37, 38, 39, 72, 73, 10, 11, 22, 23, 44, 45, 46, 47, 88, 89, 12, 13, 26, 27, 52, 53, 54, 55, 104, 105, 14, 15, 30, 31, 60, 61, 62

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

OFFSET

0,3

COMMENTS

m is a local maximum iff m == 9 modulo 10 (see A017377).

A257831 seen as binary numbers: A007088(a(n)) = A257831(n). - Reinhard Zumkeller, May 10 2015

LINKS

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

EXAMPLE

n=27 -> '2''7' -> '10''111' -> '10111' -> 23: a(27)=23.

See also A257831.

MATHEMATICA

Table[FromDigits[Flatten[IntegerDigits[#, 2]&/@IntegerDigits[n]], 2], {n, 80}] (* Harvey P. Dale, Aug 30 2014 *)

PROG

(Haskell)

import Data.Maybe (mapMaybe)

a080719 = foldr (\b v -> 2 * v + b) 0 .

concat . mapMaybe (flip lookup bin) . a031298_row

where bin = zip [0..9] a030308_tabf

-- Reinhard Zumkeller, May 10 2015

(Python)

def A080719(n):

....return int(''.join((format(int(d), 'b') for d in str(n))), 2)

# Chai Wah Wu, May 10 2015

CROSSREFS

Cf. A007088.

Cf. A257831, A030308, A031298.

KEYWORD

nonn,base

AUTHOR

Reinhard Zumkeller, Mar 06 2003

EXTENSIONS

a(0)=0 prepended and offset changed by 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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