A161153 -id:A161153 - cp's OEIS frontend

A161152 Positive integers n such that {the number of (non-leading) 0's in the binary representation of n} is coprime to n.

1, 2, 5, 6, 8, 9, 11, 13, 14, 17, 19, 20, 21, 23, 25, 27, 29, 30, 32, 33, 35, 37, 38, 39, 41, 43, 44, 45, 47, 49, 50, 51, 52, 53, 55, 56, 57, 59, 61, 62, 66, 67, 68, 69, 71, 72, 73, 77, 79, 81, 83, 85, 86, 87, 89, 91, 92, 93, 95, 96, 97, 101, 103, 106, 107, 109, 111, 113, 115

Offset: 1

Leroy Quet, Jun 03 2009

1 is the only integer of the form 2^k - 1 (k>=0) included in this sequence, because such integers contain no binary 0's, and 0 is considered here to be coprime only to 1.

			13 is in the sequence because the number of non-leading 0 s in the binary representation of 13 is 1 (13_10 = 1101_2) and gcd(1, 13) = 1. - _Indranil Ghosh_, Mar 08 2017

Indranil Ghosh, Table of n, a(n) for n = 1..1000

Cf. A023416, A094387, A161153, A161154, A161155, A161156.

Select[Range[115], GCD[DigitCount[#, 2, 0], #] == 1 &] (* Indranil Ghosh, Mar 08 2017 *)

b(n) = if(n<1, 0, b(n\2) + 1 - n%2);
for (n=1, 115, if(gcd(b(n),n)==1, print1(n", "))); \\ Indranil Ghosh, Mar 08 2017

from math import gcd
i=j=1
while j<=100:
    if gcd(bin(i)[2:].count("0"),i)==1:
        print(j, i)
        j+=1
    i+=1 # Indranil Ghosh, Mar 08 2017

Extended by Ray Chandler, Jun 11 2009

A161154 Positive integers n such that both {the number of (non-leading) 0's in the binary representation of n} is coprime to n and {the number of 1's in the binary representation of n} is coprime to n.

Original entry on oeis.org

1, 2, 5, 8, 9, 11, 13, 14, 17, 19, 23, 25, 27, 29, 32, 33, 35, 37, 38, 39, 41, 43, 44, 45, 47, 49, 50, 51, 52, 53, 56, 57, 59, 61, 62, 67, 71, 73, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 107, 109, 113, 117, 119, 121, 125, 128, 131, 133, 134, 135, 137, 139, 141

Offset: 1

Leroy Quet, Jun 03 2009

1 is the only integer of the form 2^k - 1 (k>=0) included in this sequence, because such integers contain no binary 0's, and 0 is considered here to be coprime only to 1.

Indranil Ghosh, Table of n, a(n) for n = 1..1000

Cf. A094387, A161152, A161153, A161155, A161156.

bcpQ[n_]:=Module[{ones=DigitCount[n,2,1],zeros=DigitCount[n,2,0]}, And@@ CoprimeQ[ {ones,zeros},n]]; Select[Range[150],bcpQ] (* Harvey P. Dale, Feb 19 2012 *)

b0(n) = if(n<1, 0, b0(n\2) + 1 - n%2);
b1(n) = if(n<1, 0, b1(n\2) + n%2);
for (n=1, 141, if(gcd(b0(n),n)==1 && gcd(b1(n),n)==1, print1(n", "))) \\ Indranil Ghosh, Mar 08 2017

from math import gcd
i=j=1
while j<=100:
    if gcd(bin(i)[2:].count("0"),i)==1==gcd(bin(i)[2:].count("1"),i):
        print(j, i)
        j+=1
    i+=1 # Indranil Ghosh, Mar 08 2017

Extended by Ray Chandler, Jun 11 2009

A161155 Positive integers n such that {the number of (non-leading) 0's in the binary representation of n} is coprime to n, {the number of 1's in the binary representation of n} is coprime to n and {the number of digits in the binary representation of n} is coprime to n.

Original entry on oeis.org

1, 5, 9, 11, 13, 17, 19, 23, 27, 29, 35, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 85, 87, 89, 93, 95, 97, 101, 103, 107, 109, 113, 117, 121, 125, 131, 133, 135, 137, 139, 141, 143, 147, 149, 151, 153, 157, 161, 163, 165, 167, 169, 173, 175, 177, 179, 181

Offset: 1

Leroy Quet, Jun 03 2009

1 is the only integer of the form 2^k - 1 (k>=0) included in this sequence, because such integers contain no binary 0's, and 0 is considered here to be coprime only to 1.

Indranil Ghosh, Table of n, a(n) for n = 1..1000

Cf. A094387, A161152, A161153, A161154, A161156.

Select[Range[181], GCD[DigitCount[#,2,0] , #]==1 && GCD[DigitCount[#,2,1],#]==1 && GCD[Length[IntegerDigits[#,2]],#]==1 &] (* Indranil Ghosh, Mar 08 2017 *)

b0(n) = if(n<1, 0, b0(n\2) + 1 - n%2);
b1(n) = if(n<1, 0, b1(n\2) + n%2);
for (n=1, 181, if(gcd(b0(n), n) == 1 && gcd(b1(n), n) == 1 && gcd(#digits(n, 2), n) == 1, print1(n", "))) \\ Indranil Ghosh, Mar 08 2017

from math import gcd
i=j=1
while j<=100:
    if gcd(bin(i)[2:].count("0"),i)==1 and gcd(bin(i)[2:].count("1"),i)==1 and gcd(len(bin(i)[2:]),i)==1:
        print(j, i)
        j+=1
    i+=1 # Indranil Ghosh, Mar 08 2017

Extended by Ray Chandler, Jun 11 2009

A161156 Positive integers n such that {the number of (non-leading) 0's in the binary representation of n} is coprime to n, and {the number of 1's in the binary representation of n} is coprime to n, but {the number of digits in the binary representation of n} is not coprime to n.

Original entry on oeis.org

2, 8, 14, 25, 32, 33, 38, 39, 44, 45, 50, 51, 52, 56, 57, 62, 77, 91, 119, 128, 134, 146, 148, 152, 158, 164, 176, 182, 188, 194, 196, 206, 208, 214, 218, 224, 236, 242, 244, 248, 254, 267, 279, 291, 297, 309, 327, 333, 339, 351, 357, 369, 375, 381, 387, 393

Offset: 1

Leroy Quet, Jun 03 2009

1 is the only integer of the form 2^k - 1 (k>=0) which is coprime to the number of 0's in its binary representation, because such integers contain no binary 0's, and 0 is considered here to be coprime only to 1.

Indranil Ghosh, Table of n, a(n) for n = 1..1000

Cf. A094387, A161152, A161153, A161154, A161155.

Select[Range[393], GCD[DigitCount[#, 2, 0] , #]==1 && GCD[DigitCount[#, 2, 1], #] == 1 && GCD[Length[IntegerDigits[#, 2]], #] != 1 &] (* Indranil Ghosh, Mar 08 2017 *)

b0(n) = if(n<1, 0, b0(n\2) + 1 - n%2);
b1(n) = if(n<1, 0, b1(n\2) + n%2);
for (n=1, 393, if(gcd(b0(n), n) == 1 && gcd(b1(n), n) == 1 && gcd(#digits(n, 2), n) != 1, print1(n", "))); \\ Indranil Ghosh, Mar 08 2017

from math import gcd
i=j=1
while j<=1000:
    if gcd(bin(i)[2:].count("0"),i)==1 and gcd(bin(i)[2:].count("1"),i)==1 and gcd(len(bin(i)[2:]),i)!=1:
        print(i, end=", ")
        j+=1
    i+=1 # Indranil Ghosh, Mar 08 2017

Extended by Ray Chandler, Jun 11 2009

cp's OEIS Frontend

A161152 Positive integers n such that {the number of (non-leading) 0's in the binary representation of n} is coprime to n.

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A161154 Positive integers n such that both {the number of (non-leading) 0's in the binary representation of n} is coprime to n and {the number of 1's in the binary representation of n} is coprime to n.

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A161155 Positive integers n such that {the number of (non-leading) 0's in the binary representation of n} is coprime to n, {the number of 1's in the binary representation of n} is coprime to n and {the number of digits in the binary representation of n} is coprime to n.

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Programs

Mathematica

PARI

Python

Extensions

A161156 Positive integers n such that {the number of (non-leading) 0's in the binary representation of n} is coprime to n, and {the number of 1's in the binary representation of n} is coprime to n, but {the number of digits in the binary representation of n} is not coprime to n.

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Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions