A094387 -id:A094387 - cp's OEIS frontend

A339079 a(n) is the least number which is coprime to its binary weight (A094387) with a gap n to the next term of A094387, or 0 if such a number does not exist.

1, 5, 19, 113, 872, 2357, 619, 831479, 645109, 28011357, 97768316, 377282539, 469754781, 403248635900

Offset: 1

Amiram Eldar, Nov 22 2020

a(15) > 6 * 10^12, if it exists.

			a(1) = 1 since both 1 and 2 = 1 + 1 are coprime to their binary weight, and they are the least pair of consecutive numbers with this property.
a(2) = 5 since 5 and 7 = 5 + 2 are coprime to their binary weight, and 6 is not since gcd(6, A000120(6)) = 2, and they are the least pair with a difference 2 with this property.

Cf. A000120, A094387, A339078 (decimal analog).

copQ[n_] := CoprimeQ[n, DigitCount[n, 2, 1]]; s[mx_] := Module[{c = 0, n1 = 1, n2, seq, d}, seq = Table[0, {mx}]; n2 = n1 + 1; While[c < mx, While[! copQ[n2], n2++]; d = n2 - n1; If[d <= mx && seq[[d]] == 0, c++; seq[[d]] = n1]; n1 = n2; n2++]; seq]; s[9]

A339076 Numbers which are coprime to their digital sum (A007953).

Original entry on oeis.org

1, 10, 11, 13, 14, 16, 17, 19, 23, 25, 29, 31, 32, 34, 35, 37, 38, 41, 43, 47, 49, 52, 53, 56, 58, 59, 61, 65, 67, 71, 73, 74, 76, 79, 83, 85, 89, 91, 92, 94, 95, 97, 98, 100, 101, 103, 104, 106, 107, 109, 113, 115, 119, 121, 122, 124, 125, 127, 128, 131, 137

Offset: 1

Amiram Eldar, Nov 22 2020

Numbers k such that gcd(k, A007953(k)) = 1.
Olivier (1975, 1976) proved that the asymptotic density of this sequence is 9/(2*Pi^2) = 0.455945... (A088245).
None of the terms are divisible by 3.
The powers of 10 (A011557) are terms. These are also the only Niven numbers (A005349) in this sequence.
Includes all the prime numbers above 7.

			10 is a term since A007953(10) = 1 + 0 = 1, and gcd(10, 1) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Curtis Cooper and Robert E. Kennedy, On the set of positive integers which are relatively prime to their digital sum and its complement, J. Inst. Math. & Comp. Sci. (Math. Ser.), Vol. 10 (1997), pp. 173-180.
Christian Mauduit, Carl Pomerance, and András Sárközy, On the distribution in residue classes of integers with a fixed sum of digits, The Ramanujan Journal, Vol. 9, No. 1-2 (2005), pp. 45-62; alternative link.
Michel Olivier, Sur la probabilité que n soit premier à la somme de ses chiffres, C. R. Math. Acad. Sci. Paris, Série A, Vol. 280 (1975), pp. 543-545.
Michel Olivier, Fonctions g-additives et formule asymptotique pour la propriété (n, f(n)) = q, Acta Arithmetica, Vol. 31, No. 4 (1976), pp. 361-384; alternative link.

Cf. A005349, A007953, A088245.
Subsequence of A001651.
Subsequence: A011557.
Binary version: A094387.

Select[Range[200], CoprimeQ[#, Plus @@ IntegerDigits[#]] &]

A332880 If n = Product (p_j^k_j) then a(n) = numerator of Product (1 + 1/p_j).

Original entry on oeis.org

1, 3, 4, 3, 6, 2, 8, 3, 4, 9, 12, 2, 14, 12, 8, 3, 18, 2, 20, 9, 32, 18, 24, 2, 6, 21, 4, 12, 30, 12, 32, 3, 16, 27, 48, 2, 38, 30, 56, 9, 42, 16, 44, 18, 8, 36, 48, 2, 8, 9, 24, 21, 54, 2, 72, 12, 80, 45, 60, 12, 62, 48, 32, 3, 84, 24, 68, 27, 32, 72

Offset: 1

Ilya Gutkovskiy, Feb 28 2020

Numerator of sum of reciprocals of squarefree divisors of n.

(6/Pi^2) * A332881(n)/a(n) is the asymptotic density of numbers that are coprime to their digital sum in base n+1 (see A094387 and A339076 for bases 2 and 10). - Amiram Eldar, Nov 24 2022

			1, 3/2, 4/3, 3/2, 6/5, 2, 8/7, 3/2, 4/3, 9/5, 12/11, 2, 14/13, 12/7, 8/5, 3/2, 18/17, ...

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A001615, A008683, A017665, A028235, A028236, A048250, A076512, A306695, A308443, A308462, A332881 (denominators), A332882.
Cf. also A348734, A348735.
Cf. A059956, A065463, A082020, A094387, A339076.

a:= n-> numer(mul(1+1/i[1], i=ifactors(n)[2])):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 28 2020

Table[If[n == 1, 1, Times @@ (1 + 1/#[[1]] & /@ FactorInteger[n])], {n, 1, 70}] // Numerator
Table[Sum[MoebiusMu[d]^2/d, {d, Divisors[n]}], {n, 1, 70}] // Numerator

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A332880(n) = numerator(A001615(n)/n);

Numerators of coefficients in expansion of Sum_{k>=1} mu(k)^2*x^k/(k*(1 - x^k)).
a(n) = numerator of Sum_{d|n} mu(d)^2/d.
a(n) = numerator of psi(n)/n.
a(p) = p + 1, where p is prime.
a(n) = A001615(n) / A306695(n) = A001615(n) / gcd(n, A001615(n)). - Antti Karttunen, Nov 15 2021
From Amiram Eldar, Nov 24 2022: (Start)
Asymptotic means:
Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A332881(k) = 15/Pi^2 = 1.519817... (A082020).
Limit_{m->oo} (1/m) * Sum_{k=1..m} A332881(k)/a(k) = Product_{p prime} (1 - 1/(p*(p+1))) = 0.704442... (A065463). (End)

A358977 Numbers that are coprime to the sum of their primorial base digits (A276150).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 34, 35, 37, 38, 39, 41, 43, 46, 47, 49, 53, 54, 55, 57, 58, 59, 61, 62, 63, 67, 69, 71, 73, 74, 78, 79, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 98, 101, 102, 103, 106, 107, 109, 110

Offset: 1

Amiram Eldar, Dec 07 2022

Numbers k such that gcd(k, A276150(k)) = 1.
The primorial numbers (A002110) are terms. These are also the only primorial base Niven numbers (A333426) in this sequence.
Includes all the prime numbers.
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 7, 59, 603, 6047, 60861, 608163, 6079048, 60789541, 607847981, 6080015681... . Conjecture: The asymptotic density of this sequence exists and equals 6/Pi^2 = 0.607927... (A059956), the same as the density of A094387.

			3 is a term since A276150(3) = 2, and gcd(3, 2) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A059956, A276150, A333426.
Subsequences: A000040, A002110.
Similar sequences: A094387, A339076, A358975, A358976, A358978.

With[{max = 4}, bases = Prime@Range[max, 1, -1]; nmax = Times @@ bases - 1; sumdig[n_] := Plus @@ IntegerDigits[n, MixedRadix[bases]]; Select[Range[nmax], CoprimeQ[#, sumdig[#]] &]]

is(n) = {my(p=2, s=0, m=n, r); while(m>0, r = m%p; s+=r; m\=p; p = nextprime(p+1)); gcd(n, s)==1; }

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

Original entry on oeis.org

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

A161153 Positive integers that are coprime to their number of digits in binary representation.

Original entry on oeis.org

1, 3, 4, 5, 7, 9, 11, 13, 15, 16, 17, 18, 19, 21, 22, 23, 24, 26, 27, 28, 29, 31, 35, 37, 41, 43, 47, 49, 53, 55, 59, 61, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 90, 92, 93, 94, 95, 96, 97, 99, 100, 101, 102, 103, 104, 106

Offset: 1

Leroy Quet, Jun 03 2009

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

Cf. A094387, A161152, A161154, A161155, A161156.

Select[Range[106], GCD[Length[IntegerDigits[#, 2]], #]==1 &] (* Indranil Ghosh, Mar 08 2017 *)

for(n=1, 106, if(gcd(#digits(n, 2), n)==1, print1(n,", "))) \\ Indranil Ghosh, Mar 08 2017

from math import gcd
i=1
j=1
while j<=100:
    if 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

Name edited by Michel Marcus, Apr 30 2021

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

A358975 Numbers that are coprime to their digital sum in base 3 (A053735).

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 17, 19, 23, 27, 29, 31, 37, 41, 43, 47, 49, 51, 53, 55, 59, 61, 67, 69, 71, 73, 79, 81, 83, 85, 89, 91, 97, 101, 103, 107, 109, 113, 119, 121, 123, 125, 127, 129, 131, 137, 139, 141, 143, 147, 149, 151, 153, 155, 157, 159, 161, 163, 167, 169

Offset: 1

Amiram Eldar, Dec 07 2022

Numbers k such that gcd(k, A053735(k)) = 1.
All the terms are odd since if k is even then A053735(k) is even and so gcd(k, A053735(k)) >= 2.
Olivier (1975, 1976) proved that the asymptotic density of this sequence is 4/Pi^2 = 0.40528... (A185199).
The powers of 3 (A000244) are terms. These are also the only ternary Niven numbers (A064150) in this sequence.
Includes all the odd prime numbers (A065091).

			3 is a term since A053735(3) = 1, and gcd(3, 1) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Christian Mauduit, Carl Pomerance, and András Sárközy, On the distribution in residue classes of integers with a fixed sum of digits, The Ramanujan Journal, Vol. 9, No. 1-2 (2005), pp. 45-62; alternative link.
Michel Olivier, Sur la probabilité que n soit premier à la somme de ses chiffres, C. R. Math. Acad. Sci. Paris, Série A, Vol. 280 (1975), pp. 543-545.
Michel Olivier, Fonctions g-additives et formule asymptotique pour la propriété (n, f(n)) = q, Acta Arithmetica, Vol. 31, No. 4 (1976), pp. 361-384; alternative link.

Cf. A064150, A053735, A185199, A332880.
Subsequences: A000244, A065091.
Similar sequences: A094387, A339076, A358976, A358977, A358978.

q[n_] := CoprimeQ[n, Plus @@ IntegerDigits[n, 3]]; Select[Range[200], q]

PARI
```
is(n) = gcd(n, sumdigits(n, 3)) == 1;
```

cp's OEIS Frontend

A339079 a(n) is the least number which is coprime to its binary weight (A094387) with a gap n to the next term of A094387, or 0 if such a number does not exist.

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A339076 Numbers which are coprime to their digital sum (A007953).

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A332880 If n = Product (p_j^k_j) then a(n) = numerator of Product (1 + 1/p_j).

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A358977 Numbers that are coprime to the sum of their primorial base digits (A276150).

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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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A161153 Positive integers that are coprime to their number of digits in binary representation.

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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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