A324644 -id:A324644 - cp's OEIS frontend

A351559 a(n) = A048675(gcd(sigma(n), A019565(n))).

0, 2, 1, 0, 1, 2, 1, 0, 0, 2, 3, 8, 9, 2, 3, 0, 1, 2, 1, 0, 1, 2, 3, 0, 0, 10, 1, 8, 5, 2, 1, 0, 1, 2, 3, 32, 1, 6, 1, 0, 9, 2, 1, 8, 33, 2, 3, 0, 0, 2, 3, 0, 1, 6, 3, 0, 1, 2, 3, 8, 1, 2, 33, 0, 1, 2, 65, 0, 1, 2, 3, 0, 1, 2, 1, 12, 1, 10, 5, 0, 16, 2, 3, 0, 1, 18, 7, 0, 1, 2, 9, 8, 1, 2, 7, 0, 1, 2, 35, 0, 65, 2, 33

Offset: 1

Antti Karttunen, Feb 19 2022

Cf. A000203, A004198, A007947, A019565, A080398, A351557, A351560.

Cf. also A324644, A351558.

Table[If[# == 1, 0, Total[#2*2^PrimePi[#1] & @@@ FactorInteger[#]]/2] &@ GCD[DivisorSigma[1, n], Times @@ Prime@ Flatten@ Position[Reverse@ IntegerDigits[n, 2], 1]], {n, 103}] (* Michael De Vlieger, Feb 20 2022 *)

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A351559(n) = A048675(gcd(sigma(n), A019565(n)));

a(n) = A048675(A351557(n)) = A048675(gcd(sigma(n), A019565(n))).

a(n) = n AND A351560(n), where AND is bitwise-and, A004198.

A324646 a(n) = gcd(n, A276086(n-1)).

Original entry on oeis.org

1, 2, 3, 2, 1, 6, 1, 2, 3, 10, 1, 6, 1, 2, 15, 2, 1, 18, 1, 10, 3, 2, 1, 6, 25, 2, 3, 2, 1, 30, 1, 2, 3, 2, 7, 18, 1, 2, 3, 10, 1, 42, 1, 2, 15, 2, 1, 6, 7, 50, 3, 2, 1, 18, 5, 14, 3, 2, 1, 30, 1, 2, 21, 2, 1, 6, 1, 2, 3, 70, 1, 18, 1, 2, 75, 2, 7, 6, 1, 10, 3, 2, 1, 42, 5, 2, 3, 2, 1, 90, 7, 2, 3, 2, 1, 6, 1, 98, 3, 10, 1, 6, 1, 2, 105

Offset: 1

Antti Karttunen, Mar 11 2019

nonn

Cf. A002110, A276086, A324198, A324644.

A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A324646(n) = gcd(n,A276086(n-1));

a(n) = gcd(n, A276086(n-1)).

a(A002110(n)) = A002110(n) for all n >= 0.

A351557 a(n) = gcd(sigma(n), A019565(n)).

Original entry on oeis.org

1, 3, 2, 1, 2, 3, 2, 1, 1, 3, 6, 7, 14, 3, 6, 1, 2, 3, 2, 1, 2, 3, 6, 1, 1, 21, 2, 7, 10, 3, 2, 1, 2, 3, 6, 13, 2, 15, 2, 1, 14, 3, 2, 7, 26, 3, 6, 1, 1, 3, 6, 1, 2, 15, 6, 1, 2, 3, 6, 7, 2, 3, 26, 1, 2, 3, 34, 1, 2, 3, 6, 1, 2, 3, 2, 35, 2, 21, 10, 1, 11, 3, 6, 1, 2, 33, 30, 1, 2, 3, 14, 7, 2, 3, 30, 1, 2, 3, 78

Offset: 1

Antti Karttunen, Feb 19 2022

nonn

Cf. A000203, A007947, A019565, A080398.

Cf. also A324644, A351549, A351556, A351559.

Table[GCD[DivisorSigma[1, n], Times @@ Prime@ Flatten@ Position[Reverse@ IntegerDigits[n, 2], 1]], {n, 99}] (* Michael De Vlieger, Feb 20 2022 *)

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A351557(n) = gcd(sigma(n), A019565(n));

a(n) = gcd(A000203(n), A019565(n)) = gcd(A080398(n), A019565(n)).
a(n) = A007947(a(n)).
a(n) = A019565(A351559(n)).

A355456 Greatest common divisor of sigma(n), A003961(n), and A276086(n).

Original entry on oeis.org

1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 1, 5, 1, 3, 5, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 3, 1, 9, 1, 1, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 5, 3, 1, 7, 1, 3, 1, 1, 7, 1, 1, 3, 1, 3, 1, 5, 1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 3, 5, 9, 1, 1, 1, 3, 1, 3, 1, 1, 1, 3, 1, 7, 1, 1, 1, 3, 1

Offset: 1

Antti Karttunen, Jul 13 2022

nonn

Cf. A000203, A003961, A276086, A324644, A342671, A355442, A355002 (terms k such that a(k) shares a prime factor with k).

Cf. also A323653, A351459.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A355442(n) = gcd(A003961(n), A276086(n));
A355456(n) = gcd(sigma(n), A355442(n));

a(n) = gcd(A000203(n), A355442(n)).

a(n) = gcd(A324644(n), A342671(n)) = gcd(A276086(n), A342671(n)) = gcd(A003961(n), A324644(n)).

A379489 a(n) = gcd(n,A003961(n))gcd(sigma(n),A276086(n)) - gcd(n,A276086(n))gcd(sigma(n),A003961(n)), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1), and A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 0, -1, 0, 5, 0, 1, 12, -2, -6, 5, 2, 1, 0, 15, 0, 17, 0, 9, -102, -1, 6, 5, 0, -24, 0, -5, 0, 29, 12, 1, 12, 3, 6, 35, 62, 1, 12, 11, 0, 41, -18, 1, 18, 15, 6, 5, 2, -6, -72, 3, 6, 17, 0, -3, -6, -5, 42, 29, 84, 1, 0, -19, 0, 35, 0, 1, 12, 3, -42, 17, 30, 1, 0, -65, 34, 59, 18, 9, -12, -2, 60, 41, 14, -3, 0, 15

Offset: 1

Antti Karttunen, Jan 02 2025

sign

Cf. A000203, A003961, A276086, A379486 (positions of 0's), A379487, A379488.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A379489(n) = { my(s=sigma(n),x=A003961(n),y=A276086(n)); (gcd(n,x)*gcd(s,y))-(gcd(n,y)*gcd(s,x)); };

a(n) = A379487(n) - A379488(n) = A322361(n)*A324644(n) - A324198(n)*A342671(n).

A324534 The smallest common prime factor of sigma(n) and A276086(n), or 1 if no such prime exists.

Original entry on oeis.org

1, 3, 2, 1, 2, 1, 2, 3, 1, 3, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 3, 2, 5, 1, 3, 2, 1, 2, 1, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 1, 2, 3, 2, 3, 2, 1, 1, 3, 2, 7, 2, 5, 2, 3, 2, 3, 2, 7, 2, 3, 2, 1, 2, 1, 2, 3, 2, 3, 2, 5, 2, 3, 2, 5, 2, 7, 2, 3, 1, 3, 2, 7, 2, 3, 2, 3, 2, 1, 2, 3, 2, 3, 2, 7, 2, 3, 2, 7, 2, 1, 2, 3, 2

Offset: 1

Antti Karttunen, Mar 11 2019

nonn

Cf. A000203, A020639, A276086, A324644.

A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A020639(n) = if(1==n, n, factor(n)[1, 1]);
A324534(n) = A020639(gcd(sigma(n),A276086(n)));

a(n) = A020639(A324644(n)) = A020639(gcd(A000203(n),A276086(n))).

A369445 Numerator of sigma(n) / A276086(n), where A276086 is the primorial base exp-function.

Original entry on oeis.org

1, 1, 2, 7, 1, 12, 4, 1, 13, 2, 2, 28, 7, 8, 4, 31, 1, 39, 2, 14, 16, 4, 4, 12, 31, 14, 4, 56, 1, 72, 16, 3, 8, 6, 8, 13, 19, 4, 4, 2, 1, 96, 22, 4, 13, 8, 8, 124, 57, 31, 12, 14, 3, 24, 36, 8, 8, 2, 2, 24, 31, 32, 52, 127, 2, 144, 34, 6, 16, 16, 4, 39, 37, 38, 62, 4, 16, 24, 8, 62, 121, 2, 2, 32, 54, 44, 4, 4, 1, 234, 8, 8, 64

Offset: 1

Antti Karttunen, Jan 25 2024

Cf. A000203, A276086, A324644, A369446 (denominators), A369261.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A369445(n) = { my(s=sigma(n)); (s/gcd(s,A276086(n))); };

a(n) = A000203(n) / A324644(n).

A379491 Multiperfect numbers k for which gcd(k,A003961(k))gcd(sigma(k),A276086(k)) is equal to gcd(k,A276086(k))gcd(sigma(k),A003961(k)), where A003961(n) is fully multiplicative with a(prime(i)) = prime(i+1), and A276086 is the primorial base exp-function.

Original entry on oeis.org

1, 6, 28, 496, 8128, 30240, 32760, 2178540, 23569920, 33550336, 45532800, 142990848, 459818240, 1379454720, 8589869056, 43861478400, 51001180160, 66433720320, 137438691328, 153003540480, 403031236608, 704575228896, 181742883469056, 6088728021160320, 14942123276641920, 20158185857531904, 275502900594021408, 622286506811515392

Offset: 1

Antti Karttunen, Jan 02 2025

nonn

Intersection of A007691 and A379486.
Cf. A000203, A003961, A276086, A322361, A324198, A324644, A342671, A379485, A379487, A379488, A379492 (complement in A007691).
Subsequences: A323653 (conjectured), A336702 (apart from any hypothetical odd terms > 1).

isA379491(n) = (!(sigma(n)%n) && A379485(n));

A324645 a(n) = gcd(d(n), A276086(n)), where d(n) gives the number of divisors (A000005).

Original entry on oeis.org

1, 1, 2, 3, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 5, 2, 1, 2, 3, 2, 1, 2, 1, 1, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 6, 1, 2, 5, 1, 3, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 6, 7, 2, 1, 2, 3, 2, 1, 2, 1, 2, 1, 6, 3, 2, 1, 2, 5, 5, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 6, 9, 2, 1, 2, 1, 2

Offset: 1

Antti Karttunen, Mar 11 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..30030

Cf. A000005, A009191, A276086, A324644.

A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A324645(n) = gcd(numdiv(n),A276086(n));

a(n) = gcd(A000005(n), A276086(n)).

A355002 Numbers k that share a prime factor with A355456(k).

Original entry on oeis.org

135, 140, 285, 435, 455, 675, 700, 855, 885, 910, 945, 980, 1120, 1185, 1305, 1335, 1365, 1425, 1435, 1485, 1540, 1635, 1755, 1820, 1995, 2085, 2175, 2235, 2275, 2295, 2380, 2565, 2574, 2655, 2660, 2685, 2870, 2905, 2985, 3045, 3105, 3135, 3185, 3220, 3311, 3375, 3395, 3435, 3500, 3555, 3585, 3640, 3705, 3915, 4005

Offset: 1

Antti Karttunen, Jul 13 2022

nonn

Numbers k such that A324198(k) and A342671(k) are not relatively prime.

Cf. A000203, A003961, A276086, A324644, A342671, A351459, A355442, A355456.

Subsequence of A324584 and of A349166.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A355442(n) = gcd(A003961(n), A276086(n));
A355456(n) = gcd(sigma(n), A355442(n));
isA355002(n) = (1!=gcd(n,A355456(n)));

cp's OEIS Frontend

A351559 a(n) = A048675(gcd(sigma(n), A019565(n))).

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A324646 a(n) = gcd(n, A276086(n-1)).

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A351557 a(n) = gcd(sigma(n), A019565(n)).

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A355456 Greatest common divisor of sigma(n), A003961(n), and A276086(n).

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A379489 a(n) = gcd(n,A003961(n))*gcd(sigma(n),A276086(n)) - gcd(n,A276086(n))*gcd(sigma(n),A003961(n)), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1), and A276086 is the primorial base exp-function.

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A324534 The smallest common prime factor of sigma(n) and A276086(n), or 1 if no such prime exists.

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A369445 Numerator of sigma(n) / A276086(n), where A276086 is the primorial base exp-function.

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A379491 Multiperfect numbers k for which gcd(k,A003961(k))*gcd(sigma(k),A276086(k)) is equal to gcd(k,A276086(k))*gcd(sigma(k),A003961(k)), where A003961(n) is fully multiplicative with a(prime(i)) = prime(i+1), and A276086 is the primorial base exp-function.

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A324645 a(n) = gcd(d(n), A276086(n)), where d(n) gives the number of divisors (A000005).

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A379489 a(n) = gcd(n,A003961(n))gcd(sigma(n),A276086(n)) - gcd(n,A276086(n))gcd(sigma(n),A003961(n)), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1), and A276086 is the primorial base exp-function.

A379491 Multiperfect numbers k for which gcd(k,A003961(k))gcd(sigma(k),A276086(k)) is equal to gcd(k,A276086(k))gcd(sigma(k),A003961(k)), where A003961(n) is fully multiplicative with a(prime(i)) = prime(i+1), and A276086 is the primorial base exp-function.