A351545 -id:A351545 - cp's OEIS frontend

A351547 a(n) = sigma(n) / A351545(n).

1, 1, 4, 7, 6, 4, 8, 15, 13, 2, 12, 28, 14, 8, 24, 31, 18, 13, 20, 6, 32, 4, 24, 12, 31, 14, 40, 56, 30, 8, 32, 63, 48, 2, 48, 91, 38, 20, 56, 90, 42, 32, 44, 84, 78, 8, 48, 124, 57, 31, 72, 98, 54, 40, 72, 120, 16, 10, 60, 24, 62, 32, 104, 127, 12, 16, 68, 14, 96, 16, 72, 195, 74, 38, 124, 140, 96, 56, 80, 186, 121

Offset: 1

Antti Karttunen, Feb 16 2022

nonn

Cf. A000203, A003961, A351545, A351546, A354997.

Cf. A286385, A342671, A349161, A349162.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A351547(n) = { my(s=sigma(n),f=factor(s),u=A003961(n)); s/prod(k=1,#f~,if(!(u%f[k,1]) && (f[k,2]>=valuation(u,f[k,1])), f[k,1]^f[k,2], 1)); };

a(n) = A000203(n) / A351545(n).

a(n) = A351546(n) * A354997(n). - Antti Karttunen, Jul 09 2022

A351544 a(n) is the largest unitary divisor of sigma(n) such that its every prime factor also divides A003961(n), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

Original entry on oeis.org

1, 3, 1, 1, 1, 3, 1, 3, 1, 9, 1, 1, 1, 3, 1, 1, 1, 3, 1, 21, 1, 9, 1, 15, 1, 3, 5, 1, 1, 9, 1, 9, 1, 27, 1, 1, 1, 3, 1, 9, 1, 3, 1, 3, 1, 9, 1, 1, 1, 3, 1, 1, 1, 15, 1, 3, 5, 9, 1, 21, 1, 3, 1, 1, 7, 9, 1, 9, 1, 9, 1, 15, 1, 3, 1, 1, 1, 3, 1, 3, 1, 9, 1, 1, 1, 3, 5, 9, 1, 9, 1, 3, 1, 9, 1, 9, 1, 9, 13, 7, 1, 27

Offset: 1

Antti Karttunen, Feb 16 2022

nonn

Cf. A000203, A003961, A351545, A351546.

Cf. A342671, A349161, A349162.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A351544(n) = { my(s=sigma(n),f=factor(s),u=A003961(n)); prod(k=1,#f~,if(!(u%f[k,1]), f[k,1]^f[k,2], 1)); };

a(n) = Product_{p^e || A000203(n)} p^(e*[p divides A003961(n)]), where [ ] is the Iverson bracket, returning 1 if p is a divisor of A003961(n), and 0 otherwise. Here p^e is the largest power of prime p dividing sigma(n).
a(n) = A000203(n) / A351546(n).
For all n >= 1, a(n) is a multiple of A351545(n).

A354997 a(n) = A351547(n) / A351546(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 5, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 3, 1, 1, 1, 9, 1, 1, 1, 7, 1, 1, 1, 3, 1

Offset: 1

Antti Karttunen, Jul 09 2022

nonn

Cf. A000203, A003961, A351545, A351546, A351547.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A351546(n) = { my(f=factor(sigma(n)),u=A003961(n)); prod(k=1,#f~,f[k,1]^((0!=(u%f[k,1]))*f[k,2])); };
A351547(n) = { my(s=sigma(n),f=factor(s),u=A003961(n)); s/prod(k=1,#f~,if(!(u%f[k,1]) && (f[k,2]>=valuation(u,f[k,1])), f[k,1]^f[k,2], 1)); };
A354997(n) = (A351547(n) / A351546(n));

a(n) = A351547(n) / A351546(n).

cp's OEIS Frontend

A351547 a(n) = sigma(n) / A351545(n).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A351544 a(n) is the largest unitary divisor of sigma(n) such that its every prime factor also divides A003961(n), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A354997 a(n) = A351547(n) / A351546(n).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula