A351547 -id:A351547 - cp's OEIS frontend

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

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).

A351546 a(n) is the largest unitary divisor of sigma(n) coprime with 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, 1, 4, 7, 6, 4, 8, 5, 13, 2, 12, 28, 14, 8, 24, 31, 18, 13, 20, 2, 32, 4, 24, 4, 31, 14, 8, 56, 30, 8, 32, 7, 48, 2, 48, 91, 38, 20, 56, 10, 42, 32, 44, 28, 78, 8, 48, 124, 57, 31, 72, 98, 54, 8, 72, 40, 16, 10, 60, 8, 62, 32, 104, 127, 12, 16, 68, 14, 96, 16, 72, 13, 74, 38, 124, 140, 96, 56, 80, 62, 121, 14, 84

Offset: 1

Antti Karttunen, Feb 16 2022

nonn

			For n = 672 = 2^5 * 3^1 * 7^1, and the largest unitary divisor of the sigma(672) [= 2^5 * 3^2 * 7^1] coprime with A003961(672) = 13365 = 3^5 * 5^1 * 11^1 is 2^5 * 7^1 = 224, therefore a(672) = 224.

Cf. A000203, A003961, A351544, A351547, A351551, A351552, A353666 [gcd(a(n),n)], A353667, A353668, A353633, A354997.

Cf. A342671, A349161, A349162.

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])); };

a(n) = Product_{p^e || A000203(n)} p^(e*[p does not divide A003961(n)]), where [ ] is the Iverson bracket, returning 0 if p is a divisor of A003961(n), and 1 otherwise. Here p^e is the largest power of each prime p dividing sigma(n).
a(n) = A000203(n) / A351544(n).
a(n) = A353666(n) * A353668(n) = A351547(n) / A354997(n). - Antti Karttunen, Jul 09 2022

A351545 a(n) is the largest unitary divisor of sigma(n) such that its every prime factor p also divides A003961(n), and valuation(sigma(n),p) >= valuation(A003961(n),p).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 16 2022

nonn

Cf. A000203, A003961, A351544, A351547.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A351545(n) = { my(s=sigma(n),f=factor(s),u=A003961(n)); 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) = Product_{p^e || A000203(n)} p^(e*[p divides A003961(n) but p^(1+e) does not divide A003961(n)]), where [ ] is the Iverson bracket, returning 1 if the condition holds, and 0 otherwise. Here p^e is the largest power of prime p dividing sigma(n).
a(n) = A000203(n) / A351547(n).
For all n >= 1, a(n) is a divisor of A351544(n), which is a divisor of A000203(n).

cp's OEIS Frontend

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

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A351546 a(n) is the largest unitary divisor of sigma(n) coprime with A003961(n), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

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A351545 a(n) is the largest unitary divisor of sigma(n) such that its every prime factor p also divides A003961(n), and valuation(sigma(n),p) >= valuation(A003961(n),p).

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