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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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.

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

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Author

Antti Karttunen, Feb 16 2022

Keywords

Examples

			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.

Links

Crossrefs

Cf. A000203, A003961, A351544, A351547, A351551, A351552, A353666 [gcd(a(n),n)], A353667, A353668, A353633, A354997.
Cf. A342671, A349161, A349162.

Programs

  • PARI
    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])); };

Formula

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

A353667 a(n) = n / gcd(n, A351546(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 3, 13, 7, 5, 16, 17, 18, 19, 10, 21, 11, 23, 6, 25, 13, 27, 1, 29, 15, 31, 32, 11, 17, 35, 36, 37, 19, 39, 4, 41, 21, 43, 11, 15, 23, 47, 12, 49, 50, 17, 26, 53, 27, 55, 7, 57, 29, 59, 15, 61, 31, 63, 64, 65, 33, 67, 34, 23, 35, 71, 72, 73, 37, 75, 19, 77, 39, 79, 40, 81, 41, 83, 3
Offset: 1

Views

Author

Antti Karttunen, May 04 2022

Keywords

Links

Crossrefs

Cf. A000203, A003961, A351546, A353666, A353668.
Differs from A098988 for the first time at n = 28, 30, 40, 60, 66, 84, 90, 102, 120, ...

Programs

  • PARI
    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])); };
A353667(n) = (n / gcd(n, A351546(n)));

Formula

a(n) = n / A353666(n) = n / gcd(n, A351546(n)).

A372572 Lexicographically earliest infinite sequence such that a(i) = a(j) => A009194(i) = A009194(j), A322361(i) = A322361(j) and A342671(i) = A342671(j), for all i, j >= 1.

Original entry on oeis.org

1, 2, 1, 1, 1, 3, 1, 2, 1, 4, 1, 5, 1, 4, 6, 1, 1, 7, 1, 8, 1, 4, 1, 9, 1, 4, 10, 11, 1, 12, 1, 13, 14, 4, 15, 16, 1, 4, 1, 17, 1, 3, 1, 18, 6, 4, 1, 5, 1, 2, 14, 19, 1, 20, 1, 21, 10, 4, 1, 22, 1, 4, 1, 1, 23, 3, 1, 24, 14, 25, 1, 26, 1, 4, 27, 28, 29, 3, 1, 4, 1, 4, 1, 30, 1, 4, 31, 32, 1, 33, 34, 18, 1, 4, 35, 36, 1, 2, 37, 23
Offset: 1

Views

Author

Antti Karttunen, May 24 2024

Keywords

Comments

Restricted growth sequence transform of the triple [A009194(n), A322361(n), A342671(n)].
For all i, j:
a(i) = a(j) => A349167(i) = A349167(j),
a(i) = a(j) => A353666(i) = A353666(j),
a(i) = a(j) => A372565(i) = A372565(j).

Links

Crossrefs

Cf. A009194, A322361, A342671, A349167, A353666, A372565.

Programs

  • PARI
    up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
Aux372572(n) = [gcd(n, sigma(n)), gcd(n, A003961(n)), gcd(sigma(n), A003961(n))];
v372572 = rgs_transform(vector(up_to, n, Aux372572(n)));
A372572(n) = v372572[n];
Showing 1-3 of 3 results.

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