cp's OEIS Frontend

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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A342671 a(n) = gcd(sigma(n), A003961(n)), where A003961 is fully multiplicative with a(prime(k)) = prime(k+1), and sigma is the sum of divisors of n.

Original entry on oeis.org

1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 21, 1, 3, 1, 15, 1, 3, 5, 1, 1, 3, 1, 9, 1, 3, 1, 1, 1, 3, 1, 9, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 1, 15, 1, 3, 5, 3, 1, 21, 1, 3, 1, 1, 7, 3, 1, 9, 1, 3, 1, 15, 1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 5, 9, 1, 3, 1, 3, 1, 3, 1, 9, 1, 3, 13, 7, 1, 3, 1, 3, 1
Offset: 1

Views

Author

Antti Karttunen, Mar 20 2021

Keywords

Links

Crossrefs

Cf. A000203, A003961, A161942, A286385, A341529, A342672, A342673, A348992, A349161, A349162, A349163, A349164, A349165 (positions of 1's), A349166 (of terms > 1), A349167, A349756, A350071 [= a(n^2)], A355828 (Dirichlet inverse).
Cf. A349169, A349745, A355833, A355924 (applied onto prime shift array A246278).
Cf. also A336850, A354827/A354828, A355932.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A342671(n) = gcd(sigma(n), A003961(n));

Formula

a(n) = gcd(A000203(n), A003961(n)).
a(n) = gcd(A000203(n), A286385(n)) = gcd(A003961(n), A286385(n)).
a(n) = A341529(n) / A342672(n).
From Antti Karttunen, Jul 21 2022: (Start)
a(n) = A003961(n) / A349161(n).
a(n) = A000203(n) / A349162(n).
a(n) = A161942(n) / A348992(n).
a(n) = A003961(A349163(n)) = A003961(n/A349164(n)).
(End)

A355934 a(n) = sigma(n) / gcd(sigma(n), sigma(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, 2, 7, 3, 1, 2, 3, 13, 9, 6, 14, 7, 1, 1, 31, 9, 39, 5, 21, 4, 9, 4, 1, 31, 7, 10, 14, 15, 3, 16, 9, 4, 27, 1, 7, 19, 5, 14, 9, 21, 1, 11, 6, 39, 3, 8, 62, 3, 31, 3, 49, 9, 5, 9, 1, 5, 45, 30, 7, 31, 12, 26, 127, 7, 3, 17, 63, 8, 3, 36, 39, 37, 19, 62, 35, 4, 7, 20, 93, 11, 63, 14, 28, 27, 11, 5, 9, 45, 117
Offset: 1

Views

Author

Antti Karttunen, Jul 22 2022

Keywords

Comments

Denominator of ratio A003973(n) / A000203(n). See comments in A355933.

Links

Crossrefs

Cf. A000203, A003961, A003973, A355932, A355933 (numerators), A355940, A355941 (positions of 1's).
Cf. also A336849, A349162.

Programs

  • Mathematica
    f[p_, e_] := ((q = NextPrime[p])^(e + 1) - 1)/(q - 1); a[1] = 1; a[n_] := Denominator[Times @@ f @@@ FactorInteger[n] / DivisorSigma[1, n]]; Array[a, 100] (* Amiram Eldar, Jul 22 2022 *)
  • PARI
    A003973(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); sigma(factorback(f)); };
A355934(n) = { my(u=sigma(n)); (u/gcd(A003973(n), u)); };

Formula

a(n) = A000203(n) / A355932(n) = A000203(n) / gcd(A000203(n), A003973(n)).

A355933 a(n) = A003973(n) / gcd(sigma(n), A003973(n)), where A003973(n) = sigma(A003961(n)) and A003961 is fully multiplicative with a(p) = nextprime(p).

Original entry on oeis.org

1, 4, 3, 13, 4, 2, 3, 8, 31, 16, 7, 39, 9, 2, 2, 121, 10, 124, 6, 52, 9, 14, 5, 4, 57, 12, 39, 39, 16, 8, 19, 52, 7, 40, 2, 31, 21, 8, 27, 32, 22, 3, 12, 13, 124, 5, 9, 363, 7, 76, 5, 117, 10, 26, 14, 4, 9, 64, 31, 26, 34, 19, 93, 1093, 12, 7, 18, 130, 15, 8, 37, 248, 40, 28, 171, 78, 7, 18, 21, 484, 71, 88, 15, 117
Offset: 1

Views

Author

Antti Karttunen, Jul 22 2022

Keywords

Comments

Numerator of ratio A003973(n) / A000203(n). This sequence gives the numerators when presented in its lowest terms, while A355934 gives the denominators. As both A000203 and A003973 are multiplicative sequences, their ratio is also: 1, 4/3, 3/2, 13/7, 4/3, 2/1, 3/2, 8/3, 31/13, 16/9, 7/6, 39/14, 9/7, 2/1, 2/1, 121/31, 10/9, 124/39, 6/5, etc.

Links

Crossrefs

Cf. A000203, A003961, A003973, A355932, A355934 (denominators).
Cf. also A341525, A349161.

Programs

  • Mathematica
    f[p_, e_] := ((q = NextPrime[p])^(e + 1) - 1)/(q - 1); a[1] = 1; a[n_] := Numerator[Times @@ f @@@ FactorInteger[n] / DivisorSigma[1, n]]; Array[a, 100] (* Amiram Eldar, Jul 22 2022 *)
  • PARI
    A003973(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); sigma(factorback(f)); };
A355933(n) = { my(u=A003973(n)); (u/gcd(sigma(n), u)); };

Formula

a(n) = A003973(n) / A355932(n) = A003973(n) / gcd(A000203(n), A003973(n)).
Showing 1-3 of 3 results.

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