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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A348946 a(n) = gcd(sigma(n), A348944(n)), where A348944 is the arithmetic mean of A003959 and A034448, and sigma is the sum of divisors function.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 3, 13, 18, 12, 28, 14, 24, 24, 1, 18, 39, 20, 42, 32, 36, 24, 12, 31, 42, 2, 56, 30, 72, 32, 3, 48, 54, 48, 1, 38, 60, 56, 18, 42, 96, 44, 84, 78, 72, 48, 4, 57, 93, 72, 98, 54, 6, 72, 24, 80, 90, 60, 168, 62, 96, 104, 1, 84, 144, 68, 126, 96, 144, 72, 3, 74, 114, 124, 140, 96, 168, 80, 6, 1, 126
Offset: 1

Views

Author

Antti Karttunen, Nov 05 2021

Keywords

Comments

This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 36 = 2^2 * 3^2, where a(36) = 1 <> 91 = 7*13 = a(4)*a(9).

Links

Crossrefs

Cf. A000203, A003959, A034448, A348944, A348945, A348947, A348948.
Cf. also A344695, A348047, A348503, A348733.

Programs

  • Mathematica
    f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := (p + 1)^e; f3[p_, e_] := p^e + 1; a[1] = 1; a[n_] := GCD[Times @@ f1 @@@ (f = FactorInteger[n]), (Times @@ f2 @@@ f + Times @@ f3 @@@ f)/2]; Array[a, 100] (* Amiram Eldar, Nov 05 2021 *)
  • PARI
    A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A034448(n) = { my(f = factor(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); };
A348946(n) = gcd(sigma(n), ((1/2)*(A003959(n)+A034448(n))));

Formula

a(n) = gcd(A000203(n), A348944(n)).
a(n) = gcd(A000203(n), A348945(n)) = gcd(A348944(n), A348945(n));
a(n) = A348944(n) / A348947(n) = A000203(n) / A348948(n).

A348947 a(n) = A348944(n) / gcd(sigma(n), A348944(n)), where A348944 is the arithmetic mean of A003959 and A034448, and sigma is the sum of divisors function.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 49, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 23, 1, 1, 1, 1, 46, 1, 1, 1, 97, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 49, 1, 1, 1, 1, 1, 23, 1, 6, 1, 1, 1, 1, 1, 1, 1, 397, 1, 1, 1, 1, 1, 1, 1, 87, 1, 1, 1, 1, 1, 1, 1, 49, 169, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 46, 1, 1, 1, 227
Offset: 1

Views

Author

Antti Karttunen, Nov 05 2021

Keywords

Comments

Numerator of ratio A348944(n) / A000203(n).
This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 36 = 2^2 * 3^2, where a(36) = 97 <> 1 = a(4)*a(9).

Links

Crossrefs

Cf. A000203, A003959, A034448, A348944, A348945, A348946, A348948 (denominators).

Programs

  • Mathematica
    f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := (p + 1)^e; f3[p_, e_] := p^e + 1; a[1] = 1; a[n_] := (s = (Times @@ f2 @@@ (f = FactorInteger[n]) + Times @@ f3 @@@ f) / 2) / GCD[Times @@ f1 @@@ f, s]; Array[a, 100] (* Amiram Eldar, Nov 05 2021 *)
  • PARI
    A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A034448(n) = { my(f = factor(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); };
A348944(n) = ((1/2)*(A003959(n)+A034448(n)));
A348947(n) = { my(u=A348944(n)); (u/gcd(sigma(n),u)); };

Formula

a(n) = A348944(n) / A348946(n) = A348944(n) / gcd(A000203(n), A348944(n)).

A348985 Numerator of ratio sigma(n) / A325973(n), where A325973 is the arithmetic mean of {sum of squarefree divisors} and {sum of unitary divisors}.

Original entry on oeis.org

1, 1, 1, 7, 1, 1, 1, 5, 13, 1, 1, 7, 1, 1, 1, 31, 1, 13, 1, 7, 1, 1, 1, 5, 31, 1, 5, 7, 1, 1, 1, 7, 1, 1, 1, 91, 1, 1, 1, 5, 1, 1, 1, 7, 13, 1, 1, 31, 57, 31, 1, 7, 1, 5, 1, 5, 1, 1, 1, 7, 1, 1, 13, 127, 1, 1, 1, 7, 1, 1, 1, 65, 1, 1, 31, 7, 1, 1, 1, 31, 121, 1, 1, 7, 1, 1, 1, 5, 1, 13, 1, 7, 1, 1, 1, 7, 1, 57, 13
Offset: 1

Views

Author

Antti Karttunen, Nov 06 2021

Keywords

Comments

This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 108 = 4*27, where a(108) = 70 <> 35 = 7*5 = a(4)*(27).

Links

Crossrefs

Cf. A000203, A048250, A034448, A325973, A325974, A348984, A348986 (denominators).
Differs from A348048 for the first time at n=108, where a(108) = 70, while A348048(108) = 35.
Cf. also A348948.

Programs

  • Mathematica
    f1[p_, e_] := p + 1; f2[p_, e_] := p^e + 1; s[1] = 1; s[n_] := (Times @@ f1 @@@ (f = FactorInteger[n]) + Times @@ f2 @@@ f)/2; a[n_] := Numerator[DivisorSigma[1, n]/s[n]]; Array[a, 100] (* Amiram Eldar, Nov 06 2021 *)
  • PARI
    A325973(n) = (1/2)*sumdiv(n, d, d*(issquarefree(d) + (1==gcd(d, n/d))));
A348985(n) = { my(s=sigma(n)); (s/gcd(s, A325973(n))); };

Formula

a(n) = A000203(n) / A348984(n) = sigma(n) / gcd(sigma(n), A325973(n)).
Showing 1-3 of 3 results.

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