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.

Showing 1-4 of 4 results.

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).

A348945 a(n) = A348944(n) - sigma(n), where A348944 is the arithmetic mean of A003959 and A034448, and sigma is the sum of divisors function.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 18, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 6, 0, 0, 0, 0, 75, 0, 0, 0, 6, 0, 0, 0, 18, 0, 0, 0, 0, 0, 0, 0, 72, 0, 0, 0, 0, 0, 18, 0, 24, 0, 0, 0, 0, 0, 0, 0, 270, 0, 0, 0, 0, 0, 0, 0, 66, 0, 0, 0, 0, 0, 0, 0, 108, 48, 0, 0, 0, 0, 0, 0, 36, 0, 0, 0, 0, 0, 0, 0, 300, 0, 0, 0, 10
Offset: 1

Views

Author

Antti Karttunen, Nov 05 2021

Keywords

Links

Crossrefs

Cf. A000203, A003959, A034448, A048146, A348029, A348944, A348946.

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] = 0; a[n_] := (Times @@ f2 @@@ (f = FactorInteger[n]) + Times @@ f3 @@@ f) / 2 - Times @@ f1 @@@ f; 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)));
A348945(n) = (A348944(n)-sigma(n));

Formula

a(n) = A348944(n) - A000203(n) = ((1/2) * (A003959(n)+A034448(n))) - A000203(n).
a(n) = (1/2) * (A348029(n)-A048146(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)).

A348948 a(n) = sigma(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, 5, 1, 1, 1, 1, 1, 1, 1, 31, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 20, 1, 1, 1, 1, 21, 1, 1, 1, 91, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 31, 1, 1, 1, 1, 1, 20, 1, 5, 1, 1, 1, 1, 1, 1, 1, 127, 1, 1, 1, 1, 1, 1, 1, 65, 1, 1, 1, 1, 1, 1, 1, 31, 121, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 21, 1, 1, 1, 217
Offset: 1

Views

Author

Antti Karttunen, Nov 05 2021

Keywords

Comments

Denominator 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) = 91 <> 1 = a(4)*a(9).

Links

Crossrefs

Cf. A000203, A003959, A034448, A348944, A348945, A348946, A348947 (numerators).

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 @@ f1 @@@ (f = FactorInteger[n])) / GCD[s, (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])); };
A348944(n) = ((1/2)*(A003959(n)+A034448(n)));
A348948(n) = { my(s=sigma(n)); (s/gcd(s,A348944(n))); };

Formula

a(n) = A000203(n) / A348946(n) = A000203(n) / gcd(A000203(n), A348944(n)).
Showing 1-4 of 4 results.

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