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 1-5 of 5 results.

A348733 a(n) = gcd(A003959(n), A034448(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e and A034448 (usigma) is multiplicative with a(p^e) = (p^e)+1.

Original entry on oeis.org

1, 3, 4, 1, 6, 12, 8, 9, 2, 18, 12, 4, 14, 24, 24, 1, 18, 6, 20, 6, 32, 36, 24, 36, 2, 42, 4, 8, 30, 72, 32, 3, 48, 54, 48, 2, 38, 60, 56, 54, 42, 96, 44, 12, 12, 72, 48, 4, 2, 6, 72, 14, 54, 12, 72, 72, 80, 90, 60, 24, 62, 96, 16, 1, 84, 144, 68, 18, 96, 144, 72, 18, 74, 114, 8, 20, 96, 168, 80, 6, 2, 126, 84, 32
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 1444 = 2^2 * 19^2, where a(1444) = 10 != 1*2 = a(4)*a(361). See A348740 for the list of such positions.

Links

Crossrefs

Cf. A003959, A034448, A348732, A348734, A348735, A348740.
Cf. also A344695, A348047, A348503, A348946 for similar, almost multiplicative sequences.

Programs

  • Mathematica
    f1[p_, e_] := (p + 1)^e; f2[p_, e_] := p^e + 1; a[1] = 1; a[n_] := GCD[Times @@ f1 @@@ (f = FactorInteger[n]), Times @@ f2 @@@ 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])); };
A348733(n) = gcd(A003959(n), A034448(n));

Formula

a(n) = gcd(A003959(n), A034448(n)).
a(n) = gcd(A003959(n), A348732(n)) = gcd(A034448(n), A348732(n)).
a(n) = A003959(n) / A348734(n) = A034448(n) / A348735(n).

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).

A348505 a(n) = usigma(n) / gcd(sigma(n), usigma(n)), where sigma is the sum of divisors function, A000203, and usigma is the unitary sigma, A034448.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 3, 10, 1, 1, 5, 1, 1, 1, 17, 1, 10, 1, 5, 1, 1, 1, 3, 26, 1, 7, 5, 1, 1, 1, 11, 1, 1, 1, 50, 1, 1, 1, 3, 1, 1, 1, 5, 10, 1, 1, 17, 50, 26, 1, 5, 1, 7, 1, 3, 1, 1, 1, 5, 1, 1, 10, 65, 1, 1, 1, 5, 1, 1, 1, 6, 1, 1, 26, 5, 1, 1, 1, 17, 82, 1, 1, 5, 1, 1, 1, 3, 1, 10, 1, 5, 1, 1, 1, 11, 1, 50, 10, 130
Offset: 1

Views

Author

Antti Karttunen, Oct 29 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 72 = 8*9, where a(72) = 6 != 3*10 = a(8) * a(9).

Links

Crossrefs

Cf. A000203, A005117, A034448, A048146, A063880, A348503, A348504, A348506 (positions of ones).
Cf. also A344697, A348049.

Programs

  • Mathematica
    f1[p_, e_] := p^e + 1; f2[p_, e_] := (p^(e + 1) - 1)/(p - 1); a[1] = 1; a[n_] := (usigma = Times @@ f1 @@@ (fct = FactorInteger[n])) / GCD[usigma, Times @@ f2 @@@ fct]; Array[a, 100] (* Amiram Eldar, Oct 29 2021 *)
  • PARI
    A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A348505(n) = { my(u=A034448(n)); (u/gcd(u, sigma(n))); };

Formula

a(n) = A034448(n) / A348503(n) = A034448(n) / gcd(A000203(n), A034448(n)).

A367991 The sum of the divisors of the squarefree part of n.

Original entry on oeis.org

1, 3, 4, 1, 6, 12, 8, 3, 1, 18, 12, 4, 14, 24, 24, 1, 18, 3, 20, 6, 32, 36, 24, 12, 1, 42, 4, 8, 30, 72, 32, 3, 48, 54, 48, 1, 38, 60, 56, 18, 42, 96, 44, 12, 6, 72, 48, 4, 1, 3, 72, 14, 54, 12, 72, 24, 80, 90, 60, 24, 62, 96, 8, 1, 84, 144, 68, 18, 96, 144, 72
Offset: 1

Views

Author

Amiram Eldar, Dec 07 2023

Keywords

Comments

First differs from A348503 at n = 72 and from A344695 at n = 108.
The sum of the infinitary divisors (A077609) of n that are squarefree (A005117). - Amiram Eldar, Jun 03 2025

Links

Crossrefs

Cf. A000203, A000290, A005117, A007913, A048250, A077609, A367990.
Cf. A002117, A013662.
Cf. A344695, A348503.

Programs

  • Mathematica
    f[p_, e_] := If[OddQ[e], p + 1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2]%2, f[i,1]+1, 1));}

Formula

Multiplicative with a(p^e) = p + 1 if e is odd and 1 otherwise.
a(n) = A000203(A007913(n)) = A048250(A007913(n)).
a(n) = A048250(n)/A367990(n).
a(n) >= 1, with equality if and only if n is a square (A000290).
a(n) <= A000203(n), with equality if and only if n is squarefree (A005117).
Dirichlet g.f.: zeta(2*s) * Product_{p prime} (1 + 1/p^(s-1) + 1/p^s).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(4)/zeta(3) = 0.900392677639... .

A348504 a(n) = sigma(n) / gcd(sigma(n), usigma(n)), where sigma is the sum of divisors function, A000203, and usigma is the unitary sigma, A034448.

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, 10, 7, 1, 1, 1, 21, 1, 1, 1, 91, 1, 1, 1, 5, 1, 1, 1, 7, 13, 1, 1, 31, 57, 31, 1, 7, 1, 10, 1, 5, 1, 1, 1, 7, 1, 1, 13, 127, 1, 1, 1, 7, 1, 1, 1, 13, 1, 1, 31, 7, 1, 1, 1, 31, 121, 1, 1, 7, 1, 1, 1, 5, 1, 13, 1, 7, 1, 1, 1, 21, 1, 57
Offset: 1

Views

Author

Antti Karttunen, Oct 29 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 72 = 8*9, where a(72) = 13 != 5*13 = a(8) * a(9).

Links

Crossrefs

Cf. A000203, A005117 (positions of ones), A034448, A048146, A348503, A348505.
Differs from A344696 for the first time at n=72, where a(72) = 13, while A344696(72) = 65. Cf. also A348048.

Programs

  • Mathematica
    f1[p_, e_] := p^e + 1; f2[p_, e_] := (p^(e + 1) - 1)/(p - 1); a[1] = 1; a[n_] := (sigma = Times @@ f2 @@@ (fct = FactorInteger[n])) / GCD[sigma, Times @@ f1 @@@ fct]; Array[a, 100] (* Amiram Eldar, Oct 29 2021 *)
  • PARI
    A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A348504(n) = { my(u=sigma(n)); (u/gcd(u, A034448(n))); };

Formula

a(n) = A000203(n) / A348503(n) = A000203(n) / gcd(A000203(n), A034448(n)).
Showing 1-5 of 5 results.

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