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).

A332880 If n = Product (p_j^k_j) then a(n) = numerator of Product (1 + 1/p_j).

Original entry on oeis.org

1, 3, 4, 3, 6, 2, 8, 3, 4, 9, 12, 2, 14, 12, 8, 3, 18, 2, 20, 9, 32, 18, 24, 2, 6, 21, 4, 12, 30, 12, 32, 3, 16, 27, 48, 2, 38, 30, 56, 9, 42, 16, 44, 18, 8, 36, 48, 2, 8, 9, 24, 21, 54, 2, 72, 12, 80, 45, 60, 12, 62, 48, 32, 3, 84, 24, 68, 27, 32, 72
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 28 2020

Keywords

Comments

Numerator of sum of reciprocals of squarefree divisors of n.
(6/Pi^2) * A332881(n)/a(n) is the asymptotic density of numbers that are coprime to their digital sum in base n+1 (see A094387 and A339076 for bases 2 and 10). - Amiram Eldar, Nov 24 2022

Examples

			1, 3/2, 4/3, 3/2, 6/5, 2, 8/7, 3/2, 4/3, 9/5, 12/11, 2, 14/13, 12/7, 8/5, 3/2, 18/17, ...

Links

Crossrefs

Cf. A001615, A008683, A017665, A028235, A028236, A048250, A076512, A306695, A308443, A308462, A332881 (denominators), A332882.
Cf. also A348734, A348735.
Cf. A059956, A065463, A082020, A094387, A339076.

Programs

  • Maple
    a:= n-> numer(mul(1+1/i[1], i=ifactors(n)[2])):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 28 2020
  • Mathematica
    Table[If[n == 1, 1, Times @@ (1 + 1/#[[1]] & /@ FactorInteger[n])], {n, 1, 70}] // Numerator
Table[Sum[MoebiusMu[d]^2/d, {d, Divisors[n]}], {n, 1, 70}] // Numerator
  • PARI
    A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A332880(n) = numerator(A001615(n)/n);

Formula

Numerators of coefficients in expansion of Sum_{k>=1} mu(k)^2*x^k/(k*(1 - x^k)).
a(n) = numerator of Sum_{d|n} mu(d)^2/d.
a(n) = numerator of psi(n)/n.
a(p) = p + 1, where p is prime.
a(n) = A001615(n) / A306695(n) = A001615(n) / gcd(n, A001615(n)). - Antti Karttunen, Nov 15 2021
From Amiram Eldar, Nov 24 2022: (Start)
Asymptotic means:
Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A332881(k) = 15/Pi^2 = 1.519817... (A082020).
Limit_{m->oo} (1/m) * Sum_{k=1..m} A332881(k)/a(k) = Product_{p prime} (1 - 1/(p*(p+1))) = 0.704442... (A065463). (End)

A348734 Numerator of Product((p+1)^e / ((p^e)+1)), when n = Product(p^e), with p primes, and e their exponents.

Original entry on oeis.org

1, 1, 1, 9, 1, 1, 1, 3, 8, 1, 1, 9, 1, 1, 1, 81, 1, 8, 1, 9, 1, 1, 1, 3, 18, 1, 16, 9, 1, 1, 1, 81, 1, 1, 1, 72, 1, 1, 1, 3, 1, 1, 1, 9, 8, 1, 1, 81, 32, 18, 1, 9, 1, 16, 1, 3, 1, 1, 1, 9, 1, 1, 8, 729, 1, 1, 1, 9, 1, 1, 1, 24, 1, 1, 18, 9, 1, 1, 1, 81, 128, 1, 1, 9, 1, 1, 1, 3, 1, 8, 1, 9, 1, 1, 1, 81, 1, 32, 8
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) = 360 != 1800 = 9*200 = a(4)*a(361). See A348740 for the list of such positions.

Links

Crossrefs

Cf. A003959, A034448, A348733, A348735 (denominators), A348740.

Programs

  • Mathematica
    f[p_, e_] := (p + 1)^e/(p^e + 1); a[1] = 1; a[n_] := Numerator[Times @@ f @@@ FactorInteger[n]]; 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])); }; \\ After code in A034448
A348734(n) = { my(u=A003959(n)); (u/gcd(u, A034448(n))); };
  • PARI
    A348734(n) = { my(f = factor(n)); numerator(prod(k=1, #f~, ((1+f[k, 1])^f[k, 2])/(1+(f[k, 1]^f[k, 2])))); };

Formula

a(n) = A003959(n) / gcd(A003959(n), A034448(n)).

A332882 If n = Product (p_j^k_j) then a(n) = numerator of Product (1 + 1/p_j^k_j).

Original entry on oeis.org

1, 3, 4, 5, 6, 2, 8, 9, 10, 9, 12, 5, 14, 12, 8, 17, 18, 5, 20, 3, 32, 18, 24, 3, 26, 21, 28, 10, 30, 12, 32, 33, 16, 27, 48, 25, 38, 30, 56, 27, 42, 16, 44, 15, 4, 36, 48, 17, 50, 39, 24, 35, 54, 14, 72, 9, 80, 45, 60, 2, 62, 48, 80, 65, 84, 24, 68, 45, 32, 72
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 28 2020

Keywords

Comments

Numerator of sum of reciprocals of unitary divisors of n.

Examples

			1, 3/2, 4/3, 5/4, 6/5, 2, 8/7, 9/8, 10/9, 9/5, 12/11, 5/3, 14/13, 12/7, 8/5, 17/16, ...

Links

Crossrefs

Cf. A017665, A028235, A028236, A034448, A077610, A306633, A319676, A323166, A332880, A332883 (denominators).
Cf. also A348734, A348735.

Programs

  • Maple
    a:= n-> numer(mul(1+i[1]^i[2], i=ifactors(n)[2])/n):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 28 2020
  • Mathematica
    Table[If[n == 1, 1, Times @@ (1 + 1/#[[1]]^#[[2]] & /@ FactorInteger[n])], {n, 1, 70}] // Numerator
Table[Sum[If[GCD[d, n/d] == 1,  1/d, 0], {d, Divisors[n]}], {n, 1, 70}] // Numerator
  • PARI
    a(n) = numerator(sumdiv(n, d, if (gcd(d, n/d)==1, 1/d))); \\ Michel Marcus, Feb 28 2020

Formula

a(n) = numerator of Sum_{d|n, gcd(d, n/d) = 1} 1/d.
a(n) = numerator of usigma(n)/n.
a(p) = p + 1, where p is prime.
a(n) = A034448(n) / A323166(n). - Antti Karttunen, Nov 13 2021
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A332883(k) = zeta(2)/zeta(3) = 1.368432... (A306633). - Amiram Eldar, Nov 21 2022

A348740 Positions k where A348733(k) is not multiplicative.

Original entry on oeis.org

1444, 3249, 3364, 4332, 4563, 6498, 7220, 7569, 9126, 10092, 10108, 12996, 13924, 15138, 15884, 16245, 16820, 17689, 18252, 18772, 21125, 21660, 22743, 22815, 23104, 23548, 24548, 24964, 25992, 27436, 30276, 30324, 31329, 31684, 31941, 32490, 33212, 35378, 35739, 36100, 36504, 37004, 37845, 38988, 41209, 41772
Offset: 1

Views

Author

Antti Karttunen, Nov 05 2021

Keywords

Comments

Numbers k with a factorization into coprime x and k/x with A348733(x) * A348733(k/x) <> A348733(k).

Crossrefs

Cf. A003959, A034448, A348733, A348734, A348735.
Cf. also A344702.

Programs

  • Mathematica
    f1[p_, e_] := (p + 1)^e; f2[p_, e_] := p^e + 1; a1[1] = 1; a1[n_] := GCD[Times @@ f1 @@@ (f = FactorInteger[n]), Times @@ f2 @@@ f]; f3[p_, e_] := a1[p^e]; a2[n_] := Times @@ f3 @@@ FactorInteger[n]; Position[Table[a2[n] - a1[n], {n, 1, 42000}], ?(# != 0 &)] // Flatten (* _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));
A348733mult(n) = { my(f = factor(n)); prod(k=1, #f~, A348733(f[k, 1]^f[k, 2])); };
isA348740(n) = (A348733(n)!=A348733mult(n));
Showing 1-5 of 5 results.

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