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 1-6 of 6 results.

A347960 Numbers k for which A348036(k) > A007947(k).

Original entry on oeis.org

36, 72, 100, 108, 144, 180, 196, 200, 216, 225, 252, 288, 300, 324, 360, 392, 396, 400, 432, 450, 468, 484, 500, 504, 540, 576, 588, 600, 612, 648, 675, 676, 684, 700, 720, 756, 784, 792, 800, 828, 864, 900, 936, 968, 972, 980, 1000, 1008, 1044, 1080, 1089, 1100, 1116, 1125, 1152, 1156, 1176, 1188, 1200, 1224, 1260, 1296
Offset: 1

Views

Author

Antti Karttunen, Oct 19 2021

Keywords

Comments

Numbers k such that A348039(k) > 1.
Numbers k such that A348037(k) < A003557(k).
Numbers k such that A327564(k) > A348038(k).
Differs from A036785 and A338539 for the first time at n=20, where a(n) = 450, as A036785(20) = A338539(20) = 441 is not included in this sequence.

Crossrefs

Cf. A003557, A003959, A003968, A007947, A327564, A348036, A348037, A348038, A348039.

Programs

  • Mathematica
    f[p_, e_] := p*(p + 1)^(e - 1); q[n_] := GCD[n, Times @@ f @@@ (fct = FactorInteger[n])] > Times @@ First /@ fct; Select[Range[1300], q] (* Amiram Eldar, Oct 20 2021 *)
  • PARI
    A003968(n) = {my(f=factor(n)); for (i=1, #f~, p= f[i, 1]; f[i, 1] = p*(p+1)^(f[i, 2]-1); f[i, 2] = 1); factorback(f); }
A007947(n) = factorback(factorint(n)[, 1]);
A348036(n) = gcd(n, A003968(n));
isA347960(n) = (A348036(n)>A007947(n));

A348037 a(n) = n / gcd(n, A003968(n)), where A003968 is multiplicative with a(p^e) = p*(p+1)^(e-1).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 2, 1, 1, 1, 8, 1, 3, 1, 2, 1, 1, 1, 4, 5, 1, 9, 2, 1, 1, 1, 16, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 3, 1, 1, 8, 7, 5, 1, 2, 1, 9, 1, 4, 1, 1, 1, 2, 1, 1, 3, 32, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 5, 2, 1, 1, 1, 8, 27, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 16, 1, 7, 3, 5, 1, 1, 1, 4, 1
Offset: 1

Views

Author

Antti Karttunen, Oct 19 2021

Keywords

Links

Crossrefs

Differs from A003557 at the positions given by A347960.
Cf. A003959, A003968, A333634, A348038, A348039, A348499 (positions of 1's).

Programs

  • Mathematica
    f[p_, e_] := p*(p + 1)^(e - 1); a[n_] := n / GCD[n, Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Oct 20 2021 *)
  • PARI
    A003968(n) = { my(f=factor(n)); for (i=1, #f~, p= f[i, 1]; f[i, 1] = p*(p+1)^(f[i, 2]-1); f[i, 2] = 1); factorback(f); }
A348037(n) = (n/gcd(n, A003968(n)));

Formula

a(n) = n / A348036(n) = n / gcd(n, A003968(n)).
a(n) = A003557(n) / A348039(n).

A348038 a(n) = A003968(n) / gcd(n, A003968(n)), where A003968 is multiplicative with a(p^e) = p*(p+1)^(e-1).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 9, 4, 1, 1, 3, 1, 1, 1, 27, 1, 4, 1, 3, 1, 1, 1, 9, 6, 1, 16, 3, 1, 1, 1, 81, 1, 1, 1, 2, 1, 1, 1, 9, 1, 1, 1, 3, 4, 1, 1, 27, 8, 6, 1, 3, 1, 16, 1, 9, 1, 1, 1, 3, 1, 1, 4, 243, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 6, 3, 1, 1, 1, 27, 64, 1, 1, 3, 1, 1, 1, 9, 1, 4, 1, 3, 1, 1, 1, 81, 1, 8, 4, 9, 1
Offset: 1

Views

Author

Antti Karttunen, Oct 20 2021

Keywords

Links

Crossrefs

Differs from A327564 at the positions given by A347960.
Cf. A003968, A005117 (positions of 1's), A327564, A348036, A348037, A348039.

Programs

  • Mathematica
    f[p_, e_] := p*(p + 1)^(e - 1); a[n_] := (m = Times @@ f @@@ FactorInteger[n]) / GCD[n, m]; Array[a, 100] (* Amiram Eldar, Oct 20 2021 *)
  • PARI
    A003968(n) = {my(f=factor(n)); for (i=1, #f~, p= f[i, 1]; f[i, 1] = p*(p+1)^(f[i, 2]-1); f[i, 2] = 1); factorback(f); }
A348038(n) = { my(u=A003968(n)); (u/gcd(n, u)); };

Formula

a(n) = A003968(n) / A348036(n) = A003968(n) / gcd(n, A003968(n)).
a(n) = A327564(n) / A348039(n).

A348039 a(n) = gcd(A003557(n), A327564(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 6
Offset: 1

Views

Author

Antti Karttunen, Oct 19 2021

Keywords

Comments

There is interesting regularity in the scatter plot.

Links

Crossrefs

Cf. A003557, A003959, A003968, A007947, A327564, A348036, A348037, A348038.
Cf. A347960 (positions of terms > 1).

Programs

  • Mathematica
    {1}~Join~Array[GCD @@ Map[Times @@ # &, Transpose@ Map[{#1^(#2 - 1), (#1 + 1)^(#2 - 1)} & @@ # &, FactorInteger[#]]] &, 105, 2] (* Michael De Vlieger, Oct 20 2021 *)
  • PARI
    A003968(n) = {my(f=factor(n)); for (i=1, #f~, p= f[i, 1]; f[i, 1] = p*(p+1)^(f[i, 2]-1); f[i, 2] = 1); factorback(f); }
A007947(n) = factorback(factorint(n)[, 1]);
A348036(n) = gcd(n, A003968(n));
A348039(n) = (A348036(n)/A007947(n));
  • PARI
    A003557(n) = (n/factorback(factorint(n)[, 1]));
A327564(n) = { my(f=factor(n)); for(k=1, #f~, f[k, 1]++; f[k, 2]--); factorback(f); }; \\ From A327564
A348039(n) = gcd(A003557(n), A327564(n));

Formula

a(n) = gcd(A003557(n), A327564(n)).
a(n) = A348036(n) / A007947(n).
a(n) = A003557(n) / A348037(n).
a(n) = A327564(n) / A348038(n).

A348499 Numbers k such that A003968(k) is a multiple of k, where A003968 is multiplicative with a(p^e) = p*(p+1)^(e-1).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 72, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113, 114, 115, 118, 119
Offset: 1

Views

Author

Antti Karttunen, Oct 29 2021

Keywords

Comments

This is a subsequence of A333634. See comments in A347892.

Links

Crossrefs

Union of A005117 and A347892 (terms that are not squarefree).
Subsequence of A333634.
Positions of ones in A348037.
Cf. A003968, A348030, A348036, A347960.

Programs

  • Mathematica
    f[p_, e_] := p*(p + 1)^(e - 1); s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[120], Divisible[s[#], #] &] (* Amiram Eldar, Oct 29 2021 *)
  • PARI
    A003968(n) = {my(f=factor(n)); for (i=1, #f~, p= f[i, 1]; f[i, 1] = p*(p+1)^(f[i, 2]-1); f[i, 2] = 1); factorback(f); }
isA348499(n) = !(A003968(n)%n);

A348030 a(n) = A003968(n) - n, where A003968 is multiplicative with a(p^e) = p*(p+1)^(e-1).

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 10, 3, 0, 0, 6, 0, 0, 0, 38, 0, 6, 0, 10, 0, 0, 0, 30, 5, 0, 21, 14, 0, 0, 0, 130, 0, 0, 0, 36, 0, 0, 0, 50, 0, 0, 0, 22, 15, 0, 0, 114, 7, 10, 0, 26, 0, 42, 0, 70, 0, 0, 0, 30, 0, 0, 21, 422, 0, 0, 0, 34, 0, 0, 0, 144, 0, 0, 15, 38, 0, 0, 0, 190, 111, 0, 0, 42, 0, 0, 0, 110, 0, 30, 0, 46
Offset: 1

Views

Author

Antti Karttunen, Oct 19 2021

Keywords

Comments

Möbius transform of A348029(n), which is A003959(n) - sigma(n).

Links

Crossrefs

Cf. A003959, A003968, A005117 (positions of zeros), A005596, A008683, A104141, A348029, A348036.

Programs

  • Mathematica
    f[p_, e_] := p*(p + 1)^(e - 1); a[n_] := Times @@ f @@@ FactorInteger[n] - n; Array[a, 100] (* Amiram Eldar, Oct 20 2021 *)
  • PARI
    A003968(n) = {my(f=factor(n)); for (i=1, #f~, p= f[i, 1]; f[i, 1] = p*(p+1)^(f[i, 2]-1); f[i, 2] = 1); factorback(f); }
A348030(n) = (A003968(n)-n);

Formula

a(n) = A003968(n) - n.
a(n) = Sum_{d|n} A008683(n/d) * A348029(d).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + 1/(p^3 - p^2 - p)) - 1 = A104141/A005596 - 1 = 0.625665... . - Amiram Eldar, May 29 2025
Showing 1-6 of 6 results.

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