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.

A344700 Numbers k for which A306927(k) [= A001615(k)-k] is a multiple of A344705(k) [= A001615(k)-A001065(k)], and their quotient is nonnegative.

Original entry on oeis.org

1, 6, 24, 28, 168, 496, 864, 1080, 1520, 1836, 2016, 2088, 2112, 2520, 2912, 2976, 3000, 3024, 3240, 3800, 8128, 9000, 11088, 11232, 11448, 14160, 14688, 16920, 17028, 18360, 19872, 20520, 20880, 25280, 25488, 27552, 29376, 30800, 31200, 31320, 31968, 35400, 39240, 44064, 48768, 49896, 50760, 51480, 51660, 52200, 55680
Offset: 1

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Author

Antti Karttunen, May 28 2021

Keywords

Comments

Numbers k for which A344704(k) = A344705(k), i.e., numbers k such that gcd(A001615(k)-k, A001615(k)-A001065(k)) = A001615(k) - A001065(k).
Note that A306927(k) is always nonnegative, but A344705(k) = A033879(k) + A306927(k) gets also negative values. Number k is perfect only when A033879(k) = A344705(k) - A306927(k) = 0, that is, when A344705(k) = A306927(k), which necessitates that A306927(k) should be a multiple of A344705(k), and their quotient should be nonnegative (actually = +1).
In the range 1 .. 2^31 there are 782 such numbers, of which only the initial 1 is odd.

Links

Crossrefs

Cf. A000203, A001065, A001615, A033879, A244963, A306927, A344704, A344705, A344752 (gives the quotient A306927(k)/A344705(k) computed for these terms), A344753.
Cf. A000396 (subsequence).
Cf. also A344754, A344755.

Programs

  • 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
isA344700(n) = { my(t=A001615(n), s=sigma(n), u = (n+t)-s); (gcd(t-n,u)==u); };
\\ Alternatively as:
isA344700(n) = { my(t=A001615(n), s=sigma(n), u = (n+t)-s); ((u>0)&&(0==((t-n)%u))); };

A344752 Quotient A306927(k) / A344705(k) computed for such k >= 1 that the quotient is a nonnegative natural number. (The k are given by A344700.)

Original entry on oeis.org

0, 1, 2, 1, 3, 1, 12, 21, 2, 3, 36, 4, 4, 61, 2, 3, 5, 243, 36, 2, 1, 70, 345, 90, 5, 5, 38, 11, 3, 131, 25, 87, 172, 2, 9, 5, 228, 5, 20, 43, 18, 5, 10, 304, 3, 1035, 31, 1301, 7, 172, 8, 554, 60, 15, 295, 59, 14, 150, 110, 7, 2439, 258, 371, 5, 549, 8, 13, 15, 63, 1134, 24, 900, 23, 50, 7, 4, 27, 1292, 254, 6681, 5, 18
Offset: 1

Views

Author

Antti Karttunen, May 28 2021

Keywords

Crossrefs

Cf. A000203, A001615, A306927, A344700, A344705.

Programs

  • 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
isA344700_and_give_quotient(n) = { my(t=A001615(n), s=sigma(n), u = (n+t)-s); if(((u>0)&&(0==((t-n)%u))), ((t-n)/u), 0); };
for(n=1,2^17,x=isA344700_and_give_quotient(n); if(x>0||(1==n), print1(x,", ")));

A344753 a(n) = sigma(n) + psi(n) - 2n = Sum_{d|n, d

Original entry on oeis.org

0, 2, 2, 5, 2, 12, 2, 11, 7, 16, 2, 28, 2, 20, 18, 23, 2, 39, 2, 38, 22, 28, 2, 60, 11, 32, 22, 48, 2, 84, 2, 47, 30, 40, 26, 91, 2, 44, 34, 82, 2, 108, 2, 68, 60, 52, 2, 124, 15, 83, 42, 78, 2, 120, 34, 104, 46, 64, 2, 192, 2, 68, 74, 95, 38, 156, 2, 98, 54, 148, 2, 195, 2, 80, 94, 108, 38, 180, 2, 170, 67, 88, 2
Offset: 1

Views

Author

Antti Karttunen, May 28 2021

Keywords

Comments

Sigma is the sum of divisors (A000203), and psi is Dedekind psi-function (A001615). Coincides with the latter only on perfect numbers (A000396).

Links

Crossrefs

Cf. A001065, A001615, A008683, A008966, A033879, A173557, A306927, A344705, A342001, A344705, A344754, A344755, A344997, A344998.
Cf. A000396 [where a(n) = A001615(n)], A005100 [a(n) < A001615(n)], A005101 [a(n) > A001615(n)].
Cf. also A051709, A345001.
Cf. A013661, A082020.

Programs

  • Mathematica
    a[n_] := Sum[d + If[SquareFreeQ[n/d], d, 0], {d, Most[Divisors[n]]}];
Array[a, 100] (* Jean-François Alcover, Jun 12 2021 *)
  • PARI
    A344753(n) = sumdiv(n,d,(d

Formula

a(n) = Sum_{d|n, dA008966(n/d) * d).
a(n) = A001065(n) + A306927(n).
a(n) = A001615(n) - A033879(n).
a(n) = A344705(n) + 2*A001065(n) - n.
For squarefree n, a(n) = 2*A001065(n).
a(n) = A344997(n) / A173557(n) = A344998(n) / A342001(n). - Antti Karttunen, Jun 06 2021
Sum_{k=1..n} a(k) = c * n^2 / 2 + O(n*log(n)), where c = Pi^2/6 + 15/Pi^2 - 2 = 1.164751... . - Amiram Eldar, Dec 08 2023

Extensions

New primary definition added by Antti Karttunen, Jun 06 2021

A344704 a(n) = gcd(A001615(n)-n, sigma(n)-(A001615(n)+n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 4, 1, 2, 3, 1, 1, 3, 1, 2, 1, 2, 1, 12, 1, 2, 1, 20, 1, 6, 1, 1, 3, 2, 1, 1, 1, 2, 1, 2, 1, 6, 1, 4, 3, 2, 1, 4, 1, 1, 3, 2, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 11, 1, 1, 6, 1, 10, 3, 2, 1, 3, 1, 2, 1, 4, 1, 6, 1, 2, 1, 2, 1, 4, 1, 2, 3, 4, 1, 18, 7, 4, 1, 2, 5, 12, 1, 5, 3, 1, 1, 6, 1, 2, 3
Offset: 1

Views

Author

Antti Karttunen, May 28 2021

Keywords

Links

Crossrefs

Cf. A000203, A001615, A244963, A306927, A344700, A344705.

Programs

  • 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
A344704(n) = gcd(A001615(n)-n, sigma(n)-(A001615(n)+n));

Formula

a(n) = gcd(A306927(n), n-A244963(n)) = gcd(A001615(n)-n, sigma(n)-(A001615(n)+n)).
Showing 1-4 of 4 results.

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