A344704 -id:A344704 - cp's OEIS frontend

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.

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

Antti Karttunen, May 28 2021

nonn

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.

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.

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))); };

A344705 a(n) = n + A001615(n) - sigma(n), where A001615 is the Dedekind psi-function, and sigma(n) gives the sum of divisors of n; difference between psi and the sum of proper divisors.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 5, 8, 10, 11, 8, 13, 14, 15, 9, 17, 15, 19, 14, 21, 22, 23, 12, 24, 26, 23, 20, 29, 30, 31, 17, 33, 34, 35, 17, 37, 38, 39, 22, 41, 42, 43, 32, 39, 46, 47, 20, 48, 47, 51, 38, 53, 42, 55, 32, 57, 58, 59, 36, 61, 62, 55, 33, 65, 66, 67, 50, 69, 70, 71, 21, 73, 74, 71, 56, 77, 78, 79, 38, 68, 82

Offset: 1

Antti Karttunen, May 28 2021

First negative term occurs as a(1440) = -18.

Antti Karttunen, Table of n, a(n) for n = 1..16560
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000203, A001065, A001615, A033879, A244963, A291209 (positions of zeros), A344700, A344704, A344753.

Cf. A013661, A082020.

f[p_, e_] := (p^(e+1) - 1)/(p-1); a[1] = 1; a[n_] := Module[{fct = FactorInteger[n]}, n * (Times @@ (1 + 1/fct[[;; , 1]]) + 1) - Times @@ f @@@ fct]; Array[a, 100] (* Amiram Eldar, Dec 08 2023 *)

A001615(n) = (n * sumdivmult(n, d, issquarefree(d)/d));
A344705(n) = ((n + A001615(n)) - sigma(n));

a(n) = A001615(n) - A001065(n) = n - A244963(n) = n + A001615(n) - sigma(n).
a(n) = A033879(n) + A306927(n).
a(n) = n + A344753(n) - 2*A001065(n).
Sum_{k=1..n} a(k) = c * n^2 / 2 + O(n*log(n)), where c = 15/Pi^2 + 1 - Pi^2/6 = 0.874883... . - Amiram Eldar, Dec 08 2023

cp's OEIS Frontend

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.

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