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

1,2

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

Antti Karttunen, Table of n, a(n) for n = 1..782 (terms < 2^31)

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Prog

(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))); };

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.

Sequence in context: A226476 A216793 A294900 * A064510 A335736 A228383

Adjacent sequences: A344697 A344698 A344699 * A344701 A344702 A344703

Keyword

nonn

Author

Antti Karttunen, May 28 2021

Status

approved