A180090
Sigma-powerful numbers: powerful numbers n such that sigma(n) is also powerful. An incomplete version of A337045.
Original entry on oeis.org
81, 343, 400, 1705636, 3648100, 13645088, 25992000, 26680500, 29184800, 80802000, 110215125, 178054848, 180093375, 213444000, 310144500, 408632609, 575664500, 1055340196, 1120504500, 1476326929, 1667329664, 2066544500
Offset: 1
sigma(3^4)=11^2, sigma(7^3)=2^4*5^2, sigma(2^4*5^2)=31^2, sigma(2^2*653^2)=7^2*13^2*19^2.
A337044
Numbers k such that both k and sigma(k)=A000203(k) are powerful, i.e., are terms of A001694.
Original entry on oeis.org
1, 81, 343, 400, 9261, 27783, 32400, 137200, 189728, 224939, 972000, 1705636, 2205472, 3087000, 3591200, 3648100, 3704400, 7968032, 11113200, 13645088, 15350724, 15367968, 18220059, 21161304, 24240600, 25992000, 26680500, 29184800, 32832900, 48586824, 51595489
Offset: 1
-
for(k=1, 60000000, if(ispowerful(k) && ispowerful(sigma(k)), print1(k, ", ")))
-
\\ See Corneth link \\ David A. Corneth, Aug 14 2020
A349109
Powerful numbers (A001694) whose sum of powerful divisors (including 1) is also powerful.
Original entry on oeis.org
1, 64, 243, 441, 1764, 9800, 15552, 28224, 41616, 60516, 82369, 88200, 189728, 226576, 329476, 336200, 648675, 741321, 968256, 1317904, 1428025, 1707552, 1943236, 2039184, 2056356, 2381400, 2446227, 2798929, 2965284, 2986568, 4372281, 5189400, 5271616, 6508832
Offset: 1
64 = 2^6 is a term since it is powerful and the sum of its powerful divisors, A183097(64) = 1 + 4 + 8 + 16 + 32 + 64 = 125 = 5^3 is also powerful.
-
powQ[n_] := n == 1 || AllTrue[FactorInteger[n][[;;,2]], # > 1 &]; f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - p; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := powQ[n] && powQ[s[n]]; Select[Range[7*10^6], q]
-
isok(n) = ispowerful(n) && ispowerful(sumdiv(n, d, d*ispowerful(d))); \\ Michel Marcus, Nov 08 2021
-
is(k) = {my(f = factor(k)); ispowerful(f) && ispowerful(prod(i = 1, #f~, (f[i,1]^(f[i,2]+1) - 1)/(f[i,1] - 1) - f[i,1]));} \\ Amiram Eldar, Sep 14 2024
Showing 1-3 of 3 results.
Comments