A387154
The least number k that is not n-free whose sum of n-free divisors is larger than 2*k.
Original entry on oeis.org
401120980260, 360360, 55440, 110880, 100800, 120960, 241920, 483840, 967680, 1935360, 3870720, 7741440, 15482880, 30965760, 61931520, 123863040, 247726080, 495452160, 990904320, 1981808640, 3963617280, 7927234560, 15854469120, 31708938240, 63417876480, 126835752960
Offset: 2
For n = 2, the numbers k such that A048250(k) > 2*k include all the squarefree abundant numbers (A087248). The least nonsquarefree number (A013929) k such that A048250(k) > 2*k is 401120980260 = 2^2*3*5*7*11*13*17*19*23*29*31.
For n = 3, the numbers k such that A073185(k) > 2*k include all the cubefree abundant numbers (A357695). The least noncubefree number (A046099) k such that A073185(k) > 2*k is A357700(1) = 360360 = 2^3*3^2*5*7*11*13.
Cf.
A004709,
A005117,
A013929,
A025487,
A046099,
A046100,
A048250,
A087248,
A073185,
A357695,
A357700,
A385006,
A387155.
-
a[n_] := If[n < 7, {401120980260, 360360, 55440, 110880, 100800}[[n-1]], 945 * 2^n]; Array[a, 26, 2]
-
a(n) = if(n < 7, [401120980260, 360360, 55440, 110880, 100800][n-1], 945 * 2^n);
A387155
The number of n-free abundant numbers below the least number k that is not n-free whose sum of n-free divisors is larger than 2*k.
Original entry on oeis.org
22148167706, 52012, 10828, 24601, 23660, 29114, 58967, 118828, 238600, 478099, 957324, 1916191, 3834167, 7669094, 15335488, 30667762, 61337894, 122679755, 245357929, 490718137, 981456651, 1962956352, 3925957422, 7851819466, 15703524589, 31406984903, 62813576969
Offset: 2
a(2) = 22148167706 because there are 22148167706 squarefree numbers k such that A048250(k) > 2*k (i.e., terms of A087248) that are less than the least nonsquarefree number k that has this property, A387154(2) = 401120980260.
a(3) = 52012 because there are 52012 cubefree numbers k such that A073185(k) > 2*k (i.e., terms of A357695) that are less than the least noncubefree number k that has this property, A387154(3) = 360360.
-
freeQ[n_, k_] := AllTrue[FactorInteger[n][[;; , 2]], # < k &];
sigma[n_, k_] := Times @@ ((First[#]^(Min[Last[#], k - 1] + 1) - 1)/(First[#] - 1) & /@ FactorInteger[n]);
a[n_] := Module[{m = 2, c = 0}, While[True, If[sigma[m, n] > 2*m, c++; If[!freeQ[m, n], Break[]]]; m++]; c-1];
-
isfree(n, k) = if(n == 1, 1, my(e = factor(n)[,2]); for(i=1, #e, if(e[i] >= k, return(0))); 1);
sigmafree(n, k) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^(min(f[i,2],k-1)+1)-1)/(f[i,1]-1));}
a(n) = {my(m = 2, c = 0); while(1, if(sigmafree(m, n) > 2*m, c++; if(!isfree(m, n), break)); m++); c-1;}
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