A307963
Larger of coreful amicable numbers pair: numbers (m, n) such that csigma(m) = csigma(n) = m + n, where csigma(n) is the sum of the coreful divisors of n (A057723).
Original entry on oeis.org
2218040, 4924700, 6654120, 15526280, 14774100, 28834520, 37706680, 42142760, 34472900, 46578840, 51014920, 64323160, 68759240, 64021100, 82067480, 86503560, 90939640, 95375720, 104247880, 83719900, 113120040, 117556120, 93569300, 126428280, 103418700, 130864360
Offset: 1
-
f[p_,e_] := (p^(e+1)-1)/(p-1)-1; csigma[1]=1; csigma[n_] := Times @@ (f @@@ FactorInteger[n]); s={}; Do[m = csigma[n] - n; If[m > n && csigma[m] - m == n, AppendTo[s, m]], {n, 1, 10^8}]; s
A308029
Numbers whose sum of coreful divisors is equal to the sum of non-coreful divisors.
Original entry on oeis.org
6, 1638, 55860, 168836850, 12854283750
Offset: 1
Divisors of 1638 are 1, 2, 3, 6, 7, 9, 13, 14, 18, 21, 26, 39, 42, 63, 78, 91, 117, 126, 182, 234, 273, 546, 819, 1638. The coreful ones are 546, 1638 and 1 + 2 + 3 + 6 + 7 + 9 + 13 + 14 + 18 + 21 + 26 + 39 + 42 + 63 + 78 + 91 + 117 + 126 + 182 + 234 + 273 + 819 = 546 + 1638 = 2184.
- G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer. 37 (1983), 277-307. (Annotated scanned copy)
-
with(numtheory): P:=proc(q) local a, k, n; for n from 1 to q do
a:=mul(k, k=factorset(n)); if sigma(n)=2*a*sigma(n/a)
then print(n); fi; od; end: P(10^7);
-
f[p_, e_] := (p^(e + 1) - 1)/(p - 1); fc[p_, e_] := f[p, e] - 1; csigmaQ[n_] := Times @@ (fc @@@ FactorInteger[n]) == Times @@ (f @@@ FactorInteger[n])/2; Select[Range[2, 10^5], csigmaQ] (* Amiram Eldar, May 11 2019 *)
-
rad(n) = factorback(factorint(n)[, 1]); \\ A007947
s(n) = my(rn=rad(n)); rn*sigma(n/rn); \\ A057723
isok(n) = 2*s(n) == sigma(n); \\ Michel Marcus, May 11 2019
A307986
Amicable pairs {x, y} such that y is the sum of the divisors of x that are not divided by every prime factor of x and vice versa.
Original entry on oeis.org
42, 54, 198, 204, 582, 594, 142310, 168730, 1077890, 1099390, 1156870, 1292570, 1511930, 1598470, 1669910, 2062570, 2236570, 2429030, 2728726, 3077354, 4246130, 4488910, 4532710, 5123090, 5385310, 5504110, 5812130, 6135962, 6993610, 7158710, 7288930, 8221598
Offset: 1
Divisors of x = 42 are 1, 2, 3, 6, 7, 14, 21, 42 and prime factors are 2, 3, 7. Among the divisors, 42 is the only one that is divisible by every prime factor, so we have 1 + 2 + 3 + 6 + 7 + 14 + 21 = 54 = y.
Divisors of y = 54 are 1, 2, 3, 6, 9, 18, 27, 54 and prime factors are 2, 3. Among the divisors, 6, 18, 54 are the only ones that are divisible by every prime factor, so we have 1 + 2 + 3 + 9 + 27 = 42 = x.
-
with(numtheory): P:=proc(q) local a,b,c,k,n; for n from 2 to q do
a:=mul(k,k=factorset(n)); b:=sigma(n)-a*sigma(n/a);
a:=mul(k,k=factorset(b)); c:=sigma(b)-a*sigma(b/a);
if c=n and b<>c then print(n); fi; od; end: P(10^8);
-
f[p_, e_] := (p^(e + 1) - 1)/(p - 1); fc[p_, e_] := f[p, e] - 1; ncs[n_] := Times @@ (f @@@ FactorInteger[n]) - Times @@ (fc @@@ FactorInteger[n]); seq = {}; Do[m = ncs[n]; If[m > 1 && m != n && n == ncs[m], AppendTo[seq, n]], {n, 2, 10^6}]; seq (* Amiram Eldar, May 11 2019 *)
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