A264155
a(n) is the smallest integer m such that n is the least exponent k satisfying sigma(m)^k divides m.
Original entry on oeis.org
1, 24, 40, 384, 486, 6144, 640, 18688, 39366, 91136, 10240, 23482368, 958464, 52612659, 163840, 375717888, 9568256, 1502871552, 2621440, 353370112, 186646528
Offset: 1
-
fk(s, m) = {my(j = 1); while(denominator(s^j/m) != 1, j++); j;}
rad(n) = factorback(factorint(n)[, 1]);
a(n) = {my(k = 1, ok = 0, sk); while (!ok, sk = sigma(k); if ((denominator(sk/rad(k)) == 1) && (fk(sk, k) == n), ok = 1, k++; ); ); k; } \\ corrected by Michel Marcus, Feb 14 2019
A274205
Numbers such that the sum of divisors is twice the sum of the exponential divisors.
Original entry on oeis.org
6, 24, 54, 216, 1638, 6552, 14256, 55860, 80262, 276822, 321048, 502740, 1107288, 1396500, 1724976, 12568500, 13564278, 20165460, 54257112, 168836850, 181489140, 504136500, 675347400, 4537228500, 28533427650, 60950102850, 114133710600, 162252212850, 243800411400, 649008851400, 734916514878
Offset: 1
Divisors of 6 are 1, 2, 3 and 6 which sum to 12. The only exponential divisor is 6. Finally 12 / 6 = 2.
Divisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24 which sum to 60. Exponential divisors are 6, 24 and their sum is 30. Finally 60 / 30 = 2.
-
with(numtheory): P:=proc(q) local a,b,c,d,j,k,n,ok;
for n from 2 to q do a:=ifactors(n)[2]; b:=sort([op(divisors(n))]); c:=0;
for k from 2 to nops(b) do d:=ifactors(b[k])[2]; if nops(d)=nops(a) then
ok:=1; for j from 1 to nops(d) do if not type(a[j][2]/d[j][2],integer) then ok:=0; break; fi; od;
if ok=1 then c:=c+b[k]; fi; fi; od; if sigma(n)=2*c then print(n); fi; od; end: P(10^9);
-
Select[Range[10^6], 2 Times @@ Map[Sum[First[#]^d, {d, Divisors@ Last@ #}] &, FactorInteger@ #] == DivisorSigma[1, #] &] (* Michael De Vlieger, Jun 16 2016 *)
Comments