A340834 - OEIS

%I #64 Nov 17 2024 18:55:50

%S 12,1222,1437286,3441373,1032893366969

%N Fixed points of A341885.

%C Numbers n such that A341885(n) = n.

%C Includes 2*p*q if p and q are primes such that p^2-4*p*q+q^2+p+q+6 = 0.  This includes 12 for p=2, q=3, 1222 for p=13,q=47, 1437286 for p=439, q=1637, and 76498942675946443126 for p=3201392659, q=11947760057.

%C Another term: 6538810199342921107066977217325653068509 = 13 * 4401624135264074597*114272683103433355069. - _Chai Wah Wu_, Feb 25 2021

%F A341885(a(n)) = a(n).

%e a(2) = 1222 is a term because 1222 = 2*13*47 and A341885(1222) = 2*3/2 + 13*14/2 + 47*48/2 = 1222.

%p f:= proc(n) local F,t;

%p   F:= ifactors(n)[2];

%p   add(t[1]*(t[1]+1)/2*t[2],t=F)

%p end proc:

%p select(t -> f(t)=t, [$1..4000000]);

%t Block[{a = {}}, Monitor[Do[If[# == i, AppendTo[a, i]] &@ Total[PolygonalNumber@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[i]]], {i, 2, 4*10^6}], i]; a] (* _Michael De Vlieger_, Feb 22 2021 *)

%o (Python)

%o from sympy import factorint

%o A340834_list = [n for n in range(2,10**4) if n == sum(k*m*(m+1)//2 for m,k in factorint(n).items())] # _Chai Wah Wu_, Feb 25 2021

%Y Cf. A341885.

%K nonn,more

%O 1,1

%A _J. M. Bergot_ and _Robert Israel_, Feb 22 2021

%E a(5) from _Martin Ehrenstein_, Mar 07 2021