I have noticed that out of 9 Heegner numbers:
{1,2,3,7,11,19,43,67,163}
the group of four leftmost of them and the group of four rightmost of
them could be constructed together on the similar recurrence principle:

a[n] = 5*(5*(EuilerPhi[((a[n-1]+a[n-5])/5 + 11)/5] + Prime[n-5]) - 11)
- a[n-4]

Simplify[RecurrenceTable[{a[n] ==5*(5*(EuilerPhi[((a[n-1]+a[n-5])/5 +
11)/5] + Prime[n-5]) - 11) - a[n-4],a[1]==1,a[2]==2,a[3]==3,a[4]==7, a[5]==19},a[n],{n, 1, 8}]]

{1,2,3,7,19,43,67,163}

Of course above recurrence is not much productive since it doesn't
cover the fifth Heegner number and also requires five (out of total
eight covered Heegner's numbers) to be given in the recurrence's
initial conditions.

Unfortunately RSolve (see below) is not capable to find the explicit
direct solution for the above recurrence

$Assumptions = n \[Element] Integers

RSolve[{a[n]==-a[-4+n]+5(-11+5EulerPhi[1/5(11+1/5(a[-5+n]+a[-1+n]))]+Pri me[-5+n]),a[1]==1,a[2]==2,a[3]==3,a[4]==7,a[5]==19},a[n],n]

However, I, seems to be, was able to figure out "by hand" some sort of
explicit direct solution, which gives four leftmost (as a[1], a[2],
a[3] and a[4]) and the four rightmost (as a[6], a[7], a[8] and a[9])
Heegner numbers:

a[n] =
EulerPhi[Prime[Mod[4,n]!!]]+Floor[n/5]+((1+Floor[n/5])^2)!*((1+Sqrt[3])^ (n-(1+2*(Floor[n/5]))!)-(1-Sqrt[3])^(n-(1+2*(Floor[n/5]))!))/(2*Sqrt[3])


Simplify[Table[EulerPhi[Prime[Mod[4,n]!!]]+Floor[n/5]+((1+Floor[n/5])^2)! *((1+Sqrt[3])^(n-(1+2*(Floor[n/5]))!)-(1-Sqrt[3])^(n-(1+2*(Floor[n/5]))!) )/(2*Sqrt[3]), {n,1,9}]]

{1,2,3,7,31,19,43,67,163}

(Note that unfortunately a[5]=31 in above is "off" - the actual Heegner
number for index 5 should be not 31 but 11 ...)

and as follows:

a[n-1] = EulerPhi[Prime[Mod[4,(n-1)]!!]]+Floor[(n-1)/5]+((1+Floor[(n-1)/5])^2)!*( (1+Sqrt[3])^((n-1)-(1+2*(Floor[(n-1)/5]))!)-(1-Sqrt[3])^((n-1)-(1+2*(Flo or[(n-1)/5]))!))/(2*Sqrt[3])

a[n-4] = EulerPhi[Prime[Mod[4,(n-4)]!!]]+Floor[(n-4)/5]+((1+Floor[(n-4)/5])^2)!*( (1+Sqrt[3])^((n-4)-(1+2*(Floor[(n-4)/5]))!)-(1-Sqrt[3])^((n-4)-(1+2*(Flo or[(n-4)/5]))!))/(2*Sqrt[3])

a[n-5]= EulerPhi[Prime[Mod[4,(n-5)]!!]]+Floor[(n-5)/5]+((1+Floor[(n-5)/5])^2)!*( (1+Sqrt[3])^((n-5)-(1+2*(Floor[(n-5)/5]))!)-(1-Sqrt[3])^((n-5)-(1+2*(Flo or[(n-5)/5]))!))/(2*Sqrt[3])

Would it be possible to combine (utilize) above explicit and recurrent
formulas (please keep in mind that indexes for the last four Heegner
numbers are off by one if to compare explicit and recurrent formulas)
towards deriving (either explicit or recurrent) formula which would
cover all nine Heegner numbers?

Thanks,
Alexander R. Povolotsky

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