A376222 Numbers k such that Sum_{i=1..q-1} d(i)^i is prime, where d(1)

4, 21, 27, 39, 57, 77, 183, 189, 203, 205, 219, 237, 253, 371, 387, 391, 417, 489, 565, 611, 655, 667, 669, 675, 687, 749, 767, 799, 831, 849, 897, 921, 955, 1007, 1047, 1135, 1189, 1207, 1349, 1371, 1379, 1407, 1421, 1461, 1469, 1497, 1513, 1569, 1633, 1643, 1659

Offset: 1

Author

Michel Lagneau, Sep 16 2024

Keywords

• nonn

Comments

The corresponding primes are in A376223.

Examples

			39 is a term because the 3 first divisors of 39 are {1,3,13} and 1^1 + 3^2 + 13^3 = 2207 is prime.
189 is a term since the 7 first divisors of 189 are {1, 3, 7, 9, 21, 27, 63} and 1^1+3^2+7^3+9^4+21^5+27^6+63^7 = 3939372150671 is prime.
		

Crossrefs

Cf. A180852, A376223.

Programs

Maple

with(numtheory):nn:=1700:
for n from 1 to nn do:
d:=divisors(n):n0:=nops(d):s:=sum(‘d[k]^k’, ‘k’=1..n0-1):
   if isprime(s)
    then
     printf(`%d,`,n):
    else
   fi:
od:

Mathematica

Select[Range[1700],PrimeQ[Sum[Part[Divisors[#],i]^i,{i,DivisorSigma[0,#]-1}]] &] (* Stefano Spezia, Sep 16 2024 *)

PARI

isok(k) = my(d=divisors(k)); isprime(sum(j=1, #d-1, d[j]^j)); \\ Michel Marcus, Sep 16 2024