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A278913 a(n) is the smallest number k with prime sum of divisors such that tau(k) = n-th prime.

%I A278913 #29 Jul 20 2025 15:03:51
%S A278913 2,4,16,64,9765625,4096,65536,262144,
%T A278913 1471383076677527699142172838322885948765175969,
%U A278913 10264895304762966931257013446474591264089923314972889033759201,1073741824,18701397461209715023927088008788055619800417991632621566284510161
%N A278913 a(n) is the smallest number k with prime sum of divisors such that tau(k) = n-th prime.
%C A278913 tau(n) = A000005(n) = the number of divisors of n.
%C A278913 a(11) = 1073741824; a(n) > A023194(10000) = 5896704025969 for n = 9, 10 and n >= 12.
%H A278913 Davin Park, <a href="/A278913/b278913.txt">Table of n, a(n) for n = 1..50</a>
%F A278913 a(n) = A123487(n)^(prime(n)-1). - _Davin Park_, Dec 10 2016
%e A278913 a(3) = 16 because 16 is the smallest number with prime values of sum of divisors (sigma(16) = 31) such that tau(16) = 5 = 3rd prime.
%t A278913 A278913[n_] := NestWhile[NextPrime, 2, ! PrimeQ[Cyclotomic[Prime[n], #]] &]^(Prime[n] - 1) (* _Davin Park_, Dec 28 2016 *)
%o A278913 (Magma) A278913:=func<n|exists(r){k:k in[1..10000000] | IsPrime(SumOfDivisors(k)) and NumberOfDivisors(k) eq NthPrime(n)} select r else 0>; [A278913(n): n in[1..8]];
%o A278913 (PARI) a(n) = {my(k=1); while(! (isprime(sigma(k)) && isprime(p=numdiv(k)) && (primepi(p) == n)), k++); k;} \\ _Michel Marcus_, Dec 03 2016
%Y A278913 Cf. A000005, A000203, A023194, A278914.
%K A278913 nonn
%O A278913 1,1
%A A278913 _Jaroslav Krizek_, Nov 30 2016
%E A278913 More terms from _Davin Park_, Dec 08 2016