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A141198 a(n) is the number of divisors of n that are each one more than a power of a prime.

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%I A141198 #19 Jun 22 2025 11:39:05
%S A141198 0,1,1,2,1,3,0,3,2,3,0,5,0,2,2,3,1,5,0,5,1,1,0,7,1,2,2,4,0,6,0,4,2,2,
%T A141198 1,7,0,2,1,6,0,5,0,3,3,1,0,8,0,4,2,3,0,6,1,5,1,1,0,10,0,2,2,4,2,4,0,4,
%U A141198 1,4,0,10,0,2,2,3,0,4,0,7,2,2,0,9,2,1,1,4,0,9,0,2,1,1,1,9,0,3,3,6,0,5,0
%N A141198 a(n) is the number of divisors of n that are each one more than a power of a prime.
%C A141198 1 is considered here to be a power of a prime. 0 is not considered here to be a power of a prime.
%H A141198 Amiram Eldar, <a href="/A141198/b141198.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1049 from Diana Mecum)
%e A141198 The divisors of 9 are 1, 3 and 9. 1 is one more than 0, not a power of a prime. 3 is one more than 2, a power of a prime. And 9 is one more than 8, a power of a prime. There are therefore 2 such divisors that are each one more than a power of a prime. So a(9) = 2.
%t A141198 a[n_] := DivisorSum[n, 1 &, # == 2 || PrimePowerQ[#-1] &]; Array[a, 100] (* _Amiram Eldar_, Jun 22 2025 *)
%o A141198 (PARI) a(n) = sumdiv(n, d, d == 2 || isprimepower(d - 1)); \\ _Amiram Eldar_, Jun 22 2025
%Y A141198 Cf. A000961, A141197.
%K A141198 nonn
%O A141198 1,4
%A A141198 _Leroy Quet_, Jun 12 2008
%E A141198 Corrected and extended by _Diana L. Mecum_, Jul 05 2007