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approved

a(n) is the number of numbers k <= prime(n)^2 and such that A075860(k) = prime(n).

For n=3, prime(3)=5. The only integers k <= 5^2 and such that A075860(k)=5 are 5,6,12,18,24 and 25. Therefore, a(3)=6.

proposed

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proposed

a(n) is the number of numbers k <= prime(n)^2 and such that A075860(k) = prime(n).

2, 2, 6, 8, 2, 10, 3, 14, 6, 8, 22, 7, 8, 21, 9, 14, 12, 45, 14, 17, 45, 17, 21, 20, 18, 17, 64, 21, 54, 28, 25, 22, 22, 72, 37, 82, 26, 28, 31, 43, 36, 93, 44, 95, 38, 95, 41, 38, 33, 106, 36, 49, 111, 65, 53, 53, 49, 113, 55, 68, 138, 80, 49, 50, 152, 61, 55, 43, 73, 120, 55, 80, 63, 60, 65, 89, 81, 76, 84, 82

(PARI) fp(n, pn) = if (n == pn, n, fp(vecsum(factor(n)[, 1]), n));

f(n) = if (n==1, 0, fp(n, 0)); \\ A075860

a(n) = sum(k=1, prime(n)^2, f(k) == prime(n)); \\ Michel Marcus, Jul 04 2024

Cf. A001248, A075860.

proposed

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proposed

For all n>=1, a(n)n>=2.

allocated for Rafik Khalfia(n) is the number of numbers k <= prime(n)^2 and such that A075860(k)= prime(n).

2, 2, 6, 8, 2, 10, 3, 14, 6, 8, 22, 7, 8, 21, 9, 14, 12, 45, 14, 17, 45, 17, 21, 20, 18, 17, 64, 21, 54, 28, 25, 22, 22, 72, 37, 82, 26, 28, 31, 43, 36, 93, 44, 95, 38, 95, 41, 38, 33, 106, 36, 49, 111, 65, 53, 53, 49, 113, 55, 68, 138, 80, 49, 50, 152, 61, 55, 43, 73, 120, 55, 80, 63, 60, 65, 89, 81, 76, 84, 82

1,1

For all n>=1, a(n)n>=2.

For n=3, prime(3)=5. The only integers k <= 5^2 and such that A075860(k)=5 are 5,6,12,18,24 and 25. Therefore, a(3)=6.

f := proc (n)

option remember;

if isprime(n) then

return n

else

return procname(convert(numtheory:-factorset(n), `+`))

end if

end proc:

g := proc (n)

local count, k;

count := 0;

for k from ithprime(n) to ithprime(n)^2 do

if f(k) = ithprime(n) then

count := count + 1

end if

end do;

return count

end proc:

map(g, [$1 .. 80]);

allocated

nonn

Rafik Khalfi, Jul 04 2024

approved

editing

allocated for Rafik Khalfi

allocated

approved