A332687 a(n) = Sum_{k=1..n} ceiling(n/prime(k)).
1, 2, 4, 6, 8, 10, 13, 15, 17, 19, 22, 24, 27, 29, 32, 35, 37, 39, 42, 44, 47, 50, 53, 55, 58, 60, 63, 65, 68, 70, 74, 76, 78, 81, 84, 87, 90, 92, 95, 98, 101, 103, 107, 109, 112, 115, 118, 120, 123, 125, 128, 131, 134, 136, 139, 142, 145, 148, 151, 153
Offset: 1
Keywords
Programs
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Mathematica
Table[Sum[Ceiling[n/Prime[k]], {k, 1, n}], {n, 1, 60}] Table[n + Sum[PrimeNu[k], {k, 1, n - 1}], {n, 1, 60}] nmax = 60; CoefficientList[Series[x/(1 - x)^2 + (x/(1 - x)) Sum[x^Prime[k]/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest With[{nmax = 100}, Range[nmax] + Join[{0}, Accumulate[Table[PrimeNu[k], {k, 1, nmax - 1}]]]] (* Amiram Eldar, Sep 21 2024 *) -
PARI
a(n) = sum(k=1, n, ceil(n/prime(k))); \\ Michel Marcus, Feb 21 2020
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PARI
lista(nmax) = my(s = 1); for(n = 2, nmax, print1(s, ", "); s += omega(n-1) + 1); \\ Amiram Eldar, Sep 21 2024
Formula
G.f.: x/(1 - x)^2 + (x/(1 - x)) * Sum_{k>=1} x^prime(k) / (1 - x^prime(k)).
a(n) = n + Sum_{k=1..n-1} omega(k), where omega = A001221.
a(n) = n - omega(n) + Sum_{k=1..n} pi(floor(n/k)), where pi = A000720.
a(n) = n + A013939(n-1) for n >= 2. - Amiram Eldar, Sep 21 2024