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A370902
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Partial sums of the powerful part function (A057521).
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2
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1, 2, 3, 7, 8, 9, 10, 18, 27, 28, 29, 33, 34, 35, 36, 52, 53, 62, 63, 67, 68, 69, 70, 78, 103, 104, 131, 135, 136, 137, 138, 170, 171, 172, 173, 209, 210, 211, 212, 220, 221, 222, 223, 227, 236, 237, 238, 254, 303, 328, 329, 333, 334, 361, 362, 370, 371, 372, 373
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OFFSET
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1,2
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LINKS
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FORMULA
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a(n) = c_1 * n^(3/2) / 3 + c_2 * n^(4/3) / 4 + O(n^(6/5)), where c_1 = A328013 and c_2 are positive constants (Tóth, 2017).
c_2 = zeta(2/3) * Product_{p prime} (1 + 1/p^(4/3) - 2/p^2 - 1/p^(7/3) + 1/p^3) = -2.59305556147555965163... (László Tóth, personal communication). - Amiram Eldar, Mar 07 2024
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MATHEMATICA
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f[p_, e_] := If[e == 1, 1, p^e]; pfp[n_] := Times @@ f @@@ FactorInteger[n]; pfp[1] = 1; Accumulate[Array[pfp[#] &, 100]]
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PROG
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(PARI) pfp(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1, 1, f[i, 1]^f[i, 2])); }
lista(kmax) = {my(s = 0); for(k = 1, kmax, s += pfp(k); print1(s, ", "))};
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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