A324706 The sum of the tri-unitary divisors of n.

1, 3, 4, 5, 6, 12, 8, 15, 10, 18, 12, 20, 14, 24, 24, 17, 18, 30, 20, 30, 32, 36, 24, 60, 26, 42, 40, 40, 30, 72, 32, 33, 48, 54, 48, 50, 38, 60, 56, 90, 42, 96, 44, 60, 60, 72, 48, 68, 50, 78, 72, 70, 54, 120, 72, 120, 80, 90, 60, 120, 62, 96, 80, 85, 84, 144

Offset

1,2

Comments

A divisor d of n is tri-unitary if the greatest common bi-unitary divisor of d and n/d is 1.

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Graeme L. Cohen, On an integer's infinitary divisors, Mathematics of Computation, Vol. 54, No. 189 (1990), pp. 395-411.

Pentti Haukkanen, On the k-ary convolution of arithmetical functions, The Fibonacci Quarterly, Vol. 38, No. 5 (2000) pp. 440-445.

Formula

Multiplicative with a(p^3) = 1 + p + p^2 + p^3, a(p^6) = 1 + p^2 + p^4 + p^6, and a(p^e) = 1 + p^e otherwise.

Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 - 1/p^3 + 1/p^4 - 2/p^6 + 2/p^8 - 1/p^9 - 1/p^12 + 1/p^13) = 0.72189237802... . - Amiram Eldar, Nov 24 2022

Mathematica

f[p_, e_] := If[e == 3, (p^4-1)/(p-1), If[e==6, (p^8-1)/(p^2-1), p^e+1]]; a[1]=1; a[n_]:= Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) A324706(n) = { my(f = factor(n)); prod(i=1, #f~, if(3==f[i, 2], sigma(f[i, 1]^f[i, 2]), if(6==f[i, 2], ((f[i, 1]^8)-1)/((f[i, 1]^2)-1), 1+(f[i, 1]^f[i, 2])))); }; \\ Antti Karttunen, Mar 12 2019

Crossrefs

Cf. A000203, A034448, A049417, A072691, A188999, A322485, A324707, A324708, A324709.

Sequence in context: A241405 A322485 A327668 * A049417 A331110 A188999

Adjacent sequences: A324703 A324704 A324705 * A324707 A324708 A324709

Keyword

nonn,mult

Author

Amiram Eldar, Mar 11 2019

Status

approved