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

Author

Amiram Eldar, Mar 11 2019

Keywords

• nonn
• mult

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.

Crossrefs

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

Programs

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]

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

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