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 A344222 a(n) = Sum_{k=1..n} tau(gcd(k,n)^4), where tau(n) is the number of divisors of n. 5
 1, 6, 7, 16, 9, 42, 11, 36, 25, 54, 15, 112, 17, 66, 63, 76, 21, 150, 23, 144, 77, 90, 27, 252, 49, 102, 79, 176, 33, 378, 35, 156, 105, 126, 99, 400, 41, 138, 119, 324, 45, 462, 47, 240, 225, 162, 51, 532, 81, 294, 147, 272, 57, 474, 135, 396, 161, 198, 63, 1008, 65, 210, 275, 316, 153, 630, 71 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Seiichi Manyama, Table of n, a(n) for n = 1..10000 FORMULA a(n) = Sum_{d|n} phi(n/d) * tau(d^4). a(n) = n * Sum_{d|n} 4^omega(d) / d. If p is prime, a(p) = 4 + p. From Amiram Eldar, Nov 15 2022: (Start) Multiplicative with a(p^e) = (p^e*(p + 3) - 4)/(p - 1). Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 + 3/p^2) = 2.4997873122... . (End) MATHEMATICA Table[Sum[DivisorSigma[0, GCD[k, n]^4], {k, n}], {n, 100}] (* Giorgos Kalogeropoulos, May 13 2021 *) f[p_, e_] := (p^e*(p + 3) - 4)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 15 2022 *) PROG (PARI) a(n) = sum(k=1, n, numdiv(gcd(k, n)^4)); (PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*numdiv(d^4)); (PARI) a(n) = n*sumdiv(n, d, 4^omega(d)/d); CROSSREFS Cf. A060648, A072691, A344221, A344223, A344224, A344225. Sequence in context: A106680 A214325 A078388 * A315843 A261195 A008538 Adjacent sequences: A344219 A344220 A344221 * A344223 A344224 A344225 KEYWORD nonn,mult AUTHOR Seiichi Manyama, May 12 2021 STATUS approved

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Last modified June 5 12:07 EDT 2023. Contains 363136 sequences. (Running on oeis4.)