A389455 The largest unitary divisor of n that is a term in A048103 (not divisible by p^p for any prime p).

1, 2, 3, 1, 5, 6, 7, 1, 9, 10, 11, 3, 13, 14, 15, 1, 17, 18, 19, 5, 21, 22, 23, 3, 25, 26, 1, 7, 29, 30, 31, 1, 33, 34, 35, 9, 37, 38, 39, 5, 41, 42, 43, 11, 45, 46, 47, 3, 49, 50, 51, 13, 53, 2, 55, 7, 57, 58, 59, 15, 61, 62, 63, 1, 65, 66, 67, 17, 69, 70, 71, 9, 73, 74, 75, 19, 77, 78, 79, 5, 1, 82, 83, 21, 85

Offset

1,2

Comments

Differs from A380085 first at n=625, where a(625)=625, while A380085(625)=1.

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Formula

Multiplicative with a(p^e) = p^e if e < p, and 1 otherwise.

a(n) = n / A368333(n).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + 1/((p+1)*p^(2*p-1)) - 1/p^p) = 0.76291507838193389636... . - Amiram Eldar, Oct 05 2025

Mathematica

f[p_, e_] := If[e < p, p^e, 1]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 05 2025 *)

Prog

(PARI)
A359550(n) = { my(pp); forprime(p=2, , pp = p^p; if(!(n%pp), return(0)); if(pp > n, return(1))); };
A389455(n) = fordiv(n, d, if(1==gcd(n/d, d) && A359550(n/d), return(n/d)));
(PARI) A389455(n) = { my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] < f[i, 1], f[i, 1]^f[i, 2], 1)); }

Crossrefs

Cf. A048103 (positions k where a(k)=k), A054744 (positions of 1's), A359550, A368333, A389450 (sum of unitary divisors), A389451 (sum of divisors), A389454 (arithmetic derivative).

Cf. also A327937 (the largest divisor of n that is a term in A048103), A380085, A380088.

Sequence in context: A325978 A326049 A380085 * A357684 A382891 A072400

Adjacent sequences: A389452 A389453 A389454 * A389456 A389457 A389458

Keyword

nonn,easy,mult

Author

Antti Karttunen, Oct 04 2025

Status

approved