A351168 If n = p_1^e_1 * ... * p_k^e_k, where p_1 < ... < p_k are primes, then a(n) is obtained by replacing the last factor p_k^e_k by (p_k - 1)^e_k; a(1) = 1.

1, 1, 2, 1, 4, 4, 6, 1, 4, 8, 10, 8, 12, 12, 12, 1, 16, 8, 18, 16, 18, 20, 22, 16, 16, 24, 8, 24, 28, 24, 30, 1, 30, 32, 30, 16, 36, 36, 36, 32, 40, 36, 42, 40, 36, 44, 46, 32, 36, 32, 48, 48, 52, 16, 50, 48, 54, 56, 58, 48, 60, 60, 54, 1, 60, 60, 66, 64, 66, 60, 70, 32, 72, 72, 48

Offset

1,3

Comments

First time a term appears four or more times in a row is when n = 1684.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Jan Misali, What should we call the other bases?, Web page, no date. Inspiration for this sequence.

Formula

a(n) = n*(1 - 1/p_k)^e_k where prime factorization n = p_1^e_1 * ... * p_k^e_k with ascending p_1 < ... < p_k.

a(n) = n*(1 - 1/A006530(n))^A071178(n).

Example

The prime factorization of 44 is 2^2 * 11^1, so a(44) = 2^2 * 10^1 = 40.
The prime factorization of 50 is 2^1 * 5^2, so a(50) = 2^1 * 4^2 = 32.

Mathematica

a[n_] := Module[{f = FactorInteger[n]}, n*(1 - 1/f[[-1, 1]])^f[[-1, 2]]]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Feb 04 2022 *)

Prog

(PARI) a(n) = my(f=factor(n), r=matsize(f)[1]); if(r, f[r, 1]--); factorback(f); \\ Kevin Ryde, Feb 03 2022

Crossrefs

Cf. A006530 (greatest prime), A071178 (its exponent).

Cf. A171462 (one instance of the decrement), A003958 (all primes decremented), A351419, A351425.

Sequence in context: A193631 A190993 A326147 * A196082 A273724 A108755

Adjacent sequences: A351165 A351166 A351167 * A351169 A351170 A351171

Keyword

nonn

Author

Ben Polson, Feb 03 2022

Extensions

a(1) = 1 prepended by Michel Marcus, Feb 04 2022

Edited by N. J. A. Sloane, Feb 11 2022

Status

approved