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

Author

Ben Polson, Feb 03 2022

Keywords

• nonn

Comments

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

Examples

			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.
		

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.

Crossrefs

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

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

Programs

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 *)

PARI

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

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).

Extensions

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

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