A334748 Let p be the smallest odd prime not dividing the squarefree part of n. Multiply n by p and divide by the product of all smaller odd primes.

3, 6, 5, 12, 15, 10, 21, 24, 27, 30, 33, 20, 39, 42, 7, 48, 51, 54, 57, 60, 35, 66, 69, 40, 75, 78, 45, 84, 87, 14, 93, 96, 55, 102, 105, 108, 111, 114, 65, 120, 123, 70, 129, 132, 135, 138, 141, 80, 147, 150, 85, 156, 159, 90, 165, 168, 95, 174, 177, 28, 183, 186, 189

Offset: 1

Author

Peter Munn, May 09 2020

Keywords

• nonn
• easy

Comments

A permutation of A028983.

A007417 (which has asymptotic density 3/4) lists index n such that a(n) = 3n. The sequence maps the terms of A007417 1:1 onto A145204\{0}, defining a bijection between them.

Similarly, bijections are defined from the odd numbers (A005408) to the nonsquare odd numbers (A088828), from the positive even numbers (A299174) to A088829, from A003159 to the nonsquares in A003159, and from A325424 to the nonsquares in A036668. The latter two bijections are between sets where membership depends on whether a number's squarefree part divides by 2 and/or 3.

Examples

			84 = 21*4 has squarefree part 21 (and square part 4). The smallest odd prime absent from 21 = 3*7 is 5 and the product of all smaller odd primes is 3. So a(84) = 84*5/3 = 140.
		

Crossrefs

Permutation of A028983.

Row 3, and therefore column 3, of A331590. Cf. A334747 (row 2).

A007913, A034386, A225546, A284723 are used in formulas defining the sequence.

The formula section details how the sequence maps the terms of A003961, A019565, A070826; and how f(a(n)) relates to f(n) for f = A008833, A048675, A267116; making use of A003986.

Subsequences: A016051, A145204\{0}, A329575.

Bijections are defined that relate to A003159, A005408, A007417, A036668, A088828, A088829, A299174, A325424.

Programs

PARI

a(n) = {my(c=core(n), m=n); forprime(p=3, , if(c % p, m*=p; break, m/=p)); m;} \\ Michel Marcus, May 22 2020

Formula

a(n) = n * p / (A034386(p-1)/2), where p = A284723(A007913(n)).

a(n) = A334747(A334747(n)).

a(n) = A331590(3, n) = A225546(4 * A225546(n)).

a(2*n) = 2 * a(n).

a(A019565(n)) = A019565(n+2).

a(k * m^2) = a(k) * m^2.

a(A003961(n)) = A003961(A334747(n)).

a(A070826(n)) = prime(n+1).

A048675(a(n)) = A048675(n) + 2.

A008833(a(n)) = A008833(n).

A267116(a(n)) = A267116(n) OR 1, where OR denotes the bitwise operation A003986.

a(A007417(n)) = A145204(n+1) = 3 * A007417(n).