A284723 -id:A284723 - cp's OEIS frontend

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

Peter Munn, May 09 2020

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.

			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.

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.

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

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).

A355001 Smallest common prime factor of A003961(n) and A276086(n), or 1 if they are coprime, where A003961 is fully multiplicative with a(p) = nextprime(p), and A276086 is primorial base exp-function.

Original entry on oeis.org

1, 3, 1, 3, 1, 5, 1, 3, 5, 3, 1, 5, 1, 3, 5, 3, 1, 5, 1, 3, 5, 3, 1, 5, 1, 3, 5, 3, 1, 7, 1, 3, 1, 3, 7, 5, 1, 3, 5, 3, 1, 5, 1, 3, 5, 3, 1, 5, 1, 3, 5, 3, 1, 5, 7, 3, 5, 3, 1, 7, 1, 3, 1, 3, 7, 5, 1, 3, 5, 3, 1, 5, 1, 3, 5, 3, 1, 5, 1, 3, 5, 3, 1, 5, 7, 3, 5, 3, 1, 7, 1, 3, 1, 3, 7, 5, 1, 3, 5, 3, 1, 5, 1, 3, 5

Offset: 1

Antti Karttunen, Jul 13 2022

nonn

Cf. A003961, A020639, A276086, A284723 (even bisection), A355442, A355820, A355821 (positions of 1's).

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A020639(n) = if(1==n,n,vecmin(factor(n)[, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A355442(n) = gcd(A003961(n), A276086(n));
A355001(n) = A020639(A355442(n));

a(n) = A020639(A355442(n)) = A020639(gcd(A003961(n), A276086(n))).

A284721 Smallest odd prime that is relatively prime to 2n+1.

Original entry on oeis.org

3, 5, 3, 3, 5, 3, 3, 7, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 7, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 7, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 11, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 7, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 7, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 7, 3, 3, 5

Offset: 0

N. J. A. Sloane, Apr 04 2017

nonn

More than the usual number of terms are shown in order to distinguish this from A239278.
a(n) = smallest odd prime missing from rad(2*n+1).
If rad(2m+1) = rad(2n+1), a(m) = a(n) (cf. A007947). - Bob Selcoe, Apr 04 2017

Robert Israel, Table of n, a(n) for n = 0..10000

Similar to but different from A239278.

Cf. A002110, A007947, A284722, A284723.

f:= proc(n) local p;
 p:= 2;
 do
   p:= nextprime(p);
   if igcd(p,2*n+1)=1 then return p fi
 od
end proc:
map(f, [$0..100]); # Robert Israel, Dec 09 2024

a[n_] := Module[{p = 3}, While[Divisible[2*n + 1, p], p = NextPrime[p]]; p]; Array[a, 100, 0] (* Amiram Eldar, Dec 09 2023 *)

a(n) = my(p=3); while(gcd(2*n+1, p) != 1, p=nextprime(p+1)); p; \\ Michel Marcus, Apr 04 2017

a(n) = 3 unless n == 1 (mod 3).
For all n >= 2, a(n) < 3*log(2*n+1). Also, for all n >= 1, a(n) < 5*log(2*n+1). [Upper bound corrected by N. J. A. Sloane, Apr 15 2017. Thanks to Bob Selcoe for pointing out that the old bound failed at n=1.]
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2 * Sum_{k>=2} (prime(k) * (1/prime(k-1)# - 1/prime(k)#)) = 3.84010195463226942418..., where prime(k)# = A002110(k). - Amiram Eldar, Dec 09 2023

A371531 a(n) is the multiplicative order of A053669(n) modulo n.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 3, 2, 6, 4, 10, 2, 12, 6, 4, 4, 8, 6, 18, 4, 6, 5, 11, 2, 20, 3, 18, 6, 28, 4, 5, 8, 10, 16, 12, 6, 36, 18, 12, 4, 20, 6, 14, 10, 12, 11, 23, 4, 21, 20, 8, 6, 52, 18, 20, 6, 18, 28, 58, 4, 60, 30, 6, 16, 12, 10, 66, 16, 22, 12, 35, 6, 9, 18, 20

Offset: 1

Darío Clavijo, Mar 26 2024

nonn

Cf. A002326, A053669, A284723.

a[n_] := Module[{p = 2}, While[Divisible[n, p], p = NextPrime[p]]; MultiplicativeOrder[p, n]]; Array[a, 75] (* Amiram Eldar, Mar 26 2024 *)

f(n) = forprime(p=2, , if(n%p, return(p))); \\ A053669
a(n) = znorder(Mod(f(n), n)); \\ Michel Marcus, Mar 26 2024

from sympy.ntheory.residue_ntheory import n_order
from sympy import nextprime
def a(n):
  if n == 1: return 1
  if n & 1 == 1: return n_order(2, n)
  p = 2
  while n % p == 0:
    p = nextprime(p)
  return n_order(p, n)
print([a(n) for n in range(1, 76)])

a(2k+1) = A002326(k) for k >= 1.

a(2k) = ord(A284723(k), 2k).

cp's OEIS Frontend

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.

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A355001 Smallest common prime factor of A003961(n) and A276086(n), or 1 if they are coprime, where A003961 is fully multiplicative with a(p) = nextprime(p), and A276086 is primorial base exp-function.

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A284721 Smallest odd prime that is relatively prime to 2n+1.

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A371531 a(n) is the multiplicative order of A053669(n) modulo n.

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