A214685 a(n) is obtained from n by removing 2s, 3s, and 5s from the prime factorization of n that do not contribute to a factor of 30.

1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 11, 1, 13, 7, 1, 1, 17, 1, 19, 1, 7, 11, 23, 1, 1, 13, 1, 7, 29, 30, 31, 1, 11, 17, 7, 1, 37, 19, 13, 1, 41, 7, 43, 11, 1, 23, 47, 1, 49, 1, 17, 13, 53, 1, 11, 7, 19, 29, 59, 30, 61, 31, 7, 1, 13, 11, 67, 17, 23, 7, 71, 1, 73, 37

Offset

1,7

Comments

In this sequence, the number 30 exhibits characteristics of a prime number. It exhibits characteristics of a prime number since all extraneous 2s, 3s, and 5s have been removed from the prime factorizations of all of the numbers.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Formula

a(n) = (n*30^(v_30(n)))/(2^(v_2(n))*3^(v_3(n))*5^(v_5(n))), where v_k(n) is the k-adic valuation of n. That is, v_k(n) is the largest power of k, a, such that k^a divides n.

Sum_{k=1..n} a(k) ~ (31/144) * n^2. - Amiram Eldar, Dec 25 2023

Example

n=15, v_2(15)=0, v_3(15)=1, v_5(15)=1, v_30(15)=0, so a(15) = 30^0*15/(2^0*3^1*5^1) = 1.
n=60, v_2(60)=2, v_3(60)=1, v_5(60)=1, v_30(60)=1, so a(60) = 30^1*60/(2^2*3^1*5^1) = 30.

Maple

a:= proc(n) local i, m, r; m:=n;
      for i from 0 while irem(m, 30, 'r')=0 do m:=r od;
      while irem(m, 2, 'r')=0 do m:=r od;
      while irem(m, 3, 'r')=0 do m:=r od;
      while irem(m, 5, 'r')=0 do m:=r od;
      m*30^i
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 04 2013

Mathematica

With[{v = IntegerExponent}, a[n_] := n*30^v[n, 30]/2^v[n, 2]/3^v[n, 3]/5^v[n, 5]; Array[a, 100]] (* Amiram Eldar, Dec 09 2020 *)

Prog

(Sage)
n=100 #change n for more terms
C=[]
b=30
P = factor(b)
for i in [1..n]:
    prod = 1
    for j in range(len(P)):
        prod = prod * ((P[j][0])^(Integer(i).valuation(P[j][0])))
    C.append((b^(Integer(i).valuation(b)) * i) /prod)

Crossrefs

Cf. A214681, A214682.

Sequence in context: A318674 A284118 A165725 * A327670 A178637 A364092

Adjacent sequences: A214682 A214683 A214684 * A214686 A214687 A214688

Keyword

easy,nonn

Author

Daniel Juda, Jul 25 2012

Status

approved