A371473 a(1) = 1; for n>1, if a(n-1) is squarefree, a(n) = a(n-1) + n, otherwise a(n) = squarefree kernel of a(n-1).

1, 3, 6, 10, 15, 21, 28, 14, 23, 33, 44, 22, 35, 49, 7, 23, 40, 10, 29, 49, 7, 29, 52, 26, 51, 77, 104, 26, 55, 85, 116, 58, 91, 125, 5, 41, 78, 116, 58, 98, 14, 56, 14, 58, 103, 149, 196, 14, 63, 21, 72, 6, 59, 113, 168, 42, 99, 33, 92, 46

Offset: 1

Author

Joseph C. Y. Wong, Mar 24 2024

Keywords

• nonn

Comments

Inspired by Recaman's sequence A005132.

Some nonsquarefree numbers will not appear in this sequence. However, I conjecture that all squarefree numbers will appear. First occurrence of 2 is at a(766) = 2.

Examples

			a(1) = 1 is squarefree, so a(2) = a(1) + 2 = 3.
a(7) = 28 = 2*2*7 is not squarefree, so a(8) = 2*7 = 14.
		

Links

• Joseph C. Y. Wong, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005132, A005117, A007947.

Programs

Mathematica

rad[n_]:=Product[Part[First/@FactorInteger[n],i],{i,Length[FactorInteger[n]]}]; a[1]=1; a[n_]:=If[SquareFreeQ[a[n-1]],a[n-1]+n,rad[a[n-1]]]; Array[a,60] (* Stefano Spezia, Mar 26 2024 *)

PARI

lista(nn) = my(v = vector(nn)); v[1] = 1; for (n=2, nn, if (issquarefree(v[n-1]), v[n] = v[n-1]+n, v[n] = factorback(factor(v[n-1])[,1]));); v; \\ Michel Marcus, Mar 26 2024

Python

from numpy import prod
def primefact(a):
  factors = []
  d = 2
  while a > 1:
    while a % d == 0:
      factors.append(d)
      a /= d
    d = d + 1
  return factors
def squarefree(a):
  return sorted(list(set(primefact(a)))) == sorted(primefact(a))
sequence = [1]
a = 1
for n in range(1, 1001):
  if not squarefree(a):
    a = prod(list(set(primefact(a))))
  else:
    a += n+1
  sequence.append(a)
print(sequence)