A254625 Integers n such that core(n), core(n+1), core(n+2) are smaller than n^(1/3) where core(n) is A007913(n), the squarefree part of n.

48, 629693, 8388223, 9841094, 1728322595, 19503452898, 27558254932, 2399283556900

Offset: 1

Author

Michel Marcus, Feb 03 2015

Keywords

• nonn
• more

Comments

Theorem 2 in Rouse & Yang link proves that this sequence is infinite.

a(9) > 7*10^12. - Giovanni Resta, Jul 17 2015

Examples

			48 is a term since core(48)=3, core(49)=1, core(50)=2, these 3 values being smaller than 48^(1/3).
		

Links

• Jeremy Rouse and Yilin Yang, Three consecutive almost squares, arXiv:1502.00605 [math.NT], 2015.

Programs

PARI

isok(n) = my(cb = sqrtnint(n, 3)); (core(n) <= cb) && (core(n+1) <= cb) && (core(n+2) <= cb);

PARI

/* This program is a little sloppy in testing more points than needed near the start and end, but adding extra code to avoid this case would add to complexity without greatly affecting runtime. */
list(lim,startAt=27)=my(c0,c1,c2); for(c=sqrtnint(startAt\1,3), ceil(sqrtn(lim,3)), my(n=c^3+1,lm=(c+1)^3); while(nc, n+=3; next); c1=core(n+1); if(c1>c, n+=2; next); c0=core(n); if(c0>c, n++; next); print1(n", "); n++)) \\ Charles R Greathouse IV, Jul 16 2015

Python

from operator import mul
from functools import reduce
from sympy import factorint
def A007913(n):
    return reduce(mul,[1]+[p for p,e in factorint(n).items() if e % 2])
A254625_list, n, c0, c1, c2 = [], 1, 1, 8, 27
for _ in range(10**6):
    if max(c0,c1,c2) < n:
        A254625_list.append(n)
    n += 1
    c0, c1, c2 = c1, c2, A007913(n+2)**3 # Chai Wah Wu, Feb 08 2015

Extensions

a(5)-a(7) from Charles R Greathouse IV, Jul 17 2015

a(8) from Giovanni Resta, Jul 17 2015