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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. 1
48, 629693, 8388223, 9841094, 1728322595, 19503452898, 27558254932, 2399283556900 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

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

LINKS

Table of n, a(n) for n=1..8.

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

EXAMPLE

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

PROG

(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(n<lm, if(isprime(n+1), n+=2; next); if(isprime(n), n++; next); c2=core(n+2); if(c2>c, 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

CROSSREFS

Sequence in context: A079234 A051235 A282403 * A165643 A165047 A291865

Adjacent sequences:  A254622 A254623 A254624 * A254626 A254627 A254628

KEYWORD

nonn,more

AUTHOR

Michel Marcus, Feb 03 2015

EXTENSIONS

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

a(8) from Giovanni Resta, Jul 17 2015

STATUS

approved

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Last modified March 19 13:29 EDT 2019. Contains 321330 sequences. (Running on oeis4.)