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%I #16 Jul 17 2015 09:40:21
%S 48,629693,8388223,9841094,1728322595,19503452898,27558254932,
%T 2399283556900
%N 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.
%C Theorem 2 in Rouse & Yang link proves that this sequence is infinite.
%C a(9) > 7*10^12. - _Giovanni Resta_, Jul 17 2015
%H Jeremy Rouse and Yilin Yang, <a href="http://arxiv.org/abs/1502.00605">Three consecutive almost squares</a>, arXiv:1502.00605 [math.NT], 2015.
%e 48 is a term since core(48)=3, core(49)=1, core(50)=2, these 3 values being smaller than 48^(1/3).
%o (PARI) isok(n) = my(cb = sqrtnint(n, 3)); (core(n) <= cb) && (core(n+1) <= cb) && (core(n+2) <= cb);
%o (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. */
%o 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
%o (Python)
%o from operator import mul
%o from functools import reduce
%o from sympy import factorint
%o def A007913(n):
%o ....return reduce(mul,[1]+[p for p,e in factorint(n).items() if e % 2])
%o A254625_list, n, c0, c1, c2 = [], 1, 1, 8, 27
%o for _ in range(10**6):
%o ....if max(c0,c1,c2) < n:
%o ........A254625_list.append(n)
%o ....n += 1
%o ....c0, c1, c2 = c1, c2, A007913(n+2)**3 # _Chai Wah Wu_, Feb 08 2015
%K nonn,more
%O 1,1
%A _Michel Marcus_, Feb 03 2015
%E a(5)-a(7) from _Charles R Greathouse IV_, Jul 17 2015
%E a(8) from _Giovanni Resta_, Jul 17 2015
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