A252493 Numbers n such that n(n+1) is 13-smooth. (Related to the abc conjecture.)

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 20, 21, 24, 25, 26, 27, 32, 35, 39, 44, 48, 49, 54, 55, 63, 64, 65, 77, 80, 90, 98, 99, 104, 120, 125, 143, 168, 175, 195, 224, 242, 324, 350, 351, 363, 384, 440, 539, 624, 675, 728, 1000, 1715, 2079, 2400, 3024, 4095, 4224, 4374, 6655, 9800, 10647, 123200

Offset

1,2

Comments

Equivalently: Numbers n such that all prime factors of n and n+1 are <= 13, i.e., both are in A080197.

This sequence is complete by a theorem of Stormer, cf. A002071.

This is the 6th row of the table A138180. It has 68=A002071(6)=A145604(1)+...+ A145604(6) terms and ends with A002072(6)=123200. It is the union of all terms in rows 1 through 6 of the table A145605.

Contains A085152, A085153, A252494 as subsequences.

Links

Table of n, a(n) for n=1..68.

Abderrahmane Nitaj, The ABC conjecture homepage

OEIS Index entries for sequences related to the abc conjecture

Maple

N:= 130000: # to get all entries <= N
f:= proc(n)
uses padic;
evalb(2^ordp(n, 2)*3^ordp(n, 3)*5^ordp(n, 5)*7^ordp(n, 7)*11^ordp(n, 11)*13^ordp(n, 13) = n)
end proc:
L:= map(f, [$1..N+1]):
select(t -> L[t] and L[t+1], [$1..N]); # Robert Israel, Jan 16 2015

Mathematica

Select[Range[123456], FactorInteger[ # (# + 1)][[ -1, 1]] <= 13 &]

Prog

(PARI) for(n=1, 123456, vecmax(factor(n++, 13)[, 1])<17 && vecmax(factor(n--+(n<2), 13))<17 && print1(n", ")) \\ Skips the next n if n+1 is not 13-smooth: Twice as fast as the naïve version. Instead of vecmax(.)<17 one could use is_A080197().

Crossrefs

Cf. A002071, A145604, A138180, A145605, A002072, A085152, A085153, A252493, A252492.

Sequence in context: A246096 A098314 A052057 * A005496 A052060 A084688

Adjacent sequences: A252490 A252491 A252492 * A252494 A252495 A252496

Keyword

nonn,fini,full

Author

M. F. Hasler, Jan 16 2015

Status

approved