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

Author

M. F. Hasler, Jan 16 2015

Keywords

• nonn
• fini
• full

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

• Abderrahmane Nitaj, The ABC conjecture homepage
• OEIS Index entries for sequences related to the abc conjecture

Crossrefs

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

Programs

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 &]

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().