|
|
A002071
|
|
Number of pairs of consecutive integers x, x+1 such that all prime factors of both x and x+1 are at most the n-th prime.
(Formerly M3386 N1366)
|
|
16
|
|
|
1, 4, 10, 23, 40, 68, 108, 167, 241, 345, 482, 653, 869, 1153, 1502, 1930, 2454, 3106, 3896
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
Also: Number of positive integers x such that x(x+1) is prime(n)-smooth. - M. F. Hasler, Jan 16 2015
On an effective abc conjecture (c < rad(abc)^2), we have that a(20)-a(33) is (4839, 6040, 7441, 9179, 11134, 13374, 16167, 19507, 23367, 27949, 33233, 39283, 46166, 54150). - Lucas A. Brown, Oct 16 2022
|
|
REFERENCES
|
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
LINKS
|
|
|
FORMULA
|
|
|
MATHEMATICA
|
(* This program needs x maxima taken from A002072. *) xMaxima = A002072;
smoothNumbers[p_, max_] := Module[{a, aa, k, pp, iter}, k = PrimePi[p]; aa = Array[a, k]; pp = Prime[Range[k]]; iter = Table[{a[j], 0, PowerExpand @ Log[pp[[j]], max/Times @@ (Take[pp, j-1]^Take[aa, j-1])]}, {j, 1, k}]; Table[Times @@ (pp^aa), Sequence @@ iter // Evaluate] // Flatten // Sort]; a[n_] := Module[{sn, cnt}, sn = smoothNumbers[Prime[n], xMaxima[[n]]+1]; cnt = 0; Do[If[sn[[i]]+1 == sn[[i+1]], cnt++], {i, 1, Length[sn]-1}]; cnt]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 16}] (* Jean-François Alcover, Nov 10 2016 *)
A002072 = {1, 8, 80, 4374, 9800, 123200, 336140, 11859210, 11859210};
Table[Length[Select[Table[Max[FactorInteger[x], FactorInteger[x + 1]], {x, A002072[[n]]}], # <= Prime[n] &]], {n, 7}] (* Robert Price, Oct 29 2018 *)
|
|
PROG
|
(PARI) A002071(n)=[1, 4, 10, 23, 40, 68, 108, 167, 241, 345, 482, 653, 869, 1153, 1502][n] \\ "practical" solution. - M. F. Hasler, Jan 16 2015
(PARI) A002071(n, b=A002072, c=1, p=prime(n))={for(k=2, b(n), vecmax(factor(k++, p)[, 1])<=p && vecmax(factor(k--+(k<2), p)[, 1])<=p && c++); c} \\ b can be any upper bound for A002072, e.g., n->10^n should work, too. - M. F. Hasler, Jan 16 2015
|
|
CROSSREFS
|
Cf. A138180 (triangle of x values for each n).
|
|
KEYWORD
|
nonn,nice,hard,more
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|