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)

1, 4, 10, 23, 40, 68, 108, 167, 241, 345, 482, 653, 869, 1153, 1502, 1930, 2454, 3106, 3896

Offset

1,2

Comments

Størmer's theorem proves that a(n) is finite. - Charles R Greathouse IV, Feb 19 2013

Also: Number of positive integers x such that x(x+1) is prime(n)-smooth. - M. F. Hasler, Jan 16 2015

Also: Row lengths of A138180; partial sums of A145604. - 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

Table of n, a(n) for n=1..19.

Lucas A. Brown, stormer.py.

E. F. Ecklund and R. B. Eggleton, Prime factors of consecutive integers, Amer. Math. Monthly, 79 (1972), 1082-1089.

D. Eppstein, Smooth pairs.

D. Eppstein, Python program

D. H. Lehmer, On a problem of Størmer, Ill. J. Math., 8 (1964), 57-69.

C. Stormer, Quelques théorèmes sur l'équation de Pell x^2 - Dy^2 = +-1 et leurs applications, Skrifter Videnskabs-selskabet (Christiania), Mat.-Naturv (1897). Kl. I (2).

Wikipedia, Størmer's theorem

OEIS Index entries for sequences related to the abc conjecture

Formula

a(n) <= (2^n-1)*(prime(n)+1)/2 is implicit in Lehmer 1964. - Charles R Greathouse IV, Feb 19 2013

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. A002072, A145604, A145605, A145606.

Cf. A138180 (triangle of x values for each n).

Cf. A085152, A085153.

Cf. A285283 (equivalent for x^2 + 1). - Tomohiro Yamada, Apr 22 2017

Sequence in context: A276308 A334260 A038423 * A375749 A024980 A002766

Adjacent sequences: A002068 A002069 A002070 * A002072 A002073 A002074

Keyword

nonn,nice,hard,more

Author

N. J. A. Sloane

Extensions

Better description and more terms from David Eppstein, Mar 23 2007

a(16) from Jean-François Alcover, Nov 10 2016

a(17)-a(18) from Lucas A. Brown, Aug 23 2020

a(19) from Lucas A. Brown, Oct 16 2022

Status

approved