A002072
a(n) = smallest number m such that for all k > m, either k or k+1 has a prime factor > prime(n).
Original entry on oeis.org
1, 8, 80, 4374, 9800, 123200, 336140, 11859210, 11859210, 177182720, 1611308699, 3463199999, 63927525375, 421138799639, 1109496723125, 1453579866024, 20628591204480, 31887350832896, 31887350832896, 119089041053696, 2286831727304144, 9591468737351909375, 9591468737351909375, 9591468737351909375, 9591468737351909375, 9591468737351909375, 19316158377073923834000
Offset: 1
a(1) = 1 since for any number k greater than 1, it is impossible that k and k+1 both are powers of 2, so at least one of them has a prime factor > 2. (For m = 0 this would not hold for k = 1, k+1 = 2.)
a(2) = 8 since for any larger k, we cannot have k and k+1 both 3-smooth (cf. A003586).
31887350832897 = 3^9*7*37*41^2*61^2, 31887350832896 = 2^8*13*19*23*29^4*31, this number appears twice because there is no pair of numbers with max. factor = 67 that is larger than this number.
- 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).
- Lucas A. Brown, Python program.
- E. F. Ecklund and R. B. Eggleton, Prime factors of consecutive integers, Amer. Math. Monthly, 79 (1972), 1082-1089.
- D. H. Lehmer, On a problem of Størmer, Ill. J. Math., 8 (1964), 57-79.
- Don Reble, Python program
- Jim White, Results to P = 127
- Wikipedia, Størmer's theorem
-
smoothNumbers[p_?PrimeQ, max_Integer] := 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}]; Sort[Flatten[Table[Times @@ (pp^aa), Evaluate[ Sequence @@ iter]]]]]; a[n_] := Module[{sn = smoothNumbers[Prime[n], Ceiling[2000 + 10^n/n]], pos}, pos = Position[Differences[sn], 1][[-1, 1]]; sn[[pos]]]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 12}] (* Jean-François Alcover, Nov 17 2016, after M. F. Hasler's observation *)
-
A002072(n, a=[1, 8, 80, 4374, 9800, 123200, 336140, 11859210, 11859210, 177182720, 1611308699, 3463199999, 63927525375, 421138799639, 1109496723125, 1453579866024])=a[n] \\ "practical" solution for use in other sequences, easily extended to more values. - M. F. Hasler, Jan 16 2015
-
A2072=List(1); A002072(n)={while(#A2072 best && isSmooth(sol, P) && isSmooth(sol+1, P) && best=sol, p=primes([1, P])); for(i=1, 2^#p, i==2 && next; my(qq = 2*vecprod(vecextract(p,i-1)), qn = [qq, sqrtint(qq), 0, 1], cf = [1,0,0,1], xi, aa, x0, x1, y0, y1); until(x0, aa = (qn[2]+qn[3])\qn[4]; qn[3] = aa*qn[4] - qn[3]; qn[4] = (qn[1] - qn[3]^2) \ qn[4]; cf = [aa*cf[1]+cf[3], aa*cf[2]+cf[4], cf[1], cf[2]]; cf[1]^2 - qq*cf[2]^2 == 1 && [x0,x1, y0,y1] = [x1, cf[1], y1, cf[2]] ); isSmooth(y0, P) || next; check(xi = x0); check(x1); for (i=3, max(P\/2, 3), [x0, x1] = [x1, x1 * xi * 2 - x0]; check(x1)))/*for i*/; listput(A2072, best) } \\ Following Don Reble's Python program. - M. F. Hasler, Mar 01 2025
Corrected and extended by
Andrey V. Kulsha, Aug 10 2011, according to Jim White's computations.
A145606
Largest number x such that x and x+1 are prime(n)-smooth but not prime(n-1)-smooth.
Original entry on oeis.org
1, 8, 80, 4374, 9800, 123200, 336140, 11859210, 5142500, 177182720, 1611308699, 3463199999, 63927525375, 421138799639, 1109496723125, 1453579866024, 20628591204480, 31887350832896, 12820120234375, 119089041053696, 2286831727304144, 9591468737351909375, 17451620110781856, 166055401586083680, 49956990469100000, 4108258965739505499, 19316158377073923834000, 386539843111191224
Offset: 1
Terms a(16) onward by
Andrey V. Kulsha, Aug 10 2011, according to Jim White's computations
A228610
Numbers k such that the largest consecutive pair of prime(k)-smooth integers is the same as the largest consecutive pair of prime(k-1)-smooth integers.
Original entry on oeis.org
9, 19, 23, 24, 25, 26
Offset: 1
For n = 1, k = a(1) = 9 gives prime(k) = 23 such that the largest consecutive pair of 23-smooth integers, (11859210,11859211), is the same as the largest consecutive pair of prime(k-1)-smooth integers (19-smooth integers).
A228611
Primes p such that the largest consecutive pair of p-smooth integers is the same as the largest consecutive pair of (p-1)-smooth integers.
Original entry on oeis.org
23, 67, 83, 89, 97, 101
Offset: 1
For n = 1, a(1) = 23 is a prime such that the largest consecutive pair of 23-smooth integers, (11859210,11859211), is the same as the largest consecutive pair of 22-smooth integers (or of 19-smooth integers, 19 being the next smaller prime).
A250298
Primes p such that the largest integer m such that both m and m-1 factor into primes less than or equal to p is a perfect square, m = k^2.
Original entry on oeis.org
3, 5, 11, 13, 29, 53, 103
Offset: 1
p = 3 gives m = 3^2;
p = 5 gives m = 9^2;
p = 11 gives m = 99^2;
p = 13 gives m = 351^2;
p = 29 gives m = 13311^2;
p = 53 gives m = 1205645^2;
p = 103 gives m = 138982582999^2.
A250302
Positive integers k whose square is, for some prime p, the largest integer m such that both m and m-1 factor into primes less than or equal to p.
Original entry on oeis.org
3, 9, 99, 351, 13311, 1205645, 138982582999
Offset: 1
Here are the largest pairs of consecutive integers with prime factors p or smaller:
p : pair
--------------------------
3 : 3^2-1 and 3^2;
5 : 9^2-1 and 9^2;
11 : 99^2-1 and 99^2;
13 : 351^2-1 and 351^2;
29 : 13311^2-1 and 13311^2;
53 : 1205645^2-1 and 1205645^2;
103 : 138982582999^2-1 and 138982582999^2.
-
lista(v_002072) = {v = v_002072; for (i=1, #v, vi = v[i]; if (issquare(vi+1), print1(sqrtint(vi+1), ", ")););} \\ Michel Marcus, Feb 28 2015
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