cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

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Author

N. J. A. Sloane

Keywords

Comments

An effective abc conjecture (c < rad(abc)^2) would imply that a(27) = a(28) = ... = a(32), and a(33) = 124225935845233319439173. - Lucas A. Brown, Sep 20 2020

Examples

			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.

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

Crossrefs

Cf. A002071, A003032, A003033, A122463, A145606, A175607.
Equals A117581(n) - 1.

Programs

  • Mathematica
    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 *)
  • PARI
    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
  • PARI
    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

Formula

a(n) < 10^n/n except for n=4. (Conjectured, from experimental data.) - M. F. Hasler, Jan 16 2015

Extensions

More terms from Don Reble, Jan 11 2005
a(18)-a(26) from Fred Schneider, Sep 09 2006
Corrected and extended by Andrey V. Kulsha, Aug 10 2011, according to Jim White's computations.

A193944 Integers k such that for all j > k the largest prime factor of j*(j+1)*(j+2) exceeds the largest prime factor of k*(k+1)*(k+2).

Original entry on oeis.org

2, 8, 48, 98, 350, 440, 2430, 13310, 13454, 17575, 212380, 1205644, 2018978, 3939648, 15473808, 407498958, 138982582998, 768026327418, 1049851495966, 2682238231673, 5556134065128, 14334401249714, 201602864021438
Offset: 1

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Author

Andrey V. Kulsha, Aug 10 2011

Keywords

Comments

Heuristics show that these terms are valid, but a strict proof is yet to be done. Terms 24-26 were found with a program written by Robert Gerbicz. - Andrey V. Kulsha, Aug 29 2011

Links

Crossrefs

Cf. A003032, A193943, A193945.

A085904 Numbers k such that k, k+1 and k+2 are 7-smooth, i.e., all prime divisors <= 7 (A002473).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 14, 48
Offset: 1

Views

Author

Amarnath Murthy, Jul 09 2003

Keywords

Comments

No more terms < 3*10^7. Probably no more terms. - David Wasserman, Feb 10 2005
No more terms < 2^180. - Donovan Johnson, Oct 10 2012
There are no further terms: see A003032 (and maybe A002072). - Don Reble, Mar 14 2019

Examples

			48 is a member as 48, 49 and 50 have all prime divisors <= 7.

Crossrefs

Cf. A002473.

Programs

  • PARI
    mx=2^180+2; v=vector(4607193); c=0; for(e1=0, 180, x1=2^e1; for(e2=0, 113, x2=x1*3^e2; if(x2>mx, next(2)); for(e3=0, 77, x3=x2*5^e3; if(x3>mx, next(2)); for(e4=0, 64, x4=x3*7^e4; if(x4>mx, next(2)); c++; v[c]=x4)))); v=vecsort(v); for(i=1, 4607191, if(v[i+1]-v[i]==1, if(v[i+2]-v[i]==2, print1(v[i] ", ")))) /* Donovan Johnson, Oct 10 2012 */

Extensions

Offset corrected and missing term added by Donovan Johnson, Oct 10 2012
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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