A002071 -id:A002071 - cp's OEIS frontend

A175607 Largest number k such that the greatest prime factor of k^2 - 1 is prime(n).

3, 17, 161, 8749, 19601, 246401, 672281, 23718421, 10285001, 354365441, 3222617399, 9447152318, 127855050751, 842277599279, 2218993446251, 2907159732049, 41257182408961, 63774701665793, 25640240468751, 238178082107393, 4573663454608289, 19182937474703818751, 34903240221563713, 332110803172167361, 99913980938200001

Offset: 1

Charles R Greathouse IV, Jul 23 2010

For any prime p, there are finitely many k such that k^2-1 has p as its largest prime factor.
For every prime p, is there some k where the greatest prime factor of k^2-1 is p? Answer from Artur Jasinski, Oct 22 2010: Yes.
As mentioned by Luca and Najman, this problem is closely related to the one in A002071.
The terms give an upper bound with a method for the simultaneous computation of logarithms of small primes, see the fxtbook link. - Joerg Arndt, Jul 03 2012

Charles R Greathouse IV, Table of n, a(n) for n = 1..25
Joerg Arndt, Matters Computational (The Fxtbook), section 32.4, pp.632-633.
Florian Luca and Filip Najman, "On the largest prime factor of x^2-1", Mathematics of Computation 80:273 (2011), pp. 429-435. (Paper has errata that was posted on the MOC website.)
Filip Najman, Home Page (gives all 16223 numbers k such that k^2-1 has no prime factor greater than 97)

Cf. A214093 (largest primes p such that the greatest prime factor of p^2-1 is prime(n)).
Cf. A076605 (largest prime divisor of n^2-1).
Cf. A285283 (equivalent for k^2+1). - Tomohiro Yamada, Apr 22 2017
Cf. A006530, A005563. - M. F. Hasler, Jun 13 2018

/* up to term for p=97 */
/* S[] is the list computed by Filip Najman (16223 elements) */
S=[2,3,4, ... ,332110803172167361, 19182937474703818751];
lpf(n)={ vecmax(factor(n)[, 1]) } /* largest prime factor */
{ forprime (p=2, 97,
  t = 0;
  for (n=1,#S, if ( lpf(S[n]^2-1)==p, t=n ) );
  print1(S[t],", ");
);}
/* Joerg Arndt, Jul 03 2012 */

More terms (using Filip Najman's list) by Joerg Arndt, Jul 03 2012

A085152 All prime factors of n and n+1 are <= 5. (Related to the abc conjecture.)

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 15, 24, 80

Offset: 1

Benoit Cloitre, Jun 21 2003

Equivalently: Numbers n such that n(n+1) is 5-smooth.
The ABC conjecture would imply that if the prime factors of A, B, C are prescribed in advance, then there is only a finite number of solutions to the equation A + B = C with gcd(A,B,C)=1 (indeed it would bound C to be no more than "roughly" the product of those primes). So in particular there ought to be only finitely many pairs of adjacent integers whose prime factors are limited to {2, 3, 5} (D. Rusin).
This sequence is complete by a theorem of Stormer. See A002071. - T. D. Noe, Mar 03 2008
This is the 3rd row of the table A138180. It has 10 = A002071(3) = A145604(1)+A145604(2)+A145604(3) terms and ends with A002072(3) = 80. It is the union of all terms in rows 1 through 3 of the table A145605. It is a subsequence of A252494 and A085153. - M. F. Hasler, Jan 16 2015

OEIS Index entries for sequences related to the abc conjecture

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

Select[Range[10000], FactorInteger[ # (# + 1)][[ -1,1]] <= 5 &] (* T. D. Noe, Mar 03 2008 *)

for(n=1,99,vecmax(factor(n++)[,1])<6 && vecmax(factor(n--+(n<2))[,1])<6 && print1(n", ")) \\ This skips 2 if n+1 is not 5-smooth: twice as fast as the naive version. - M. F. Hasler, Jan 16 2015

Edited by Dean Hickerson, Jun 30 2003

A085153 All prime factors of n and n+1 are <= 7. (Related to the abc conjecture.)

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 14, 15, 20, 24, 27, 35, 48, 49, 63, 80, 125, 224, 2400, 4374

Offset: 1

Benoit Cloitre, Jun 21 2003

The ABC conjecture would imply that if the prime factors of A, B, C are prescribed in advance, then there is only a finite number of solutions to the equation A + B = C with gcd(A,B,C)=1 (indeed it would bound C to be no more than "roughly" the product of those primes). So in particular there ought to be only finitely many pairs of adjacent integers whose prime factors are limited to {2, 3, 5, 7} (D. Rusin).
This sequence is complete by a theorem of Stormer. See A002071. - T. D. Noe, Mar 03 2008
This is the 4th row of the table A138180. It has 23=A002071(4)=A145604(1)+...+ A145604(4) terms and ends with A002072(4)=4374. It is the union of all terms in rows 1 through 4 of the table A145605. It is a subsequence of A252494 and contains A085152 as a subsequence. - M. F. Hasler, Jan 16 2015
Equivalently, this is the sequence of numbers for which A074399(n) <= 7, or A252489(n) <= 4.

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

Cf. A085152, A002473, A086247.

Select[Range[10000], FactorInteger[ # (# + 1)][[ -1,1]] <= 7 &] (* T. D. Noe, Mar 03 2008 *)

for(n=1,9e6,vecmax(factor(n++)[,1])<8 && vecmax(factor(n--+(n<2))[,1])<8 && print1(n",")) \\ M. F. Hasler, Jan 16 2015

Edited by Dean Hickerson, Jun 30 2003

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

N. J. A. Sloane

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

			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

Cf. A002071, A003032, A003033, A122463, A145606, A175607.

Equals A117581(n) - 1.

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

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

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.

A138180 Irregular triangle read by rows: row n consists of all numbers x such that x and x+1 have no prime factor larger than prime(n).

Original entry on oeis.org

1, 1, 2, 3, 8, 1, 2, 3, 4, 5, 8, 9, 15, 24, 80, 1, 2, 3, 4, 5, 6, 7, 8, 9, 14, 15, 20, 24, 27, 35, 48, 49, 63, 80, 125, 224, 2400, 4374, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 20, 21, 24, 27, 32, 35, 44, 48, 49, 54, 55, 63, 80, 98, 99, 120, 125, 175, 224, 242, 384, 440, 539

Offset: 1

T. D. Noe, Mar 04 2008

A number x is p-smooth if all prime factors of x are <= p. The length of row n is A002071(n). Row n begins with 1 and ends with A002072(n). Each term of row n-1 is in row n.

The n-th row is the union of the rows 1 to n of A145605. - M. F. Hasler, Jan 18 2015

			The table reads:
1,
1, 2, 3, 8,
1, 2, 3, 4, 5, 8, 9, 15, 24, 80,  (= A085152)
1, 2, 3, 4, 5, 6, 7, 8, 9, 14, 15, 20, 24, 27, 35, 48, 49, 63, 80, 125, 224, 2400, 4374, (= A085153)
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 20, 21, 24, 27, 32, 35, 44, 48, 49, 54, 55, 63, 80, 98, 99, 120, 125, 175, 224, 242, 384, 440, 539, 2400, 3024, 4374, 9800 (= A252494),
...

See A002071.

T. D. Noe, Rows n=1..10 of triangle, flattened

Cf. A145605; A085152, A085153, A252494, A252493, A252492.

(* 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]; row[n_] := Module[{sn}, sn = smoothNumbers[Prime[n], xMaxima[[n]]+1]; Reap[Do[If[sn[[i]]+1 == sn[[i+1]], Sow[sn[[i]]]], {i, 1, Length[sn]-1}]][[2, 1]]]; Table[Print[n]; row[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Jan 16 2015, updated Nov 10 2016 *)

A138180_row=[]; A138180(n,k)={if(k, A138180(n)[k], #A138180_rowA138180_row=concat(A138180_row,vector(n)); if(#A138180_row[n], A138180_row[n], k=0; p=prime(n); A138180_row[n]=vector(A002071(n),i, until( vecmax(factor(k++)[, 1])<=p && vecmax(factor(k--+(k<2))[, 1])<=p,k++); k)))} \\ A138180(n) (w/o 2nd arg. k) returns the whole row. - M. F. Hasler, Jan 16 2015

A145604 Number of pairs of consecutive integers x, x+1 such that both are prime(n)-smooth but both are not prime(n-1)-smooth.

Original entry on oeis.org

1, 3, 6, 13, 17, 28, 40, 59, 74, 104, 137, 171, 216, 284, 349, 428, 524, 652, 790

Offset: 1

T. D. Noe, Oct 14 2008

See A145605 for a triangle of x value. See A145606 for the largest x for each n.

An effective abc conjecture (c < rad(abc)^2) would imply that a(20)-a(33) is (943, 1201, 1401, 1738, 1955, 2240, 2793, 3340, 3860, 4582, 5284, 6050, 6883, 7984). - Lucas A. Brown, Oct 16 2022

Lucas A. Brown, stormer.py.

Cf. A145605, A145606.

First differences of A002071.

a(16) from Jean-François Alcover, Nov 11 2016
a(17)-a(18) from Lucas A. Brown, Sep 20 2020
a(19) from Lucas A. Brown, Oct 16 2022

A252493 Numbers n such that n(n+1) is 13-smooth. (Related to the abc conjecture.)

Original entry on oeis.org

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

M. F. Hasler, Jan 16 2015

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.

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

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

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

Select[Range[123456], FactorInteger[ # (# + 1)][[ -1,1]] <= 13 &]

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().

A252492 The largest prime factor of n*(n+1) equals 17. (Related to the abc conjecture.)

Original entry on oeis.org

16, 17, 33, 34, 50, 51, 84, 119, 135, 153, 169, 220, 255, 272, 288, 374, 441, 560, 594, 714, 832, 935, 1088, 1155, 1224, 1274, 1700, 2057, 2430, 2499, 2600, 4913, 5831, 12375, 14399, 28560, 31212, 37179, 194480, 336140

Offset: 1

M. F. Hasler, Jan 16 2015

Equivalently, the prime factors of n and n+1 are not larger than 17, but not all smaller than 17 (in which case n is in A252493).
This sequence is complete by a theorem of Stormer, cf. A002071 and sequences A085152, A085153, A252494, A252493.
This is row 7 of A145605. It has A145604(7)=40 terms and ends with A002072(7)=336140.

OEIS Index entries for sequences related to the abc conjecture

Cf. A002473, A086247, A252493, A252494, A085153, A085152.

Select[Range[345678], FactorInteger[ # (# + 1)][[ -1,1]] == 17 &]

for(n=1,9e6,vecmax(factor(n++)[,1])<18 && vecmax(factor(n*n--)[,1])==17 && print1(n",")) \\ Skips 2 if n+1 is not 17-smooth: Twice as fast as the naïve version.

A117581 For each successive prime p, the largest integer n such that both n and n-1 factor into primes less than or equal to p.

Original entry on oeis.org

2, 9, 81, 4375, 9801, 123201, 336141, 11859211, 11859211, 177182721, 1611308700, 3463200000, 63927525376, 421138799640, 1109496723126, 1453579866025, 20628591204481, 31887350832897, 31887350832897, 119089041053697, 2286831727304145, 9591468737351909376, 9591468737351909376, 9591468737351909376, 9591468737351909376, 9591468737351909376, 19316158377073923834001

Offset: 1

Gene Ward Smith, Mar 29 2006

By a theorem of Størmer, the number of such integers is finite; moreover he provides an algorithm for finding the complete list.
Størmer came to this problem from music theory. Another way to formulate the statement of the theorem is that for any prime p, there are only a finite number of superparticular ratios R = n/(n-1) such that R factors into primes less than or equal to p. The numerator of the smallest such R for the i-th prime is the i-th element of the above sequence. For instance, 81/80, the syntonic comma, is the smallest 5-limit superparticular "comma", i.e., small ratio greater than one.
An effective abc conjecture (c < rad(abc)^2) would imply that a(21) = 2286831727304145 and a(22) = ... = a(26) = 9591468737351909376 and a(27) = ... = a(32) = 19316158377073923834001 and a(33) = 124225935845233319439174. - Lucas A. Brown, Oct 16 2022

Lucas A. Brown, stormer.py.
D. H. Lehmer, On a problem of Størmer, Ill. J. Math., 8 (1964), 57-79.
Wikipedia, Størmer's theorem.

Cf. A002071, A002072, A116486, A117582, A117583.

a(n) = A002072 + 1.

Entry edited by N. J. A. Sloane, Apr 01 2006
Corrected and extended by Don Reble, Nov 21 2006
More terms from A002072 added by Amiram Eldar, Apr 13 2025

A252494 Numbers n such that all prime factors of n and n+1 are <= 11. (Related to the abc conjecture.)

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 20, 21, 24, 27, 32, 35, 44, 48, 49, 54, 55, 63, 80, 98, 99, 120, 125, 175, 224, 242, 384, 440, 539, 2400, 3024, 4374, 9800

Offset: 1

M. F. Hasler, Jan 16 2015

This sequence is complete by a theorem of Stormer, cf. A002071.
This is the 5th row of the table A138180. It has 40=A002071(5)=A145604(1)+...+ A145604(5) terms and ends with A002072(5)=9800. It is the union of all terms in rows 1 through 5 of the table A145605.
This is a subsequence of A252493, and contains A085152 and A085153 as subsequences.

OEIS Index entries for sequences related to the abc conjecture

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

Select[Range[10000], FactorInteger[ # (# + 1)][[ -1,1]] <= 11 &]

for(n=1,9e6,vecmax(factor(n++)[,1])<12 && vecmax(factor(n--+(n<2))[,1])<12 && print1(n",")) \\ Skips 2 if n+1 is not 11-smooth: Twice as fast as the naive version.

cp's OEIS Frontend

A175607 Largest number k such that the greatest prime factor of k^2 - 1 is prime(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Extensions

A085152 All prime factors of n and n+1 are <= 5. (Related to the abc conjecture.)

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A085153 All prime factors of n and n+1 are <= 7. (Related to the abc conjecture.)

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A002072 a(n) = smallest number m such that for all k > m, either k or k+1 has a prime factor > prime(n).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A138180 Irregular triangle read by rows: row n consists of all numbers x such that x and x+1 have no prime factor larger than prime(n).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

A145604 Number of pairs of consecutive integers x, x+1 such that both are prime(n)-smooth but both are not prime(n-1)-smooth.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A252493 Numbers n such that n(n+1) is 13-smooth. (Related to the abc conjecture.)

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A252492 The largest prime factor of n*(n+1) equals 17. (Related to the abc conjecture.)