A252493 -id:A252493 - cp's OEIS frontend

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

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

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

A074399 a(n) is the largest prime divisor of n(n+1).

Original entry on oeis.org

2, 3, 3, 5, 5, 7, 7, 3, 5, 11, 11, 13, 13, 7, 5, 17, 17, 19, 19, 7, 11, 23, 23, 5, 13, 13, 7, 29, 29, 31, 31, 11, 17, 17, 7, 37, 37, 19, 13, 41, 41, 43, 43, 11, 23, 47, 47, 7, 7, 17, 17, 53, 53, 11, 11, 19, 29, 59, 59, 61, 61, 31, 7, 13, 13, 67, 67, 23, 23, 71, 71, 73, 73, 37, 19

Offset: 1

N. J. A. Sloane, Nov 29 2002

Størmer shows that a(n) tends to infinity with n. Pólya generalized this result to other polynomials.
Kotov shows that a(n) >> log log n. - Charles R Greathouse IV, Mar 26 2012
Keates and Schinzel give effective constants for the above; in particular the latter shows that lim inf a(n)/log log n >= 2/7. - Charles R Greathouse IV, Nov 12 2012
Erdős conjectures ("on very flimsy probabilistic grounds") that for every e > 0, a(n) < (log n)^(2+e) infinitely often, while a(n) < (log n)^(2-e) only finitely often. - Charles R Greathouse IV, Mar 11 2015

S. V. Kotov, The greatest prime factor of a polynomial (in Russian), Mat. Zametki 13 (1973), pp. 515-522.
K. Mahler, Über den größten Primteiler spezieller Polynome zweiten Grades, Archiv for mathematik og naturvidenskab 41:6 (1934), pp. 3-26.
Georg Pólya, Zur arithmetischen Untersuchung der Polynome, Math. Zeitschrift 1 (1918), pp. 143-148.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
P. Erdős, Problems and results on number theoretic properties of consecutive integers and related questions, Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975), Congress. Numer. XVI , pp. 25-44, Utilitas Math., Winnipeg, Man., 1976.
M. Keates, On the greatest prime factor of a polynomial (1968), pp. 301-303.
Hector Pasten, The largest prime factor of n^2+1 and improvements on subexponential ABC, arXiv:2312.03566 [math.NT] (2024)
A. Schinzel, On two theorems of Gelfond and some of their applications, Acta Arithmetica 13:2 (1967-1968), pp. 177-236.
Carl Størmer, Quelques théorèmes sur l'équation de Pell x^2 - Dy^2 = +-1 et leurs applications (in French), Skrifter udgivne af Videnskabsselskabet i Christiania: Mathematisk-naturvidenskabelig Klasse (1897).

With A037464, the bisections of A076605.
Essentially the same as A069902.
Positions of primes <= p: A085152 (p=5), A085153 (p=7), A252494 (p=11), A252493 (p=13), A252492 (p=17).
Last position of each prime: A002072.

Table[ Last[ Table[ # [[1]]] & /@ FactorInteger[n^2 - 1]], {n, 3, 160, 2}]
Table[FactorInteger[n(n+1)][[-1,1]],{n,80}] (* Harvey P. Dale, Sep 28 2021 *)

gpf(n)=my(f=factor(n)[,1]); f[#f]
a(n)=if(n<3, n+1, max(gpf(n),gpf(n+1))) \\ Charles R Greathouse IV, Sep 14 2015

a(n) = Max (A006530(2n), A006530(2n+2)).

Pasten proves that a(n) >> (log log n)^2/(log log log n), see Corollary 1.5. - Charles R Greathouse IV, Oct 14 2024

Extended by Robert G. Wilson v, Dec 02 2002

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.

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.

A285282 Numbers n such that n^2 + 1 is 13-smooth.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 18, 57, 239

Offset: 1

Tomohiro Yamada, Apr 16 2017

Equivalently: Numbers n such that all prime factors of n^2 + 1 are <= 13.
Since an odd prime factor of n^2 + 1 must be of the form 4m + 1, n^2 + 1 must be of the form 2*5^a*13^b.
This sequence is complete by a theorem of Størmer.
The largest instance 239^2 + 1 = 2*13^4 also gives the only nontrivial solution for x^2 + 1 = 2y^4 (Ljunggren).

			For n = 8, a(8)^2 + 1 = 57^2 + 1 = 3250 = 2*5^3*13.

W. Ljunggren, Zur Theorie der Gleichung x^2 + 1 = 2y^4, Avh. Norsk Vid. Akad. Oslo. 1(5) (1942), 1--27.

A. Schinzel, On two theorems of Gelfond and some of their applications, Acta Arithmetica 13 (1967-1968), 177--236.
Ray Steiner, Simplifying the Solution of Ljunggren's Equation X^2 + 1 = 2Y^4, J. Number Theory 37 (1991), 123--132, more accesible proof of Ljunggren's result.
Carl Størmer, Quelques théorèmes sur l'équation de Pell x^2 - Dy^2 = +-1 et leurs applications (in French), Skrifter Videnskabs-selskabet (Christiania), Mat.-Naturv. Kl. I Nr. 2 (1897), 48 pp.

Cf. A014442, A252493 (n(n+1) instead of n^2 + 1).

Select[Range[1000], FactorInteger[#^2 + 1][[-1, 1]] <= 13&] (* Jean-François Alcover, May 17 2017 *)

for(n=1, 9e6, if(vecmax(factor(n^2+1)[, 1])<=13, print1(n", ")))

from sympy import primefactors
def ok(n): return max(primefactors(n**2 + 1))<=13 # Indranil Ghosh, Apr 16 2017

A275156 The 108 numbers n such that n(n+1) is 17-smooth.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 20, 21, 24, 25, 26, 27, 32, 33, 34, 35, 39, 44, 48, 49, 50, 51, 54, 55, 63, 64, 65, 77, 80, 84, 90, 98, 99, 104, 119, 120, 125, 135, 143, 153, 168, 169, 175, 195, 220, 224, 242, 255, 272, 288, 324, 350, 351, 363, 374, 384, 440, 441, 539, 560, 594, 624, 675, 714, 728, 832, 935, 1000, 1088, 1155, 1224, 1274, 1700, 1715, 2057, 2079, 2400, 2430, 2499, 2600, 3024, 4095, 4224, 4374, 4913, 5831, 6655, 9800, 10647, 12375, 14399, 28560, 31212, 37179, 123200, 194480, 336140

Offset: 1

Jean-François Alcover, Nov 13 2016

This is the 7th row of the table A138180.

See A002071.

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

pMax = 17; smoothMax = 10^12; smoothNumbers[p_?PrimeQ, 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]; Select[(Sqrt[1 + 4*smoothNumbers[pMax, smoothMax]] - 1)/2, IntegerQ]

is(n)=my(t=510510); n*=n+1; while((t=gcd(n,t))>1, n/=t); n==1 \\ Charles R Greathouse IV, Nov 13 2016

A275164 The 167 numbers n such that n(n+1) is 19-smooth.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 24, 25, 26, 27, 32, 33, 34, 35, 38, 39, 44, 48, 49, 50, 51, 54, 55, 56, 63, 64, 65, 75, 76, 77, 80, 84, 90, 95, 98, 99, 104, 119, 120, 125, 132, 135, 143, 152, 153, 168, 169, 170, 175, 189, 195, 208, 209, 220, 224, 242, 255, 272, 285, 288, 323, 324, 342, 350, 351, 360, 363, 374, 384, 399, 440

Offset: 1

Jean-François Alcover, Nov 14 2016

See A002071.

The full list of 167 terms is given in the b-file (this is the 8th row of the table A138180).

Jean-François Alcover, Table of n, a(n) for n = 1..167

Cf. A002071, A145604, A138180, A145605, A002072, A085152, A085153, A252493, A252492,A275156.

pMax = 19; smoothMax = 10^15; smoothNumbers[p_?PrimeQ, 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]; Select[(Sqrt[1 + 4*smoothNumbers[pMax, smoothMax]] - 1)/2, IntegerQ]

cp's OEIS Frontend

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

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

A074399 a(n) is the largest prime divisor of n(n+1).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A285282 Numbers n such that n^2 + 1 is 13-smooth.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Python

A275156 The 108 numbers n such that n(n+1) is 17-smooth.

Views

Author

Keywords

Comments

References

Crossrefs

Programs

Mathematica

PARI