A002072 -id:A002072 - 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

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

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.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

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

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, 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

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

(* 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 *)

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

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

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

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

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

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

T. D. Noe, Oct 14 2008

Note that this sequence is not always increasing. For many n, a(n) is the same as A002072(n). See A145605 for a triangle of values.

An effective abc conjecture (c < rad(abc)^2) would imply that a(29)-a(33) is (90550606380841216610, 205142063213188103639, 53234795127882729824, 4114304445616636016031, 124225935845233319439173). - Lucas A. Brown, Sep 20 2020

Jim White, Results to P = 127

Cf. A002072, A117581, A145605, A175607.

Terms a(16) onward by Andrey V. Kulsha, Aug 10 2011, according to Jim White's computations

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

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

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

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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.

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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).

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

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A145606 Largest number x such that x and x+1 are prime(n)-smooth but not prime(n-1)-smooth.

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

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