A145606 -id:A145606 - cp's OEIS frontend

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

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.

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

A145605 Irregular triangle in which row n consists of all numbers x such that x and x+1 are both prime(n)-smooth numbers but not both prime(n-1)-smooth.

Original entry on oeis.org

1, 2, 3, 8, 4, 5, 9, 15, 24, 80, 6, 7, 14, 20, 27, 35, 48, 49, 63, 125, 224, 2400, 4374, 10, 11, 21, 32, 44, 54, 55, 98, 99, 120, 175, 242, 384, 440, 539, 3024, 9800, 12, 13, 25, 26, 39, 64, 65, 77, 90, 104, 143, 168, 195, 324, 350, 351, 363, 624, 675, 728, 1000, 1715

Offset: 1

T. D. Noe, Oct 14 2008, Nov 03 2008

The length of row n is A145604(n). The largest x in row n is A145606(n). This is sequence A138180 with only the first occurrence of each number retained. Row n begins with prime(n)-1.

A permutation of the positive integers (when seen as linear sequence). A252489(n) yields the row in which n appears in the table. - M. F. Hasler, Jan 16 2015

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

T. D. Noe, Rows n=1..15 of triangle, flattened

(* Computation using x maxima taken from A145606 *) A145606 = {1, 8, 80, 4374, 9800, 123200, 336140, 11859210, 5142500, 177182720, 1611308699, 3463199999, 63927525375, 421138799639, 1109496723125}; 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]; smoothMax[n_] := A145606[[n]]; row[n_] := Module[{sn, sn1}, sn = smoothNumbers[Prime[n], smoothMax[n] + 1] ; sn1 = smoothNumbers[Prime[n - 1], smoothMax[n] + 1] ; Select[sn, MemberQ[sn, # + 1] && Not[MemberQ[sn1, #] && MemberQ[sn1, # + 1]] &]]; row[1] = {1}; Table[ro = row[n]; Print[n, " ", ro // Short]; ro, {n, 1, 10}] // Flatten (* Jean-François Alcover, Nov 17 2016 *)

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

A185389 Largest number k such that the greatest prime factor of k^2+1 is A002313(n), the n-th prime not congruent to 3 mod 4.

Original entry on oeis.org

1, 7, 239, 268, 307, 18543, 2943, 485298, 330182, 478707, 24208144, 22709274, 2189376182, 284862638, 599832943, 19696179, 314198789, 3558066693, 69971515635443, 18986886768, 18710140581, 104279454193

Offset: 1

Charles R Greathouse IV, Feb 21 2011

For any prime p, there are finitely many k such that k^2+1 has p as its largest prime factor.

Numbers k such that k^2+1 is p-smooth appear in arctan-relations for the computation of Pi (for example, Machin's identity Pi/4 = 4*arctan(1/5) - arctan(1/239)), see the fxtbook link. [Joerg Arndt, Jul 02 2012]

Joerg Arndt, Matters Computational (The Fxtbook), section 32.5 "Arctangent relations for Pi", pp. 633-640.
Filip Najman, Smooth values of some quadratic polynomials, Glasnik Matematicki Series III 45 (2010), pp. 347-355.
Florian Luca, Primitive divisors of Lucas sequences and prime factors of x^2 + 1 and x^4 + 1, Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 31 (2004), pp. 19-24.
Filip Najman, Home Page (gives all 811 numbers x such that x^2+1 has no prime factor greater than 197)

Equivalents for other polynomials: A175607 (k^2 - 1), A145606 (k^2 + k).

A185396 Largest number x such that the greatest prime factor of x^2-2 is A038873(n), the n-th prime not congruent to 3 or 5 mod 8.

Original entry on oeis.org

2, 10, 108, 235, 1201, 390050, 314766, 4035, 364384, 50411, 25955045, 5254864, 236558593, 16958526, 20388056, 177544434, 492981885, 2275400230, 256347346, 384902923486, 324850200677887

Offset: 1

Charles R Greathouse IV, Feb 21 2011

nonn

For any prime p, there are finitely many x such that x^2-2 has p as its largest prime factor.

Filip Najman, Smooth values of some quadratic polynomials, Glasnik Matematicki Series III 45 (2010), pp. 347-355.
Filip Najman, Home Page (gives all 537 numbers x such that x^2-2 has no prime factor greater than 199)

Cf. A242488, A379348.

Equivalents for other polynomials: A175607 (x^2 - 1), A145606 (x^2 + x), A185389 (x^2 + 1).

a(21) added by Andrew Howroyd, Dec 22 2024

A193948 Integers k such that for all i > k the largest prime factor of i(i+1)(i+2)(i+3)(i+4)(i+5)(i+6) exceeds the largest prime factor of k(k+1)(k+2)(k+3)(k+4)(k+5)(k+6).

Original entry on oeis.org

4, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 184, 527, 4896, 11658, 12874, 18904, 41919, 45998, 48504, 50688, 51982, 356207, 426851, 960750, 1961725, 4604094, 8418495, 10811745, 32963628, 45249999, 569800611, 7374557947, 121153257533

Offset: 1

Andrey V. Kulsha, Aug 10 2011

nonn

Currently terms through a(16) = 12874 have been proved to be correct, while remaining terms are conjectural. Stormer's theorem (see link) provides an approach to proving their correctness. Such an approach should be easy for the first few terms. - Franklin T. Adams-Watters, Nov 07 2011

Andrey V. Kulsha, Table of n, a(n) for n = 1..39
Wikipedia, Stormer's theorem

Cf. A145606, A193947, A199407.

A185397 Largest number x such that the greatest prime factor of x^2+2 is A033203(n), the n-th prime not congruent to 5 or 7 mod 8.

Original entry on oeis.org

22, 140, 707, 21362, 4991, 1306066, 137965, 2294636, 31768298, 1557652, 340064590, 38439662, 105080665, 273502688, 543164542, 9575480365630, 391890109484, 14629598023, 80849485336, 1241646894380

Offset: 1

Charles R Greathouse IV, Feb 21 2011

nonn

For any prime p, there are finitely many x such that x^2+2 has p as its largest prime factor.

Filip Najman, Smooth values of some quadratic polynomials, Glasnik Matematicki Series III 45 (2010), pp. 347-355.
Filip Najman, Home Page (gives all 914 numbers x such that x^2+2 has no prime factor greater than 193)

Equivalents for other polynomials: A175607 (x^2 - 1), A145606 (x^2 + x), A185389 (x^2 + 1), A185396 (x^2 - 2).

A193943 Integers n such that for all i > n the largest prime factor of i(i+1) exceeds the largest prime factor of n(n+1).

Original entry on oeis.org

1, 8, 80, 4374, 9800, 123200, 336140, 11859210, 177182720, 1611308699, 3463199999, 63927525375, 421138799639, 1109496723125, 1453579866024, 20628591204480, 31887350832896, 119089041053696, 2286831727304144, 9591468737351909375, 19316158377073923834000

Offset: 1

Andrey V. Kulsha, Aug 11 2011, according to Jim White's computations

nonn

Andrey V. Kulsha, Table of n, a(n) for n = 1..23
liangbch, consecutive pairs of smooth numbers, Blog.

Cf. A002072, A145606, A193944.

Corrected 23rd term in b-file (see Blog link), Andrey V. Kulsha, Dec 22 2014

a(20) and a(21) added (from b-file) by Jon E. Schoenfield, Apr 22 2018

cp's OEIS Frontend

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

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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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A145605 Irregular triangle in which row n consists of all numbers x such that x and x+1 are both prime(n)-smooth numbers but not both prime(n-1)-smooth.

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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.

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A185389 Largest number k such that the greatest prime factor of k^2+1 is A002313(n), the n-th prime not congruent to 3 mod 4.

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A185396 Largest number x such that the greatest prime factor of x^2-2 is A038873(n), the n-th prime not congruent to 3 or 5 mod 8.

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A193948 Integers k such that for all i > k the largest prime factor of i(i+1)(i+2)(i+3)(i+4)(i+5)(i+6) exceeds the largest prime factor of k(k+1)(k+2)(k+3)(k+4)(k+5)(k+6).

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A185397 Largest number x such that the greatest prime factor of x^2+2 is A033203(n), the n-th prime not congruent to 5 or 7 mod 8.

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A193943 Integers n such that for all i > n the largest prime factor of i*(i+1) exceeds the largest prime factor of n*(n+1).

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A193943 Integers n such that for all i > n the largest prime factor of i(i+1) exceeds the largest prime factor of n(n+1).