A145604 -id:A145604 - 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

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 *)

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.

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.

A285283 Number of integers x such that the greatest prime factor of x^2 + 1 is at most A002313(n), the n-th prime not congruent to 3 mod 4.

Original entry on oeis.org

1, 4, 9, 15, 22, 32, 41, 57, 74, 94, 120, 156, 192, 232, 278, 325, 381, 448, 521, 607, 704, 811

Offset: 1

Tomohiro Yamada, Apr 16 2017

In other words, x^2 + 1 is A002313(n)-smooth.
Størmer shows that the number of such integers is finite for any n.
a(n) <= 3^n - 2^n follows from Størmer's argument.
a(n) <= (2^n-1)*(A002313(n)+1)/2 is implicit in Lehmer 1964.
Luca 2004 determines all integers x such that x^2 + 1 is 100-smooth, which is pushed to 200 by Najman 2010.

D. H. Lehmer, On a problem of Størmer, Ill. J. Math., 8 (1964), 57--69.
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, Smooth values of some quadratic polynomials, Glas. Mat. 45 (2010), 347--355. Tables are available in the author's Home Page (gives all 811 numbers x such that x^2+1 has no prime factor greater than 197).
A. Schinzel, On two theorems of Gelfond and some of their applications, Acta Arithmetica 13 (1967-1968), 177--236.
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 (2), 48 pp.

Equivalents for x(x+1): A145604.

Cf. A002313, A014442, A185389, A223702, A285282, A379346 (first differences).

a(13)-a(22) added by Andrew Howroyd, Dec 22 2024

A252489 Index of the largest prime which divides n*(n+1).

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 2, 3, 5, 5, 6, 6, 4, 3, 7, 7, 8, 8, 4, 5, 9, 9, 3, 6, 6, 4, 10, 10, 11, 11, 5, 7, 7, 4, 12, 12, 8, 6, 13, 13, 14, 14, 5, 9, 15, 15, 4, 4, 7, 7, 16, 16, 5, 5, 8, 10, 17, 17, 18, 18, 11, 4, 6, 6, 19, 19, 9, 9, 20, 20, 21, 21, 12, 8, 8, 6

Offset: 1

M. F. Hasler, Jan 16 2015

Yields the row of A145605 in which n appears, and also the first row of A138180 in which n appears.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002071, A145604, A138180, A145605, A002072, A074399, A061395.

A061395:= [1, seq(numtheory:-pi(max(numtheory:-factorset(n))), n=2..101)]:
zip(max,A061395[1..-2],A061395[2..-1]); # Robert Israel, Feb 12 2021

a[n_] := PrimePi[Max[FactorInteger[n][[-1, 1]], FactorInteger[n+1][[-1, 1]]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Dec 05 2023 *)

a(n)=primepi(vecmax(factor(n*(n+1))[,1]))

a(n) = pi(A074399(n)), where pi = A000720.

a(n) = max(A061395(n),A061395(n+1)). - Robert Israel, Feb 12 2021

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

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

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

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

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A285283 Number of integers x such that the greatest prime factor of x^2 + 1 is at most A002313(n), the n-th prime not congruent to 3 mod 4.

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