A085153 -id:A085153 - 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

A147298 Minimum of rad(m (n - m) n) for 0 < m < n, gcd(m,n) = 1, where rad(k) = A007947(k) = product of prime factors of k.

Original entry on oeis.org

2, 6, 6, 10, 30, 42, 14, 6, 30, 66, 66, 78, 182, 210, 30, 34, 102, 114, 190, 210, 462, 322, 138, 30, 130, 30, 42, 174, 870, 186, 30, 66, 510, 210, 210, 222, 1254, 546, 390, 246, 1722, 258, 946, 330, 690, 1410, 282, 42, 70, 510, 390, 742, 210, 330, 770, 570, 1218

Offset: 2

Artur Jasinski, Nov 05 2008

nonn

Function rad(k) is used in ABC conjecture applications.
For biggest values of function rad(m n (n - m)) see A147299.
For numbers m for which rad(m n (n - m)) reached minimal value see A147300.
For numbers m for which rad(m n (n - m)) reached maximal value see A147301.
Sequence in each value Log[n]/Log[A147298(n)] reached records see A147302.

Ivan Neretin, Table of n, a(n) for n = 2..1000

Cf. A007947, A085152, A085153, A147298-A147307.

A147298 := proc(n) local rad, g, L;
rad := n -> mul(k, k in numtheory:-factorset(n)):
g := (n, k) -> `if`(igcd(n, k) = 1, 1, infinity):
L := n -> [seq(g(n,k)*rad(n*k*(n-k)), k=1..n/2)]:
min(L(n)) end: seq(A147298(n), n=2..58); # Peter Luschny, Aug 06 2019

logmax = 0; aa = {}; bb = {}; cc = {}; dd = {}; ee = {}; ff = {}; gg \ = {}; Do[min = 10^100; max = 0; ile = 0; Do[If[GCD[m, n, n - m] == 1, ile = ile + 1; s = m n (n - m); k = FactorInteger[s]; g = 1; Do[g = g k[[p]][[1]], {p, 1, Length[k]}]; If[g > max, max = g; mmax = m]; If[g < min, min = g; mmin = m]], {m, 1, n - 1}]; AppendTo[aa, min]; AppendTo[bb, max]; AppendTo[cc, mmax]; AppendTo[dd, mmin]; AppendTo[gg, ile]; If[(Log[n]/Log[min]) > logmax, logmax = (Log[n]/Log[min]); AppendTo[ee, {N[logmax], n, mmin, min, mmax, max}]; Print[{N[logmax], n, mmin, min, mmax, max}]; AppendTo[ff, n]], {n, 2, 129}]; aa (*Artur Jasinski*)
Table[Min[Times @@ FactorInteger[#][[All, 1]] & /@ ((m = Select[Range[1, n - 1], GCD[n, #] == 1 &])*(n - m)*n)], {n, 2, 58}] (* Ivan Neretin, May 21 2015 *)

A147298(n)= local(m=n^2); for( a=1,n\2, gcd(a,n)>1 && next; A007947(n-a)*A007947(a)A007947(n-a)*A007947(a)); m*A007947(n)

A147307 Numbers A of the constrained search for ABC records described in A147306.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 5, 19, 41, 125, 23, 1, 1, 1, 95

Offset: 1

Artur Jasinski, Nov 09 2008

The sequences A147305, a(n) and A147307 are steered by searching for records in the ABC conjecture along increasing C confined as described in A147306, the main entry for these three sequences.

Cf. A085152, A085153, A147298-A147307.

a(n)+A147305(n) = A147306(n). gcd(a(n),A147305(n))=1.

Edited by R. J. Mathar, Aug 24 2009

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

A120498 Numbers C from the ABC conjecture.

Original entry on oeis.org

9, 32, 49, 64, 81, 125, 128, 225, 243, 245, 250, 256, 289, 343, 375, 512, 513, 539, 625, 676, 729, 961, 968, 1025, 1029, 1216, 1331, 1369, 1587, 1681, 2048, 2057, 2187, 2197, 2304, 2312, 2401, 2500, 2673, 3025, 3072, 3125, 3136, 3211, 3481, 3584, 3773, 3888

Offset: 1

R. J. Mathar, Aug 06 2006

nonn

C-values are not repeated: (A,B,C)=(13,243,256) and (A,B,C)=(81,175,256) are only represented once, by 256, in the list, for example.

			For A=1, B=63 and C=64, C=64 is in the list because 1 and 63 are coprime,
because the set of prime factors of 1, 63=3^2*7 and 64=2^6 has the product
of prime factors 3*2*7=42 and this product is smaller than 64.

R. J. Mathar and T. D. Noe, Table of n, a(n) for n=1..868
Zenon B. Batang, Squarefree integers and the abc conjecture, arXiv:2109.10226 [math.GM], 2021. See H(c) p. 3; this is the sequence of x such that H(x)>0.
Bart de Smit, Triples of small size [references the ABC@Home project which is inactive since 2015].
A. Granville and T. J. Tucker, It's As Easy As abc, Notices of the AMS, November 2002 (49:10), pp. 1224-1231.
Abderrahmane Nitaj, The ABC Conjecture Home Page.
Ivars Peterson, Math Trek, The Amazing ABC Conjecture [Internet Archive Wayback Machine]
Eric Weisstein's World of Mathematics, abc conjecture
Wikipedia, abc conjecture
OEIS Index entries for sequences related to the abc conjecture.

Cf. A007947, A085152, A085153.

Cf. A130510 (values of c in the list of "abc-hits").

rad[n_] := Times @@ First /@ FactorInteger[n]; isABC[a_, b_, c_] := (If[a + b != c || GCD[a, b] != 1, Return[0]]; r = rad[a*b*c]; If[r < c, Return[1], Return[0]]); isC[c_] := (For[a = 1, a <= Floor[c/2], a++, If[isABC[a, c - a, c] != 0, Return[1]]]; Return[0]); Select[Range[4000], isC[#] == 1 & ] (* Jean-François Alcover, Jun 24 2013, translated and adapted from Pari *)

isABC(a,b,c)={ a+b==c && gcd(a,b)==1 && A007947(a*b*c)M. F. Hasler, Jan 16 2015
isC(c)={ for(a=1, floor(c/2), if( isABC(a,c-a,c), return(1) )); return(0); }
{ for(n=1,6000, if( isC(n), print1(n,","))) }

is_A120498(c)={for(a=1,c\2, gcd(a,c-a)==1 && A007947(a*(c-a)*c)M. F. Hasler, Jan 16 2015

from itertools import count, islice
from math import prod, gcd
from sympy import primefactors
def A120498_gen(startvalue=1): # generator of terms >= startvalue
    for c in count(max(startvalue,1)):
        pc = set(primefactors(c))
        for a in range(1,(c>>1)+1):
            b = c-a
            if gcd(a,b)==1 and c>prod(set(primefactors(a))|set(primefactors(b))|pc):
                yield c
                break
A120498_list = list(islice(A120498_gen(),30)) # Chai Wah Wu, Oct 19 2023

A+B=C; gcd(A,B)=1; A007947(A*B*C) < C.

A147638 The numbers B associated with the search for records in the ABC conjecture constrained as described in A147639.

Original entry on oeis.org

3, 7, 15, 27, 63, 125, 1701

Offset: 1

Artur Jasinski, Nov 09 2008

The standard way to search for records in the ABC conjecture is to run with the C parameter through all the integers A000027. If this search space is diluted by admitting only powers of 2 as in A147639, the sequence of records changes. This sequence here lists the B such that the triples (A=A147640(n), B=a(n), C=A147639(n)) locate records for this search restricting C to powers of 2.

Abderrahmane Nitaj, The ABC conjecture homepage

Cf. A085152, A085153, A147298-A147307, A147638-A147643.

Definition and commend edited by R. J. Mathar, Aug 28 2009

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

A147300 a(n) = smallest value of parameter m such that the function rad(mn(n - m)) has minimal value n=2,3,4,..., 0 < m < n where the function rad(k) (also called radical(k)) is the product of distinct prime divisors of k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 2, 1, 2, 1, 4, 5, 1, 9, 3, 1, 1, 11, 7, 1, 9, 1, 16, 1, 1, 1, 2, 1, 1, 1, 1, 25, 4, 5, 1, 1, 25, 9, 27, 1, 1, 1, 1, 1, 1, 1, 3, 1, 5, 1, 7, 1, 1, 25, 11, 1, 13, 1, 4, 1, 1, 1, 2, 1, 4, 5, 23, 7, 8, 1, 27, 11, 1, 13, 14, 1, 1, 17, 1, 1

Offset: 2

Artur Jasinski, Nov 05 2008

nonn

The function rad(k) is used in ABC conjecture applications.
For smallest values of the function rad(m n (n - m)) see A147298.
For the largest values of the function rad(m n (n - m)) see A147299.
For numbers m at which rad(m*n*(n - m)) reaches minimal value see A147300.
For numbers m at which rad(m*n*(n - m)) reaches maximal value see A147301.
For sequence in which each value log(n)/log(A147298(n)) reaches records see A147302.

Cf. A085152, A085153, A147298-A147307.

logmax = 0; aa = {}; bb = {}; cc = {}; dd = {}; ee = {}; ff = {}; gg \ = {}; Do[min = 10^100; max = 0; ile = 0; Do[If[GCD[m, n, n - m] == 1, ile = ile + 1; s = m n (n - m); k = FactorInteger[s]; g = 1; Do[g = g k[[p]][[1]], {p, 1, Length[k]}]; If[g > max, max = g; mmax = m]; If[g < min, min = g; mmin = m]], {m, 1, n - 1}]; AppendTo[aa, min]; AppendTo[bb, max]; AppendTo[cc, mmax]; AppendTo[dd, mmin]; AppendTo[gg, ile]; If[(Log[n]/Log[min]) > logmax, logmax = (Log[n]/Log[min]); AppendTo[ee, {N[logmax], n, mmin, min, mmax, max}]; Print[{N[logmax], n, mmin, min, mmax, max}]; AppendTo[ff, n]], {n, 2, 129}]; dd (* Artur Jasinski *)

A147302 Numbers k where records occur in the expression log(k) / log(A147298(k)).

Original entry on oeis.org

2, 9, 81, 128, 2401, 4375, 6436343

Offset: 1

Artur Jasinski, Nov 05 2008

Numbers a(n) such that a(n)/R(m a(n)(a(n)-m)) > a(n-1)/R(g a(n-1)(a(n-1)-g)) 0 < m < a(n) and 0 < g < a(n-1).
This sequence is list of successive records in the abc conjecture.
No more terms up to 10^20.
For smallest values of function rad(m*n*(n-m)) see A147298.
For biggest values of function rad(m*n*(n-m)) see A147299.
For numbers m for which rad(m*n*(n-m)) reaches a minimal value see A147300.
For numbers m for which rad(m*n*(n-m)) reaches a maximal value see A147301.

Carlos Rivera, Puzzle 901. A question about consecutive integers, The Prime Puzzles and Problems Connection.

Cf. A085152, A085153, A147298-A147307.

logmax = 0; aa = {}; Do[min = 10^100; max = 0; ile = 0; Do[If[GCD[m, n, n - m] == 1, ile = ile + 1; s = m n (n - m); k = FactorInteger[s]; g = 1; Do[g = g k[[p]][[1]], {p, 1, Length[k]}]; If[g > max, max = g; mmax = m]; If[g < min, min = g; mmin = m]], {m, 1, n - 1}]; If[(Log[n]/Log[min]) > logmax, logmax = (Log[n]/Log[min]); AppendTo[aa, n]; Print[{N[logmax], n, mmin, min, mmax, max}]; AppendTo[ff, n]], {n, 2, 2500}]; aa

A147643 Numbers A associated with the records of the merit function of the ABC conjecture admitting only C which are powers of 23.

Original entry on oeis.org

7, 17, 162, 2

Offset: 1

Artur Jasinski, Nov 09 2008

If records of the ABC merit function are listed scanning only parameters C of the form 23^x as described in A147642, a(n) is the value of A associated with B=A147641(n) and C=A147642(n).

OEIS Index entries for sequences related to the abc conjecture.

Cf. A085152, A085153, A147298-A147307, A147638-A147642.

a(n) = A147642(n)-A147641(n).

Edited by M. F. Hasler, Jan 16 2015

cp's OEIS Frontend

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

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A147298 Minimum of rad(m (n - m) n) for 0 < m < n, gcd(m,n) = 1, where rad(k) = A007947(k) = product of prime factors of k.

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A147307 Numbers A of the constrained search for ABC records described in A147306.

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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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A120498 Numbers C from the ABC conjecture.

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A147638 The numbers B associated with the search for records in the ABC conjecture constrained as described in A147639.

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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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A147300 a(n) = smallest value of parameter m such that the function rad(m*n*(n - m)) has minimal value n=2,3,4,..., 0 < m < n where the function rad(k) (also called radical(k)) is the product of distinct prime divisors of k.

A147300 a(n) = smallest value of parameter m such that the function rad(mn(n - m)) has minimal value n=2,3,4,..., 0 < m < n where the function rad(k) (also called radical(k)) is the product of distinct prime divisors of k.