A002072 -id:A002072 - cp's OEIS frontend

A245486 Product of the greatest prime factor of n and the greatest prime factor of n+1.

2, 6, 6, 10, 15, 21, 14, 6, 15, 55, 33, 39, 91, 35, 10, 34, 51, 57, 95, 35, 77, 253, 69, 15, 65, 39, 21, 203, 145, 155, 62, 22, 187, 119, 21, 111, 703, 247, 65, 205, 287, 301, 473, 55, 115, 1081, 141, 21, 35, 85, 221, 689, 159, 33, 77, 133, 551, 1711, 295, 305

Offset: 1

Franklin T. Adams-Watters, Jul 23 2014

We take gpf(1) = 1 by convention.
Except for the initial 2, every member is in A006881.
2^n+1 is never divisible by 23, and when 2^n-1 is divisible by 23, it's also divisible by 89. So 46 cannot occur in the sequence. - Jack Brennen, Jul 23 2014
More generally, let m = A014664(i), i >= 2.  If m is odd, 2*A000040(i) occurs in the sequence iff A000040(i) = A006530(2^m-1), in which case it is a(2^m-1).  If m is even, 2*A000040(i) occurs in the sequence iff A000040(i) = A006530(2^(m/2)+1), in which case it is a(2^m). - Robert Israel, Jul 24 2014
If a(n) = prime(i)*prime(j), where i < j, then n <= A002072(j).  Using this, it can be shown that 3*89 does not occur in the sequence. - Robert Israel, Jul 24 2014
This sequence has an infinite limit; equivalently, each value in A006881 occurs only finitely many times in it. See A002072 for references.

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

Cf. A000040, A006530, A006881, A002072, A014664.

gpf:= n -> max(numtheory:-factorset(n)):
gpf(1):= 1:
seq(gpf(n)*gpf(n+1),n=1..100); # Robert Israel, Jul 24 2014

gpf[n_] := FactorInteger[n][[-1, 1]]; f[n_] := gpf[n] gpf[n + 1]; Array[f, 60] (* Robert G. Wilson v, Jul 23 2014 *)
Times@@@Partition[Table[FactorInteger[n][[-1,1]],{n,100}],2,1] (* Harvey P. Dale, Sep 24 2017 *)

gpf(n)=my(ps);if(n<=1,n,ps=factor(n)[,1]~;ps[#ps])
a(n) = gpf(n)*gpf(n+1)

a(n) = A006530(n) * A006530(n+1).

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

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

A228610 Numbers k such that the largest consecutive pair of prime(k)-smooth integers is the same as the largest consecutive pair of prime(k-1)-smooth integers.

Original entry on oeis.org

9, 19, 23, 24, 25, 26

Offset: 1

Don N. Page, Dec 18 2013

For each such k = a(n), the smallest superparticular ratio R = m/(m-1) such that R factors into primes less than or equal to prime(k) have all of these prime factors strictly less than prime(k).

k = a(n) here are the values of k that make a(k) = a(k-1) in A002072 and also in A117581.

			For n = 1, k = a(1) = 9 gives prime(k) = 23 such that the largest consecutive pair of 23-smooth integers, (11859210,11859211), is the same as the largest consecutive pair of prime(k-1)-smooth integers (19-smooth integers).

Cf. A002072, A117581, A228611 gives prime(k) corresponding to k here.

A228611 Primes p such that the largest consecutive pair of p-smooth integers is the same as the largest consecutive pair of (p-1)-smooth integers.

Original entry on oeis.org

23, 67, 83, 89, 97, 101

Offset: 1

Don N. Page, Dec 18 2013

For each such prime p = a(n), the smallest superparticular ratio R = m/(m-1) such that R factors into primes less than or equal to p have all of these prime factors strictly less than p.

p = a(n) here equals prime(k) for the values of k that make a(k) = a(k-1) in A002072 and also in A117581.

			For n = 1, a(1) = 23 is a prime such that the largest consecutive pair of 23-smooth integers, (11859210,11859211), is the same as the largest consecutive pair of 22-smooth integers (or of 19-smooth integers, 19 being the next smaller prime).

Cf. A002072, A117581, A228610 gives the index of the prime that is a(n) here.

A250298 Primes p such that the largest integer m such that both m and m-1 factor into primes less than or equal to p is a perfect square, m = k^2.

Original entry on oeis.org

3, 5, 11, 13, 29, 53, 103

Offset: 1

Don N. Page, Jan 15 2015

List of primes p = A000040(i) such that A117581(i) (that is, A002072(i)+1) is a perfect square.

There are no analogous primes p < 107 for which m-1 defined above is a perfect square.

			p = 3 gives m = 3^2;
p = 5 gives m = 9^2;
p = 11 gives m = 99^2;
p = 13 gives m = 351^2;
p = 29 gives m = 13311^2;
p = 53 gives m = 1205645^2;
p = 103 gives m = 138982582999^2.

Cf. A000040, A002072, A117581.

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]

A085904 Numbers k such that k, k+1 and k+2 are 7-smooth, i.e., all prime divisors <= 7 (A002473).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 14, 48

Offset: 1

Amarnath Murthy, Jul 09 2003

No more terms < 3*10^7. Probably no more terms. - David Wasserman, Feb 10 2005
No more terms < 2^180. - Donovan Johnson, Oct 10 2012
There are no further terms: see A003032 (and maybe A002072). - Don Reble, Mar 14 2019

			48 is a member as 48, 49 and 50 have all prime divisors <= 7.

Cf. A002473.

mx=2^180+2; v=vector(4607193); c=0; for(e1=0, 180, x1=2^e1; for(e2=0, 113, x2=x1*3^e2; if(x2>mx, next(2)); for(e3=0, 77, x3=x2*5^e3; if(x3>mx, next(2)); for(e4=0, 64, x4=x3*7^e4; if(x4>mx, next(2)); c++; v[c]=x4)))); v=vecsort(v); for(i=1, 4607191, if(v[i+1]-v[i]==1, if(v[i+2]-v[i]==2, print1(v[i] ", ")))) /* Donovan Johnson, Oct 10 2012 */

Offset corrected and missing term added by Donovan Johnson, Oct 10 2012

A093876 Continued fraction expansion of fourth root of 9.1.

Original entry on oeis.org

1, 1, 2, 1, 4, 75656, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 136181, 1, 2, 4, 1, 64, 1, 1, 3602, 4, 1, 12, 7, 8, 1, 2, 4267, 2, 9, 1, 22, 1, 1, 1, 1, 1, 1, 1, 4841, 35, 1, 5, 5, 1, 262344, 1, 2, 3, 1, 3, 1, 15, 3, 1, 538, 2, 4, 1, 34, 2, 1, 6, 1, 1, 1, 2, 1, 2, 63, 1, 195, 4, 1, 2, 3, 1, 2, 2, 1, 4, 2, 1, 1, 5, 1, 9, 1, 3, 30, 2, 1, 2, 1, 15, 1, 7, 8, 6, 1, 1, 2, 7, 1, 1, 1, 3, 2, 3, 2, 13, 1, 5, 1, 13, 3, 19, 1, 1, 1, 2, 6, 1, 1, 3, 3, 1, 1, 1, 4, 1, 1, 2, 1, 2

Offset: 1

N. J. A. Sloane, based on a comment from Ed Pegg Jr, May 27 2004

Noteworthy for the large initial terms.

Comes from the best consecutive 19-smooth numbers, 11859210 and 11859211 (A002072), or 7*13*19^4 and 2*3^4*5*11^4, or 91 19^4 and 10 33^4. Which leads to 9.1 ~ (33/19)^4 and a spectacular convergent for the fourth root of 9.1. - Ed Pegg Jr, Mar 09 2005

Cf. A002072.

More terms from Klaus Brockhaus, May 27 2004

cp's OEIS Frontend

A245486 Product of the greatest prime factor of n and the greatest prime factor of n+1.

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A252489 Index of the largest prime which divides 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).

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A228610 Numbers k such that the largest consecutive pair of prime(k)-smooth integers is the same as the largest consecutive pair of prime(k-1)-smooth integers.

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A228611 Primes p such that the largest consecutive pair of p-smooth integers is the same as the largest consecutive pair of (p-1)-smooth integers.

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A250298 Primes p such that the largest integer m such that both m and m-1 factor into primes less than or equal to p is a perfect square, m = k^2.

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

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Mathematica

PARI

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

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Mathematica

A085904 Numbers k such that k, k+1 and k+2 are 7-smooth, i.e., all prime divisors <= 7 (A002473).

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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).