A093548 -id:A093548 - cp's OEIS frontend

A140078 Numbers k such that k and k+1 have 4 distinct prime factors.

7314, 8294, 8645, 9009, 10659, 11570, 11780, 11934, 13299, 13629, 13845, 14420, 15105, 15554, 16554, 16835, 17204, 17390, 17654, 17765, 18095, 18290, 18444, 18920, 19005, 19019, 19095, 19227, 20349, 20405, 20769, 21164, 21489, 21735

Offset: 1

Artur Jasinski, May 07 2008

nonn

Goldston, Graham, Pintz, & Yildirim prove that this sequence is infinite. - Charles R Greathouse IV, Jun 02 2016

The subsequence of terms where k and k+1 are also squarefree is A318896. - R. J. Mathar, Jul 15 2023

David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Rev. ed. 1997), p. 161 (entry for 7314).

Seiichi Manyama, Table of n, a(n) for n = 1..10000
D. A. Goldston, S. W. Graham, J. Pintz and C. Y. Yildirim, Small gaps between almost primes, the parity problem and some conjectures of Erdos on consecutive integers, arXiv:0803.2636 [math.NT], 2008.

Similar sequences with k distinct prime factors: A074851 (k=2), A140077 (k=3), this sequence (k=4), A140079 (k=5).
Cf. A093548.
Equals A321504 \ A321494.

a = {}; Do[If[Length[FactorInteger[n]] == 4 && Length[FactorInteger[n + 1]] == 4, AppendTo[a, n]], {n, 1, 100000}]; a (* Artur Jasinski, May 07 2008 *)
Transpose[Position[Partition[PrimeNu[Range[20000]],2,1],?(#[[1]] == #[[2]] == 4&),{1},Heads->False]][[1]] (* _Harvey P. Dale, Jun 21 2013 *)
SequencePosition[PrimeNu[Range[22000]],{4,4}][[;;,1]] (* Harvey P. Dale, Jun 20 2024 *)

isok(n) = (omega(n)==4) && (omega(n+1)==4); \\ Michel Marcus, Sep 04 2015

{k: k in A033993 and k+1 in A033993}. - R. J. Mathar, Jul 19 2023

Link provided by Harvey P. Dale, Jun 21 2013

A115186 Smallest number m such that m and m+1 have exactly n prime factors (counted with multiplicity).

Original entry on oeis.org

2, 9, 27, 135, 944, 5264, 29888, 50624, 203391, 3290624, 6082047, 32535999, 326481920, 3274208000, 6929459199, 72523096064, 37694578688, 471672487935, 11557226700800, 54386217385983, 50624737509375, 275892612890624, 4870020829413375, 68091093855502335, 2280241934368767, 809386931759611904, 519017301463269375

Offset: 1

Reinhard Zumkeller, Jan 16 2006

nonn

A001222(a(n)) = A001222(a(n)+1) = n: subsequence of A045920.
a(16) > 4*10^10. - Martin Fuller, Jan 17 2006
a(n) <= A093548(n) <= A052215(n). - Zak Seidov, Jan 16 2015
Apparently, 4*a(n)+2 is the least number k such that k-2 and k+2 have exactly n+2 prime factors, counted with multiplicity. - Hugo Pfoertner, Apr 02 2024

			a(10) = 3290624 = 6427 * 2^9, 3290624+1 = 13 * 5^5 * 3^4:
A001222(3290624) = A001222(3290625) = 10.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 135, p. 46, Ellipses, Paris 2008.

Cf. A001222, A045920, A052215, A093548.

Equivalent sequences for longer runs: A113752 (3), A356893 (4).

f:= proc(n) uses priqueue; local t,x,p,i;
    initialize(pq);
    insert([-3^n, 3$n], pq);
    do
      t:= extract(pq);
      x:= -t[1];
      if numtheory:-bigomega(x-1)=n then return x-1
      elif numtheory:-bigomega(x+1)=n then return x
      fi;
      p:= nextprime(t[-1]);
      for i from n+1 to 2 by -1 while t[i] = t[-1] do
        insert([t[1]*(p/t[-1])^(n+2-i), op(t[2..i-1]), p$(n+2-i)], pq)
      od;
    od
end proc:
seq(f(i),i=1..27); # Robert Israel, Sep 30 2024

a(13)-a(15) from Martin Fuller, Jan 17 2006
a(16)-a(17) from Donovan Johnson, Apr 08 2008
a(18)-a(22) from Donovan Johnson, Jan 21 2009
a(23)-a(25) from Donovan Johnson, May 25 2013
a(26)-a(27) from Robert Israel, Sep 30 2024

A140079 Numbers n such that n and n+1 have 5 distinct prime factors.

Original entry on oeis.org

254540, 310155, 378014, 421134, 432795, 483405, 486590, 486794, 488565, 489345, 507129, 522444, 545258, 549185, 558789, 558830, 567644, 577940, 584154, 591260, 598689, 627095, 634809, 637329, 663585, 666995, 667029, 678755, 687939, 690234

Offset: 1

Artur Jasinski, May 07 2008

nonn

For the smallest number r such that r and r+1 have n distinct prime factors, see A093548.
Goldston, Graham, Pintz, & Yildirim prove that this sequence is infinite. - Charles R Greathouse IV, Jun 02 2016
Subsequence of the variant A321505 defined with "at least 5" instead of "exactly 5" distinct prime factors. See A321495 for the differences. - M. F. Hasler, Nov 12 2018
The subset of numbers where n and n+1 are also squarefree gives A318964. - R. J. Mathar, Jul 15 2023

Seiichi Manyama, Table of n, a(n) for n = 1..10000
D. A. Goldston, S. W. Graham, J. Pintz, C. Y. Yildirim., Small gaps between almost primes, the parity problem and some conjectures of Erdos on consecutive integers, arXiv:0803.2636 [math.NT], 2008.

Cf. A074851, A140077, A140078, A273897.
Cf. A093548.
Equals A321505 \ A321495.

a = {}; Do[If[Length[FactorInteger[n]] == 5 && Length[FactorInteger[n + 1]] == 5, AppendTo[a, n]], {n, 1, 100000}]; a (*Artur Jasinski*)
Transpose[SequencePosition[Table[If[PrimeNu[n]==5,1,0],{n,700000}],{1,1}]][[1]] (* The program uses the SequencePosition function from Mathematica version 10 *) (* Harvey P. Dale, Jul 25 2015 *)

is(n)=omega(n)==5 && omega(n+1)==5 \\ Charles R Greathouse IV, Jun 02 2016

{k: k in A051270 and k+1 in A051270}. - R. J. Mathar, Jul 19 2023

A074851 Numbers k such that k and k+1 both have exactly 2 distinct prime factors.

Original entry on oeis.org

14, 20, 21, 33, 34, 35, 38, 39, 44, 45, 50, 51, 54, 55, 56, 57, 62, 68, 74, 75, 76, 85, 86, 87, 91, 92, 93, 94, 95, 98, 99, 111, 115, 116, 117, 118, 122, 123, 133, 134, 135, 141, 142, 143, 144, 145, 146, 147, 152, 158, 159, 160, 161, 171, 175, 176, 177, 183, 184

Offset: 1

Benoit Cloitre, Sep 10 2002

Subsequence of A006049. - Michel Marcus, May 06 2016

			20=2^2*5 21=3*7 hence 20 is in the sequence.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A006049, A006549, A001221.
Analogous sequences for m distinct prime factors: this sequence (m=2), A140077 (m=3), A140078 (m=4), A140079 (m=5), A273879 (m=6).
Cf. A093548.
Equals A255346 \ A321502.

Filtered([1..200],n->[Size(Set(Factors(n))),Size(Set(Factors(n+1)))]=[2,2]); # Muniru A Asiru, Dec 05 2018

[n: n in [2..200] | #PrimeDivisors(n) eq 2 and #PrimeDivisors(n+1) eq 2]; // Vincenzo Librandi, Dec 05 2018

Flatten[Position[Partition[Table[If[PrimeNu[n]==2,1,0],{n,200}],2,1],{1,1}]] (* Harvey P. Dale, Mar 12 2015 *)

isok(n) = (omega(n) == 2) && (omega(n+1) == 2); \\ Michel Marcus, May 06 2016

import sympy
from sympy.ntheory.factor_ import primenu
for n in range(1,200):
    if primenu(n)==2 and primenu(n+1)==2:
        print(n, end=', '); # Stefano Spezia, Dec 05 2018

a(n) seems to be asymptotic to c*n*log(n)^2 with c=0.13...

{k: A001221(k) = A001221(k+1) = 2}. - R. J. Mathar, Jul 18 2023

A093550 a(n) is the smallest number m such that each of the numbers m-1, m and m+1 is a product of n distinct primes.

Original entry on oeis.org

34, 1310, 203434, 16467034, 1990586014, 41704979954, 102099792179230, 22192526378762466

Offset: 2

Farideh Firoozbakht, Apr 07 2004, corrected Aug 26 2006

Each term of this sequence is of the form 4k+2.

			a(5)=16467034 because each of the three numbers 16467034-1, 16467034 & 16467034+1 are products of 5 distinct primes (16467033=3*11*17*149*197, 16467034=2*19*23*83*227, 16467035=5*13*37*41*167) and 16467034 is the smallest such number.

Jacques Tramu, Puzzle 371

Cf. A093549, A052215, A093548, A248201, A248202, A248203, A248204, A259349, A259350, A259801.

a[n_] := a[n] = (For[m=1, !(Length[FactorInteger[4m+1]]==n && SquareFreeQ[4m+1] && Length[FactorInteger[4m+2]]==n && SquareFreeQ[4m+2] && Length[FactorInteger[4m+3]]==n && SquareFreeQ[4m+3]), m++ ]; 4m+2); Table[Print[a[n]]; a[n], {n, 2, 6}] (* updated by Jean-François Alcover, Jul 04 2013 *)

a(n)={my(m=1);while(!(issquarefree(m-1)&&issquarefree(m)&&issquarefree(m+1)&&omega(m-1)==n&&omega(m)==n&&omega(m+1)==n),m++);return(m);} main(size)={my(n);return(vector(size,n,a(n+1)));} /* Anders Hellström, Jul 14 2015 */

a(7) added from Jacques Tramu's web site by Farideh Firoozbakht, Aug 26 2006
a(8) from Donovan Johnson, Oct 27 2008
a(9) from James G. Merickel, Jul 24 2015

A052215 a(n) = smallest number m such that m and m+1 are the product of exactly n distinct primes.

Original entry on oeis.org

2, 14, 230, 7314, 378014, 11243154, 965009045, 65893166030, 5702759516090, 605247139068494, 78971815814237709, 22593106657425552170

Offset: 1

Erich Friedman, Jan 29 2000

Prime factors may not be repeated in m and m+1. The difference between this sequence and A093548 is that in the latter, prime factors may be repeated. So the present sequence imposes more stringent conditions than A093548, hence a(n) >= A093548(n). - N. J. A. Sloane, Nov 21 2015
A115186(n) <= A093548(n) <= a(n). - Zak Seidov, Jan 16 2015
2^63 < a(12) <= 22593106657425552170. [Donovan Johnson, Oct 23 2008]
a(12) confirmed to be the upper limit of the range above. - Bert Dobbelaere, Jun 27 2019

			14 and 15 are both the product of 2 primes.
230 is the 3rd entry because we have (230=2*5*23, 231=3*7*11).

Cf. A093548 (another version), A093549, A093550, A115186, A318896.

Subsequence of A005117.

More terms from Naohiro Nomoto, Jul 08 2001
a(7) from Farideh Firoozbakht, Apr 06 2004
a(8)-a(10) from Martin Fuller, Jan 17 2006
a(11) from Donovan Johnson, Oct 23 2008
a(12) from Bert Dobbelaere, Jun 27 2019

A273879 Numbers k such that k and k+1 have 6 distinct prime factors.

Original entry on oeis.org

11243154, 13516580, 16473170, 16701684, 17348330, 19286805, 20333495, 21271964, 21849905, 22054515, 22527141, 22754589, 22875489, 24031370, 25348070, 25774329, 28098245, 28618394, 28625960, 30259229, 31846269, 32642805

Offset: 1

Charles R Greathouse IV, Jun 02 2016

nonn

Goldston, Graham, Pintz, & Yildirim prove that this sequence is infinite (Theorem 2).

			13516580 = 2^2 * 5 * 7 * 11 * 67 * 131 and 13516581 = 3 * 13 * 17 * 19 * 29 * 37 so 13516580 is in this sequence.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
D. A. Goldston, S. W. Graham, J. Pintz, C. Y. Yildirim, Small gaps between almost primes, the parity problem and some conjectures of Erdős on consecutive integers, arXiv:0803.2636 [math.NT], 2008.
D. A. Goldston, S. W. Graham, J. Pintz, C. Y. Yildirim, Small gaps between almost primes, the parity problem and some conjectures of Erdős on consecutive integers, International Mathematics Research Notices 7 (2011), pp. 1439-1450.

Numbers k such that k and k+1 have j distinct prime factors: A006549 (j=1, apart from the first term), A074851 (j=2), A140077 (j=3), A140078 (j=4), A140079 (j=5).

Cf. A006049, A093548.

SequencePosition[PrimeNu[Range[3265*10^4]],{6,6}][[All,1]] (* Harvey P. Dale, Nov 20 2021 *)

PARI
```
is(n)=omega(n)==6 && omega(n+1)==6
```

a(1) = A138206(2). - R. J. Mathar, Jul 15 2023

{k: k in A074969 and k+1 in A074969.} - R. J. Mathar, Jul 19 2023

A093549 a(n) is the smallest number m such that each of the numbers m-1, m and m+1 has n distinct prime divisors.

Original entry on oeis.org

3, 21, 645, 37961, 1042405, 323567035, 30989984675, 10042712381261

Offset: 1

Farideh Firoozbakht, Apr 07 2004

a(n) <= A093550(n) since here the factors do not occur necessarily to the first power, e.g. a(2)-1 = 20 = 2^2*5, therefore A093550(2) > a(2). - M. F. Hasler, May 20 2014

			a(3)=645 because 644=2^2*7*23; 645=3*5*43; 646=2*17*19 and 645 is the smallest number m such that each of the numbers m-1, m and m+1 has 3 distinct prime divisors.

Cf. A093550, A052215, A093548.

a[n_] := (For[m=2, !(Length[FactorInteger[m-1]]==n && Length[FactorInteger[m]]==n&&Length[FactorInteger[m+1]]==n), m++ ];m);Do[Print[a[n]], {n, 7}]

a(n,m=2)=until(,for(k=-1,1,omega(m-k)!=n&&(m+=2-k)&&next(2));return(m)) \\ M. F. Hasler, May 20 2014

a[n_] := (For[m=2, !(Length[FactorInteger[m-1]]==n && Length[FactorInteger[m]]==n&&Length[FactorInteger[m+1]]==n), m++ ];m)

a(7) from Donovan Johnson, Apr 07 2008

a(8) from Donovan Johnson, Jan 15 2009

A321489 Numbers m such that both m and m+1 have at least 7 distinct prime factors.

Original entry on oeis.org

965009045, 1068044054, 1168008204, 1177173074, 1209907985, 1218115535, 1240268490, 1338753129, 1344185205, 1408520805, 1477640450, 1487720234, 1509981395, 1663654629, 1693460405, 1731986894, 1758259425, 1819458354, 1821278459, 1826445984, 1857332840

Offset: 1

Amiram Eldar and M. F. Hasler, Nov 12 2018

nonn

The first 300 terms of this sequence are such that m and m+1 both have exactly 7 prime divisors. See A321497 for the terms m such that m or m+1 has more than 7 prime factors: the smallest such term is 5163068910.
Numbers m and m+1 can never have a common prime factor (consider them mod p), therefore the terms are > sqrt(p(7+7)#) = A003059(A002110(7+7)). (Here we see that sqrt(p(7+8)#) is a more realistic estimate of a(1), but for smaller values of k we may have sqrt(p(2k+1)#) > m(k) > sqrt(p(2k)#), where m(k) is the smallest of two consecutive integers each having at least k prime divisors. For example, A321503(1) < sqrt(p(3+4)#) ~ A321493(1).)
From M. F. Hasler, Nov 28 2018: (Start)
The first 100 terms and beyond are all congruent to one of {14, 20, 35, 49, 50, 69, 84, 90, 104, 105, 110, 119, 125, 129, 134, 140, 144, 170, 174, 189, 195} mod 210. Here, 35, 195, 189, 14 140, 20 and 174 (in order of decreasing frequency) occur between 6 and 13 times, and {49, 50, 110, 129, 134, 144, 170} occur only once.
However, as observed by Charles R Greathouse IV, one can construct a term of this sequence congruent to any given m > 0, modulo any given n > 0.
The first terms of this sequence which are multiples of 210 are in A321497. An example of a term that is a multiple of 210 but not in A321497 is 29759526510, due to Charles R Greathouse IV. Such examples can be constructed by solving A*210 + 1 = B for A having 3 distinct prime factors not among {2, 3, 5, 7}, B having 7 distinct prime factors and gcd(B, 210*A) = 1. (End)

			a(1) = 5 * 7 * 11 * 13 * 23 * 83 * 101, a(1)+1 = 2 * 3 * 17 * 29 * 41 * 73 * 109.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 102 terms from M. F. Hasler)

Cf. A255346, A321503 .. A321506 (analog for k = 2, ..., 6 prime divisors).
Cf. A321502, A321493 .. A321497 (m and m+1 have at least but not both exactly k = 2, ..., 7 prime divisors).
Cf. A074851, A140077, A140078, A140079 (m and m+1 both have exactly k = 2, 3, 4, 5 prime divisors).
Cf. A006049, A006549, A093548.
Cf. A002110.

Select[Range[36000000], PrimeNu[#] > 6 && PrimeNu[# + 1] > 6 &]

is(n)=omega(n)>6&&omega(n+1)>6
A321489=List();for(n=965*10^6,1.8e9,is(n)&&listput(A321489,n))

a(n) ~ n. - Charles R Greathouse IV, Nov 29 2018

A321502 Numbers m such that m and m+1 have at least 2, but m or m+1 has at least 3 prime divisors.

Original entry on oeis.org

65, 69, 77, 84, 90, 104, 105, 110, 114, 119, 129, 132, 140, 153, 154, 155, 164, 165, 170, 174, 182, 185, 186, 189, 194, 195, 203, 204, 209, 219, 220, 221, 230, 231, 234, 237, 245, 246, 252, 254, 258, 259, 260, 264, 265, 266, 272, 273, 275, 279, 284, 285, 286, 290, 294, 299, 300, 305

Offset: 1

M. F. Hasler, Nov 27 2018

nonn

Since m and m+1 cannot have a common factor, m(m+1) has at least 2+3 prime divisors (= distinct prime factors), whence m+1 > sqrt(primorial(5)) ~ 48. It turns out that a(1)*(a(1)+1) = 2*3*5*11*13, i.e., the prime factor 7 is not present.

Cf. A321493, A321494, A321495, A321496, A321497 (analog for k = 3, ..., 7 prime divisors).
Cf. A074851, A140077, A140078, A140079 (m and m+1 have exactly k = 2, 3, 4, 5 prime divisors).
Cf. A255346, A321503 .. A321506, A321489 (m and m+1 have at least 2, ..., 7 prime divisors).
Cf. A006049, A006549, A093548.

select( is_A321502(n)=vecmax(n=[omega(n), omega(n+1)])>2&&vecmin(n)>1, [1..500])

Equals A255346 \ A074851.

cp's OEIS Frontend

A140078 Numbers k such that k and k+1 have 4 distinct prime factors.

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A115186 Smallest number m such that m and m+1 have exactly n prime factors (counted with multiplicity).

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A140079 Numbers n such that n and n+1 have 5 distinct prime factors.

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A074851 Numbers k such that k and k+1 both have exactly 2 distinct prime factors.

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A093550 a(n) is the smallest number m such that each of the numbers m-1, m and m+1 is a product of n distinct primes.

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A052215 a(n) = smallest number m such that m and m+1 are the product of exactly n distinct primes.

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A273879 Numbers k such that k and k+1 have 6 distinct prime factors.

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