A140078 -id:A140078 - cp's OEIS frontend

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

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

A168628 Numbers n such that n and n+-1 have 4 distinct prime factors.

Original entry on oeis.org

37961, 44485, 45695, 50141, 51429, 55131, 55385, 61335, 63365, 64295, 67165, 68265, 68475, 70005, 70091, 71709, 72709, 76153, 80445, 81549, 81719, 82041, 84435, 85491, 86241, 90364, 95381, 97383, 98021, 99085, 99385, 99429, 99789, 100165, 100491, 100595

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 01 2009

nonn

Subsequence of A140078.

Cf. A140077, A168626

f[n_]:=Length[FactorInteger[n]]; lst={};Do[If[f[n]>=4&&f[n-1]>=4&&f[n+1]>=4,AppendTo[lst,n]],{n,9!}];lst
SequencePosition[PrimeNu[Range[110000]],{4,4,4}][[All,1]]+1 (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 27 2018 *)

is(n)=omega(n)==4 && omega(n+1)==4 && omega(n-1)==4 \\ Charles R Greathouse IV, Jan 25 2025

Corrected and extended by Harvey P. Dale, Apr 27 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.

A361796 Prime numbers preceded by two consecutive numbers which are products of four distinct primes (or tetraprimes).

Original entry on oeis.org

8647, 15107, 20407, 20771, 21491, 23003, 23531, 24767, 24971, 27967, 29147, 33287, 34847, 36779, 42187, 42407, 42667, 43331, 43991, 46807, 46867, 51431, 52691, 52747, 53891, 54167, 58567, 63247, 63367, 69379, 71711, 73607, 73867, 74167, 76507, 76631, 76847, 80447, 83591, 84247, 86243

Offset: 1

Massimo Kofler, Apr 26 2023

nonn

			8647 (prime), 8646 = 2*3*11*131 and 8645 = 5*7*13*19.
15107 (prime), 15106 = 2*7*13*83 and 15105 = 3*5*19*53.
20407 (prime), 20406 = 2*3*19*179 and 20405 = 5*7*11*53.

Robert Israel, Table of n, a(n) for n = 1..1914

Cf. A000040, A046386, A140078 and A362578.

N:= 10^5: # for terms <= N
TP:= NULL:
P:= select(isprime, [2,seq(i,i=3..N/30,2)]):
for i from 1 to nops(P) do
  for j from 1 to i-1 while P[i]*P[j] <= N/6 do
    for k from 1 to j-1 while P[i]*P[j]*P[k] <= N/2 do
      TP:= TP, op(select(`<=`,map(`*`,P[1..k-1],P[i]*P[j]*P[k]),N));
od od od:
TP:= {TP}:
TTP:= TP intersect map(`-`,TP,1):
sort(convert(select(isprime, map(`+`,TTP,2)),list)); # Robert Israel, Apr 28 2023

q[n_] := FactorInteger[n][[;; , 2]] == {1, 1, 1, 1}; Select[Prime[Range[10^4]], AllTrue[# - {1, 2}, q] &] (* Amiram Eldar, Apr 26 2023 *)

A168629 Numbers n such that n,n+1 and sum of this two numbers have at least 3 distinct prime factors.

Original entry on oeis.org

1105, 1130, 1462, 1644, 1742, 1767, 2014, 2222, 2232, 2260, 2337, 2365, 2397, 2464, 2541, 2667, 2684, 2697, 2702, 2755, 2821, 2914, 3074, 3115, 3195, 3289, 3332, 3477, 3484, 3514, 3552, 3619, 3657, 3685, 3782, 3783, 3842, 3965, 4014, 4088, 4122, 4147, 4277

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 01 2009

nonn

			1105 = 5*13*17, 1106 = 2*7*79, 1105 + 1106 = 2211 = 3*11*67.

Cf. A140077, A140078, A168626, A168628.

q:= n-> andmap(x-> nops(numtheory[factorset](x))>2, [n, n+1, 2*n+1]):
select(q, [$1..4600])[];  # Alois P. Heinz, Jun 29 2021

f[n_]:=Length[FactorInteger[n]]; lst={};Do[If[f[n]>=3&&f[n+1]>=3&&f[n+n+1]>=3,AppendTo[lst,n]],{n,8!}];lst

A168630 Numbers n such that n, n+1, and the sum of those two numbers each have 4 or more distinct prime factors.

Original entry on oeis.org

46189, 50634, 69597, 76797, 90117, 97954, 108205, 115804, 127347, 138957, 144627, 159340, 164020, 166022, 166497, 166705, 167205, 167485, 173194, 174454, 181670, 186294, 190014, 193154, 198789, 211029, 212134, 214225, 217217, 221815, 222547, 224146

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 01 2009

nonn

			FactorInteger[46189]=11*13*17*19, FactorInteger[46190]=2*5*31*149, FactorInteger[46189+46190]=3*7*53*83,..

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

Cf. A140077, A140078, A168626, A168628, A168629

g:= proc(n) option remember; nops(numtheory:-factorset(n))>=4 end proc:
filter:= n -> g(n) and g(n+1) and g(2*n+1):
select(filter, [$1..300000]); # Robert Israel, May 09 2018

f[n_]:=Length[FactorInteger[n]]; lst={};Do[If[f[n]>=4&&f[n+1]>=4&&f[n+n+1]>=4,AppendTo[lst,n]],{n,9!}];lst
Select[Range[225000],Min[Thread[PrimeNu[{#,#+1,2#+1}]]]>3&](* Harvey P. Dale, Nov 11 2017 *)

Definition modified and terms extended by Harvey P. Dale, Nov 11 2017

A336658 Numbers k such that k and k+1 both have the prime signature (2,1,1,1) (A189982).

Original entry on oeis.org

11780, 20349, 24794, 33579, 36764, 37323, 38324, 38675, 38709, 42020, 44505, 47564, 47684, 51204, 52155, 53955, 55419, 56259, 64844, 68475, 71379, 71994, 75284, 77714, 79134, 80475, 81548, 81549, 83420, 85491, 86715, 87164, 87380, 90524, 92364, 94940, 95403, 95589

Offset: 1

Amiram Eldar, Jul 28 2020

nonn

Goldston et al. (2011) proved that this sequence is infinite.

Some consecutive terms are (81548, 81549), (141218, 141219), (179828, 179829). - David A. Corneth, Jul 29 2020

			11780 is a term since 11780 = 2^2 * 5 * 19 * 31 and 11781 = 3^2 * 7 * 11 * 17.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Daniel A. Goldston, Sidney W. Graham, Janos Pintz, and Cem Y. Yıldırım, Small gaps between almost primes, the parity problem, and some conjectures of Erdős on consecutive integers, International Mathematics Research Notices, Vol. 2011, No. 7 (2011), pp. 1439-1450, preprint, arXiv:0803.2636 [math.NT], 2006.

Subsequence of A140078 and A274362.

Cf. A189982.

seqQ[n_] := Sort[FactorInteger[n][[;; , 2]]] == {1, 1, 1, 2}; Select[Range[10^5], seqQ[#] && seqQ[# + 1] &]

A362578 Prime numbers followed by two consecutive numbers which are products of four distinct primes (or tetraprimes).

Original entry on oeis.org

8293, 16553, 17389, 18289, 22153, 26893, 29209, 33409, 35509, 36293, 39233, 39829, 40493, 41809, 45589, 48109, 58393, 59629, 59753, 59981, 60493, 60913, 64013, 64921, 65713, 66169, 69221, 71329, 74093, 75577, 75853, 77689, 77933, 79393, 79609, 82913, 84533, 85853, 87589, 87701, 88681

Offset: 1

Massimo Kofler, Apr 25 2023

nonn

			8293 (prime), 8294 = 2*11*13*29 and 8295 = 3*5*7*79.
16553 (prime), 16554 = 2*3*31*89 and 16555 = 5*7*11*43.
17389 (prime), 17390 = 2*5*37*47 and 17391 = 3*11*17*31.

Cf. A000040, A046386, A140078 and A361796.

q[n_] := FactorInteger[n][[;; , 2]] == {1, 1, 1, 1}; Select[Prime[Range[10^4]], AllTrue[# + {1, 2}, q] &] (* Amiram Eldar, Apr 25 2023 *)

is(n) = (omega(n)==4) && (bigomega(n)==4); \\ A046386
isok(p) = isprime(p) && is(p+1) && is(p+2); \\ Michel Marcus, Apr 25 2023

cp's OEIS Frontend

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

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A168628 Numbers n such that n and n+-1 have 4 distinct prime factors.

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

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A361796 Prime numbers preceded by two consecutive numbers which are products of four distinct primes (or tetraprimes).

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A168629 Numbers n such that n,n+1 and sum of this two numbers have at least 3 distinct prime factors.

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Maple

Mathematica

A168630 Numbers n such that n, n+1, and the sum of those two numbers each have 4 or more distinct prime factors.

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Maple

Mathematica

Extensions

A336658 Numbers k such that k and k+1 both have the prime signature (2,1,1,1) (A189982).

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Mathematica

A362578 Prime numbers followed by two consecutive numbers which are products of four distinct primes (or tetraprimes).

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Keywords

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Mathematica

PARI