A074969 -id:A074969 - cp's OEIS frontend

A364265 The first term in a chain of at least 3 consecutive numbers each with exactly 6 distinct prime factors (i.e., belonging to A074969).

323567034, 431684330, 468780388, 481098980, 577922904, 639336984, 715008644, 720990620, 726167154, 735965384, 769385252, 808810638, 822981560, 831034918, 839075510, 847765554, 879549670, 895723268, 902976710, 903293468, 904796814, 918520420, 940737005, 944087484, 982059364

Offset: 1

R. J. Mathar, Jul 16 2023

nonn

To distinguish this from A259349: "Numbers n with exactly k distinct prime factors" means numbers with A001221(n) = omega(n) = k, which specifies that in the prime factorization n = Product_{i>=1} p_i^(e_i), e_i >= 1, the exponents are ignored, and only the size of the set of the (distinct) p_i is considered. In A259349, the numbers n are products of k distinct primes, which means in the prime factorization of n, all exponents e_i are equal to 1. (If all exponents e_i = 1, the n are squarefree, i.e., in A005117.) Rephrased: the n which are products of k distinct primes have A001221(n) = omega(n) = A001222(n) = bigomega(n) = k, whereas the n which have exactly k distinct prime factors are the superset of (weaker) requirement A001221(n) = omega(n) = k. - R. J. Mathar, Jul 18 2023

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A259349 (requires squarefree). Subsequence of A273879.
Cf. A364266 (5 distinct factors).
See also A001221, A001222, A005117.
Numbers divisible by d distinct primes: A246655 (d=1), A007774 (d=2), A033992 (d=3), A033993 (d=4), A051270 (d=5), A074969 (d=6), A176655 (d=7), A348072 (d=8), A348073 (d=9).

omega := proc(n)
    nops(numtheory[factorset](n)) ;
end proc:
for k from 1 do
    if omega(k) = 6 then
        if omega(k+1) = 6 then
            if omega(k+2) = 6 then
                print(k) ;
            end if;
        end if;
    end if;
end do:

upto(n) = {my(res = List(), streak = 0); forfactored(i = 2, n, if(#i[2]~ == 6, streak++; if(streak >= 3, listput(res, i[1] - 2)), streak = 0)); res} \\ David A. Corneth, Jul 18 2023

a(1) = A138206(3).

{k: A001221(k) = A001221(k+1) = A001221(k+2) = 6}.

More terms from David A. Corneth, Jul 18 2023

A364436 Numbers that begin a run of at least 4 consecutive integers having exactly 6 distinct prime factors each (i.e., belonging to A074969).

Original entry on oeis.org

7933641735, 9338016258, 9827010633, 10744118592, 10808993635, 10928652579, 13302330390, 15300915705, 16088310249, 16408242849, 18685633314, 18721086153, 19136152098, 19819102092, 20592248544, 20826707802, 21241193334, 21296349633, 21531380583, 21727956885, 21823418253

Offset: 1

David A. Corneth, Jul 24 2023

nonn

			7933641735 is in the sequence as it starts a run of at least 4 consecutive integers each of which has exactly 6 distinct prime factors.
That is, each of 7933641735 = 3 * 5 * 23 * 83 * 461 * 601,
7933641735 + 1 = 7933641736 = 2^3 * 17 * 47 * 59 * 109 * 193,
7933641735 + 2 = 7933641737 = 7 * 29 * 31 * 41 * 97 * 317,
7933641735 + 3 = 2 * 3 * 11 * 89 * 563 * 2399 has 6 distinct prime factors.

Cf. A074969, A364265.

upto(n) = {my(res = List(), streak = 0); n+=3; forfactored(i = 1, n, if(omega(i[2]) == 6, streak++; if(streak >= 4, listput(res, i[1]-3)), streak = 0)); res}

More terms from Jinyuan Wang, Aug 12 2023

A067885 Products of exactly 6 distinct primes.

Original entry on oeis.org

30030, 39270, 43890, 46410, 51870, 53130, 62790, 66990, 67830, 71610, 72930, 79170, 81510, 82110, 84630, 85470, 91770, 94710, 98670, 99330, 101010, 102102, 103530, 106590, 108570, 110670, 111930, 114114, 115710, 117390, 122430, 123690, 124410, 125970, 128310

Offset: 1

Benoit Cloitre, Mar 02 2002

nonn

Subsequence of A074969. - R. J. Mathar, Nov 24 2009

Products of exactly k distinct primes, for k = 1 to 6: A000040, A006881. A007304, A046386, A046387, A067885.

Select[Range[125000],PrimeNu[#]==PrimeOmega[#]==6&] (* Harvey P. Dale, May 14 2014 *)

is(n)=factor(n)[,2]==[1,1,1,1,1,1]~ \\ Charles R Greathouse IV, Sep 14 2015

is(n)=omega(n)==6 && bigomega(n)==6 \\ Hugo Pfoertner, Dec 18 2018

list(lim)=lim\=1; my(v=List(), L1,L2,L3,L4,P4,P5); forprime(p=13,lim\2310, L1=lim\p; forprime(q=11,min(L1\210,p-2), L2=L1\q; forprime(r=7, min(L2\30,q-2), L3=L2\r; forprime(s=5,min(L3\6,r-2), L4=L3\s; P4=p*q*r*s; forprime(t=3, min(L4\2,s-2), P5=P4*t; forprime(u=2, min(L4\t,t-1), listput(v,P5*u))))))); Set(v) \\ Charles R Greathouse IV, Aug 27 2021

from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A067885(n):
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,6)))
    kmin, kmax = 0,1
    while f(kmax) > kmax:
        kmax <<= 1
    while kmax-kmin > 1:
        kmid = kmax+kmin>>1
        if f(kmid) <= kmid:
            kmax = kmid
        else:
            kmin = kmid
    return kmax # Chai Wah Wu, Aug 29 2024

{k: A001221(k) = A001222(k) = 6}. - R. J. Mathar, Jul 18 2023

A000977 Numbers that are divisible by at least three different primes.

Original entry on oeis.org

30, 42, 60, 66, 70, 78, 84, 90, 102, 105, 110, 114, 120, 126, 130, 132, 138, 140, 150, 154, 156, 165, 168, 170, 174, 180, 182, 186, 190, 195, 198, 204, 210, 220, 222, 228, 230, 231, 234, 238, 240, 246, 252, 255, 258, 260, 264, 266, 270, 273, 276, 280, 282, 285

Offset: 1

N. J. A. Sloane

a(n+1)-a(n) seems bounded and sequence appears to give n such that the number of integers of the form nk/(n+k) k>=1 is not equal to Sum_{ d | n} omega(d) (i.e., n such that A062799(n) is not equal to A063647(n)). - Benoit Cloitre, Aug 27 2002

The first differences are bounded: clearly a(n+1) - a(n) <= 30. - Charles R Greathouse IV, Dec 19 2011

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A000961, A007774, A033992, A033993, A051270.

Complement of A070915.

a000977 n = a000977_list !! (n-1)
a000977_list = filter ((> 2) . a001221) [1..]
-- Reinhard Zumkeller, May 03 2013

A000977 := proc(n)
if (nops(numtheory[factorset](n)) >= 3) then
   RETURN(n)
fi: end:  seq(A000977(n), n=1..500); # Jani Melik, Feb 24 2011

DeleteCases[Table[If[Count[PrimeQ[Divisors[i]], True] >= 3, i, 0], {i, 1, 274}], 0]
Select[Range[300], PrimeNu[#] >= 3 &] (* Paolo Xausa, Mar 28 2024 *)

is(n)=omega(n)>2 \\ Charles R Greathouse IV, Dec 19 2011

a(n) = n + O(n log log n / log n). - Charles R Greathouse IV, Dec 19 2011 A001221(a(n)) > 2. - Reinhard Zumkeller, May 03 2013

A033992 UNION A033993 UNION A051270 UNION A074969 UNION A176655 UNION ... - R. J. Mathar, Dec 05 2016

More terms from Vit Planocka (planocka(AT)mistral.cz), Sep 17 2002

A030231 Numbers with an even number of distinct prime factors.

Original entry on oeis.org

1, 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 33, 34, 35, 36, 38, 39, 40, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 65, 68, 69, 72, 74, 75, 76, 77, 80, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 98, 99, 100, 104, 106, 108, 111, 112, 115, 116, 117, 118, 119

Offset: 1

David W. Wilson

Gcd(A008472(a(n)), A007947(a(n)))=1; see A014963. - Labos Elemer, Mar 26 2003
Superset of A007774. - R. J. Mathar, Oct 23 2008
A076479(a(n)) = +1. - Reinhard Zumkeller, Jun 01 2013
Union of the rows of A125666 with even indices. - R. J. Mathar, Jul 19 2023

T. D. Noe, Table of n, a(n) for n = 1..1000
H. Helfgott and A. Ubis, Primos, paridad y análisis, arXiv:1812.08707 [math.NT], Dec. 2018.

Cf. A028260, A030230, A123066.
Cf. A008472, A007947, A014963.
Cf. A007774, A076479.
Cf. A008683, A000005, A001221, A023900.

a030231 n = a030231_list !! (n-1)
a030231_list = filter (even . a001221) [1..]
-- Reinhard Zumkeller, Mar 26 2013

Select[Range[200],EvenQ[PrimeNu[#]]&] (* Harvey P. Dale, Jun 22 2011 *)

j=[]; for(n=1,200,x=omega(n); if(Mod(x,2)==0,j=concat(j,n))); j

is(n)=omega(n)%2==0 \\ Charles R Greathouse IV, Sep 14 2015

From Benoit Cloitre, Dec 08 2002: (Start)
k such that Sum_{d|k} mu(d)*A000005(d) = (-1)^omega(k) = +1 where mu(d)=A008683(d), and omega(d)=A001221(d).
k such that A023900(k) > 0. (End)
Union of A007774, A033993, A074969,... - R. J. Mathar, Jul 22 2025

Corrected by Dan Pritikin (pritikd(AT)muohio.edu), May 29 2002

A125666 Table read by ascending antidiagonals: n-th row of table consists of the positive integers divisible by exactly n distinct primes.

Original entry on oeis.org

2, 6, 3, 30, 10, 4, 210, 42, 12, 5, 2310, 330, 60, 14, 7, 30030, 2730, 390, 66, 15, 8, 510510, 39270, 3570, 420, 70, 18, 9, 9699690, 570570, 43890, 3990, 462, 78, 20, 11, 223092870, 11741730, 690690, 46410, 4290, 510, 84, 21, 13, 6469693230, 281291010

Offset: 1

Leroy Quet, Jan 29 2007

Concatenated sequence is a permutation of the integers >= 2.

The chosen encoding of the table by *rising* antidiagonals is contrary to the OEIS standard which rather expects falling antidiagonals: as a consequence, displaying this sequence as a table (2nd link after the list of terms above) will list the integers with given number of prime divisors in columns rather than rows. - M. F. Hasler, Jun 06 2024

			The table begins:
  n\k|     1     2    3    4    5    6  ...
  ---+-------------------------------------
   1 |     2,    3,   4,   5,   7,   8, ...
   2 |     6,   10,  12,  14,  15, ...
   3 |    30,   42,  60,  66, ...
   4 |   210,  330, 390, ...
   5 |  2310, 2730, ...
   6 | 30030,  ...
  ...|   ...

Cf. A001221, A002110 (col 1), A246655 (row 1), A007774 (row 2), A033992 (row 3), A033993 (row 4), A051270 (row 5), A074969 (row 6), A176655 (row 7), A348072 (row 8), A348073 (row 9), A073329 (diag), compare to A048692.

f[n_, m_] := f[n, m] = Block[{c = m, k = If[m == 1, Product[Prime[i], {i, n}], f[n, m - 1] + 1]},While[Length@FactorInteger[k] != n, k++ ];k];Table[f[d - m + 1, m], {d, 10}, {m, d}] // Flatten (* Ray Chandler, Feb 08 2007 *)

A125666(n, k=0)={if(k, for(m=vecprod(primes(n)), oo, omega(m)!=n || k-- || return(m)), A125666(A004736(n), A002260(n)))} \\ M. F. Hasler, Jun 06 2024

Extended by Ray Chandler, Feb 08 2007

A135956 Members of A050937 (nonprime Fibonacci numbers with prime index) with 5 or more distinct prime factors.

Original entry on oeis.org

322615043836854783580186309282650000354271239929, 1476475227036382503281437027911536541406625644706194668152438732346449273, 22334640661774067356412331900038009953045351020683823507202893507476314037053

Offset: 1

Artur Jasinski, Dec 08 2007

Conjecture: all numbers in this sequence are product of 5 or more sum of two squares

Cf. A000045, A001605, A050937, A075737, A090819, A134787, A134851, A134852, A135953, A135954, A135955, A135957.

k = {}; Do[If[ !PrimeQ[Fibonacci[Prime[n]]], c = Length[FactorInteger[Fibonacci[Prime[n]]]]; Print[n]; If[c > 4, Print[Fibonacci[Prime[n]]]; AppendTo[k, Fibonacci[Prime[n]]]]], {n, 1, 100}]; k

A050937 INTERSECT { A051270 UNION A074969 UNION ... } = A050937 MINUS {A135955 UNION A135954 UNION A135953}. - R. J. Mathar, Jun 09 2008

Edited by R. J. Mathar, Jun 09 2008

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

A046396 Palindromes which are the product of 6 distinct primes.

Original entry on oeis.org

222222, 282282, 474474, 555555, 606606, 646646, 969969, 2040402, 2065602, 2206022, 2417142, 2646462, 2673762, 2875782, 3262623, 3309033, 4179714, 4192914, 4356534, 4585854, 4912194, 5021205, 5169615, 5174715, 5578755

Offset: 1

Patrick De Geest, Jun 15 1998

The original definition "Palindromes with exactly 6 distinct prime factors" was misleading. For example, the number 414414 = 2 * 3^2 * 7 * 11 * 13 * 23 has exactly 6 distinct prime factors, although the factor 3 occurs twice. But the listed terms show that it is not in this sequence. See sequence A373466 for the variant corresponding to that definition. - M. F. Hasler, Jun 06 2024

Cf. A046332 (similar, but for 6 prime factors counted with multiplicity).
Cf. A002113 (palindromes), A067885 (products of 6 distinct primes).
Cf. A074969 (numbers having 6 distinct prime divisors).

Select[Range[6*10^6],#==IntegerReverse[#]&&PrimeNu[#]==PrimeOmega[#]==6&] (* The program uses the IntegerReverse function from Mathematica version 10 *) (* Harvey P. Dale, Mar 17 2016 *)

A046332_upto(N, start=1, num_fact=6)={ my(L=List()); while(N >= start = nxt_A002113(start), omega(start)==num_fact && issquarefree(start) && listput(L, start)); L} \\ M. F. Hasler, Jun 06 2024

Intersection of A002113 and A067885. - M. F. Hasler, Jun 06 2024

Name edited

A373466 Palindromes with exactly 6 distinct prime divisors.

Original entry on oeis.org

222222, 282282, 414414, 444444, 474474, 555555, 606606, 636636, 646646, 666666, 696696, 828828, 888888, 969969, 2040402, 2065602, 2141412, 2206022, 2343432, 2417142, 2444442, 2572752, 2646462, 2673762, 2747472, 2848482, 2875782, 2949492, 2976792

Offset: 1

M. F. Hasler, Jun 06 2024

The term "exactly" clarifies that we don't mean "at least". But the prime divisors may occur to higher powers in the factorization, cf. Examples.

This is different from A046396 which excludes nonsquarefree terms, i.e., terms where one or more of the distinct prime factors occur to a power greater than 1, as it is possible here, cf. Examples.

			a(1) = 222222 = 2 * 3 * 7 * 11 * 13 * 37 has exactly 6 distinct prime divisors.
a(3) = 414414 = 2 * 3^2 * 7 * 11 * 13 * 23 has 6 distinct prime divisors, even though the factor 3 occurs twice in the factorization.

Cf. A002113 (palindromes), A074969 (omega(.) = 6).

Cf. A046332 (same with bigomega = 6: prime factors counted with multiplicity), A046396 (similar, but squarefree terms only), A373465 (same with omega = 5), A373467 (same with bigomega = 7).

Select[Range[3000000],PalindromeQ[#]&&Length[FactorInteger[#]]==6&] (* James C. McMahon, Jun 08 2024 *)

A373466_upto(N, start=1, num_fact=6)={ my(L=List()); while(N >= start = nxt_A002113(start), omega(start)==num_fact && listput(L, start)); L}

Intersection of A002113 and A074969.

cp's OEIS Frontend

A364265 The first term in a chain of at least 3 consecutive numbers each with exactly 6 distinct prime factors (i.e., belonging to A074969).

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A364436 Numbers that begin a run of at least 4 consecutive integers having exactly 6 distinct prime factors each (i.e., belonging to A074969).

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A067885 Products of exactly 6 distinct primes.

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A000977 Numbers that are divisible by at least three different primes.

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A030231 Numbers with an even number of distinct prime factors.

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A125666 Table read by ascending antidiagonals: n-th row of table consists of the positive integers divisible by exactly n distinct primes.

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A135956 Members of A050937 (nonprime Fibonacci numbers with prime index) with 5 or more distinct prime factors.

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