A176655 -id:A176655 - cp's OEIS frontend

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

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

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

A123321 Products of 7 distinct primes (squarefree 7-almost primes).

Original entry on oeis.org

510510, 570570, 690690, 746130, 870870, 881790, 903210, 930930, 1009470, 1067430, 1111110, 1138830, 1193010, 1217370, 1231230, 1272810, 1291290, 1345890, 1360590, 1385670, 1411410, 1438710, 1452990, 1504230, 1540770

Offset: 1

Rick L. Shepherd, Sep 25 2006

nonn

Intersection of A005117 and A046308.

Intersection of A005117 and A176655. - R. J. Mathar, Dec 05 2016

			a(1) = 510510 = 2*3*5*7*11*13*17 = A002110(7).

Rick L. Shepherd, Table of n, a(n) for n = 1..10000

Cf. A005117, A046308, A048692, Squarefree k-almost primes: A000040 (k=1), A006881 (k=2), A007304 (k=3), A046386 (k=4), A046387 (k=5), A067885 (k=6), A123322 (k=8), A115343 (k=9).

f7Q[n_]:=Last/@FactorInteger[n]=={1, 1, 1, 1, 1, 1, 1}; lst={};Do[If[f7Q[n], AppendTo[lst, n]], {n, 9!}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 26 2008 *)
Select[Range[1600000],PrimeNu[#]==7&&SquareFreeQ[#]&] (* Harvey P. Dale, Sep 19 2013 *)

is(n)=omega(n)==7 && bigomega(n)==7 \\ Hugo Pfoertner, Dec 18 2018

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A123321(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,0,1,1,7)))
    def bisection(f,kmin=0,kmax=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
    return bisection(f) # Chai Wah Wu, Aug 31 2024

More terms from Vladimir Joseph Stephan Orlovsky, Aug 26 2008

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).

Original entry on oeis.org

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

A046397 Palindromes which are the product of exactly 7 distinct primes.

Original entry on oeis.org

22444422, 24266242, 26588562, 35888853, 36399363, 43777734, 47199174, 51066015, 53588535, 53888835, 55233255, 59911995, 60066006, 62588526, 62700726, 62888826, 81699618, 87788778, 89433498, 122434221, 202040202

Offset: 1

Patrick De Geest, Jun 15 1998

The original name "Palindromes with exactly 7 distinct prime factors" did not exclude that one or more of the factors occurred to a higher power: this is sequence A373467. As the listed data show, terms of this sequence must be squarefree. - M. F. Hasler, Jun 06 2024

			The first two palindromes with 7 distinct prime factors are 20522502 = 2 * 3^2 * 7 * 11 * 13 * 17 * 67 and 21033012 = 2^2 * 3 * 7 * 11 * 13 * 17 * 103, but these are excluded since one of the prime factors occurs to a higher power.
a(1) = 22444422 = 2 * 3 * 7 * 11 * 13 * 37 * 101, which is squarefree, is therefore the first term of this sequence.

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

Cf. A046333 (similar but prime factors counted with multiplicity), A373467 (similar but counting just the distinct prime divisors).

Cf. A002113 (palindromes), A123321 (products of 7 distinct primes), A176655 (numbers with omega = 7 distinct prime divisors).

digrev:= proc(n) local L,i;
  L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
filter:= proc(n) local F;
  F:= ifactors(n)[2];
  nops(F) = 7 and map(t -> t[2],F)=[1$7]
end proc:
Res:= NULL:
count:= 0:
for d from 2  while count < 100 do
  if d::even then
    m:= d/2;
    for n from 10^(m-1) to 10^m-1 while count < 100 do
      v:= n*10^m+digrev(n);
      if filter(v) then count:= count+1; Res:= Res, v; fi;
    od;
  else
    m:= (d-1)/2;
    for n from 10^(m-1) to 10^m-1 while count < 100 do
      for y from 0 to 9 while count < 100 do
         v:= n*10^(m+1)+y*10^m+digrev(n);
         if filter(v) then count:= count+1; Res:= Res, v; fi;
    od od
  fi
od:
Res; # Robert Israel, Jan 20 2020

A046397_upto(N, start=vecprod(primes(7)), num_fact=7)={ my(L=List()); is_A002113(start)&& start--; while(N >= start = nxt_A002113(start), omega(start)==num_fact && issquarefree(start) && listput(L, start)); L} \\ M. F. Hasler, Jun 06 2024

A373467 Palindromes with exactly 7 (distinct) prime divisors.

Original entry on oeis.org

20522502, 21033012, 22444422, 23555532, 24266242, 25777752, 26588562, 35888853, 36399363, 41555514, 41855814, 42066024, 43477434, 43777734, 44888844, 45999954, 47199174, 51066015, 51666615, 52777725, 53588535, 53888835, 55233255, 59911995, 60066006, 60366306, 61777716, 62588526, 62700726

Offset: 1

M. F. Hasler, Jun 06 2024

			Obviously all terms must be palindromic; let us consider the prime factorization:
a(1) = 20522502 = 2 * 3^2 * 7 * 11 * 13 * 17 * 67 has exactly 7 distinct prime divisors, although the factor 3 appears twice in the factorization. (Without the second factor 3 the number would not be palindromic.)
a(2) = 21033012 = 2^2 * 3 * 7 * 11 * 13 * 17 * 103 has exactly 7 distinct prime divisors, although the factor 2 appears twice in the factorization. (Without the second factor 2 the number would not be palindromic.)
a(3) = 22444422 = 2 * 3 * 7 * 11 * 13 * 37 * 101 is the product of 7 distinct primes (cf. A123321), hence the first squarefree term of this sequence.

Cf. A046333 (same with bigomega = 7: counting prime factors with multiplicity), A046397 (same but only squarefree terms), A373465 (same with omega = 5), A046396 (same with omega = 6).

Cf. A002113 (palindromes), A176655 (omega(.) = 7), A123321 (products of 7 distinct primes).

A373467_upto(N, start=vecprod(primes(7)), num_fact=7)={ my(L=List()); while(N >= start = nxt_A002113(start), omega(start)==num_fact && listput(L, start)); L}

Intersection of A002113 and A176655.

A348266 k-digit numbers whose digit(s) are the number of distinct prime factors in each of the preceding k integers.

Original entry on oeis.org

22, 313, 2232, 2323, 2333, 32215, 432152, 2434332, 4222423, 43332543, 332325334, 2535332433, 4532543535234, 5435433351423

Offset: 1

Metin Sariyar, Oct 09 2021

a(12) <= 2535332433. - David A. Corneth, Oct 10 2021

a(12) >= 10^9. - Michel Marcus, Oct 11 2021

			22 is a term because omega(20) = 2 and omega(21) = 2, whose concatenation is 22.
313 is a term because preceding it omega(310) = 3, omega(311) = 1 and omega(312) = 3, and their concatenation is 313.
32215 is a term because, the number of distinct prime divisors of 32210, 32211, 32212, 32213 and 32214 are 3, 2, 2, 1, 5 and their ordered concatenation gives the next number 32215.

Cf. A001221, A000961, A007774, A033992, A033993, A051270, A074969, A176655.

Cf. A323083.

Select[Range[33000], FromDigits[PrimeNu /@ (# - Range[IntegerLength[#], 1, -1])] == # &] (* Amiram Eldar, Oct 09 2021 *)

isok(m) = {my(s="", k=m, i=1); while(1, s = concat(s, Str(omega(k))); if (eval(s) == m+i, return (i)); if (eval(s) > m+i, return(0)); k++; i++;);}
lista(nn) = my(nb); for(n=1, nn, if (nb=isok(n), print1(n+nb, ", "))); \\ Michel Marcus, Oct 09 2021

a(8)-a(9) from Amiram Eldar, Oct 09 2021
a(10)-a(11) from Michel Marcus, Oct 10 2021
a(12) confirmed by Martin Ehrenstein, Oct 28 2021
a(13)-a(14) from Martin Ehrenstein, Oct 30 2021

A176756 Sequence defined by the recurrence formula a(n+1)=sum(a(p)*a(n-p)+k,p=0..n)+l for n>=1, with here a(0)=1, a(1)=4, k=0 and l=-2.

Original entry on oeis.org

1, 4, 6, 26, 98, 438, 1970, 9294, 44698, 219766, 1096930, 5549614, 28383498, 146538150, 762627954, 3996744862, 21074272538, 111723476502, 595145562242, 3183988894350, 17100312159018, 92164073738118, 498318304290450

Offset: 0

Richard Choulet, Apr 25 2010

			a(2)=2*1*4-2=6. a(3)=2*1*6+4^2-2=26. a(4)=2*1*26+2*4*6-2=98.

Cf. A176655.

l:=-2: : k := 0 : m:=4:d(0):=1:d(1):=m: for n from 1 to 30 do d(n+1):=sum(d(p)*d(n-p)+k, p=0..n)+l:od : taylor((1-sqrt(1-4*z*(d(0)-z*d(0)^2+z*m+(k+l)*z^2/(1-z)+k*z^2/(1-z)^2)))/(2*z), z=0, 30); seq(d(n), n=0..30);

G.f f: f(z)=(1-sqrt(1-4*z*(a(0)-z*a(0)^2+z*a(1)+(k+l)*z^2/(1-z)+k*z^2/(1-z)^2)))/(2*z) (k=0, l=-2).

Conjecture: (n+1)*a(n) +2*(-3*n+1)*a(n-1) +(-3*n+11)*a(n-2) +2*(14*n-45)*a(n-3) +20*(-n+4)*a(n-4)=0. - R. J. Mathar, Feb 18 2016

A348882 Numbers that are expressible as the product of the number of distinct prime factors of preceding integers.

Original entry on oeis.org

16, 48, 72, 96, 144, 432, 576, 1296, 2592, 5184, 20736, 32805, 221184, 1555200, 11197440, 55987200, 95551488, 268738560, 302330880, 382205952, 524880000, 671846400, 6718464000, 34012224000, 155520000000, 403107840000, 6856864358400, 107495424000000, 110075314176000

Offset: 1

Metin Sariyar, Nov 02 2021

nonn

			The number of distinct prime factors of the numbers 15, 14, 13, 12, 11, 10 are respectively 2, 2, 1, 2, 1, 2 and 2*2*1*2*1*2 = 16, hence 16 is a term.

Cf. A001221, A000961, A007774, A033992, A033993, A051270, A074969, A176655, A348266.

om[n_] := om[n] = PrimeNu[n]; q[n_] := Module[{m = n, k = n - 1}, While[k > 1 && Divisible[m, om[k]], m /= om[k]; k--]; m == 1]; Select[Range[2, 10^6], q] (* Amiram Eldar, Nov 02 2021 *)

a(13)-a(17) from Amiram Eldar, Nov 02 2021

More terms from David A. Corneth, Nov 02 2021

cp's OEIS Frontend

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

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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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A123321 Products of 7 distinct primes (squarefree 7-almost primes).

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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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A046397 Palindromes which are the product of exactly 7 distinct primes.

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A373467 Palindromes with exactly 7 (distinct) prime divisors.

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A348266 k-digit numbers whose digit(s) are the number of distinct prime factors in each of the preceding k integers.

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