A033992 -id:A033992 - cp's OEIS frontend

A062509 a(n) = n^omega(n).

1, 2, 3, 4, 5, 36, 7, 8, 9, 100, 11, 144, 13, 196, 225, 16, 17, 324, 19, 400, 441, 484, 23, 576, 25, 676, 27, 784, 29, 27000, 31, 32, 1089, 1156, 1225, 1296, 37, 1444, 1521, 1600, 41, 74088, 43, 1936, 2025, 2116, 47, 2304, 49, 2500, 2601, 2704, 53, 2916

Offset: 1

Labos Elemer, Jul 13 2001

nonn

Not always equal to product of unitary divisors of n [compare with A061537]. This deviates from A061537 at 30, 42, 60, 66, etc.

			n=30: a(30) = 30^3 = 27000;
n=72: a(72) = 72^2 = 5184.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A001221, A061537, A034444, A000005, A033992.

Cf. A176029, A229109.

a062509 n = n ^ a001221 n  -- Reinhard Zumkeller, Sep 13 2013

Table[n^PrimeNu[n],{n,60}] (* Harvey P. Dale, Jan 14 2013 *)

a(n)={n^omega(n)} \\ Harry J. Smith, Aug 08 2009

a(n) = Sum_{d|n} tau(d^n)*mu(n/d). - Ridouane Oudra, Sep 17 2022

A278569 Numbers of the form p^iq^jr^k where p,q,r are distinct odd primes and i,j,k >= 1.

Original entry on oeis.org

105, 165, 195, 231, 255, 273, 285, 315, 345, 357, 385, 399, 429, 435, 455, 465, 483, 495, 525, 555, 561, 585, 595, 609, 615, 627, 645, 651, 663, 665, 693, 705, 715, 735, 741, 759, 765, 777, 795, 805, 819, 825, 855, 861, 885, 897, 903, 915, 935, 945, 957, 969, 975, 987, 1001, 1005, 1015, 1023, 1035, 1045, 1065, 1071, 1085, 1095, 1105, 1113, 1131, 1173, 1185, 1197, 1209

Offset: 1

N. J. A. Sloane, Nov 27 2016

nonn

More than the usual number of terms are included to show the difference from A216918 (the latter includes 3*5*7*11 = 1155 and all terms of A046390, A046391 etc).

If i,j,k are all equal to 1 we get A046389.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Includes A046389, subsequence of A216918.

Select[Range@ 1500, PrimeNu@ # == 3 && OddQ@ # &] (* Michael De Vlieger, Dec 05 2016 *)

A033992 INTERSECT A005408. - R. J. Mathar, Dec 05 2016

A287484 Number of squarefree k with A002110(n) <= k < A002110(n+1) such that A001221(k) = n.

Original entry on oeis.org

1, 3, 7, 19, 58, 152, 422, 995, 2359, 6294, 14507, 36370, 88198, 187786, 386993, 840033, 1901930, 3851372, 8088478, 16388857, 30001902, 56613547, 103229263, 193020113, 389750880, 759988983, 1359250012, 2350842201, 3737393021, 5748044055, 10843131073, 19774152370

Offset: 0

Michael De Vlieger, May 25 2017

Primorial A002110(n) is the smallest squarefree number with n prime factors. a(n) is a list of squarefree numbers with n prime factors greater than and including A002110(n) but less than A002110(n+1).
a(1) counts the first primes less than 6.
a(2) counts the first squarefree semiprimes (A006881) less than 30.
a(3) counts the smallest terms of A033992 less than 210, etc.

			Let p_n# = A002110(n).
a(0) = 1 since the only squarefree number between p_0# and (p_1# - 1) (i.e., 1 and 1) with 0 prime factors is 1.
a(1) = 3 since for p_1# <= k <= (p_2# - 1), i.e., 2 <= k <= 5, there are three primes {2, 3, 5}.
a(2) = 7 since we find the squarefree semiprimes {6, 10, 14, 15, 21, 22, 26} between 6 and 29 inclusive.

Eric Weisstein's World of Mathematics, Primorial
Eric Weisstein's World of Mathematics, Squarefree

Cf. A001221, A002110, A005117, A006881, A033992.

Table[Count[Range[#, Prime[n + 1] # - 1] &@ Product[Prime@ i, {i, n}], k_ /; And[SquareFreeQ@ k, PrimeOmega@ k == n]], {n, 0, 6}]

a(25)-a(31) from David A. Corneth, May 31 2017

A304647 Smallest term of A304636 that requires exactly n iterations to reach a fixed point under the x -> A181819(x) map.

Original entry on oeis.org

5, 8, 30, 360, 1801800, 2746644314348614680000, 13268350773236509446586539974366689358164301703214270074935844483572035447570761114173070859428708074413696096366645684575600000000

Offset: 0

Gus Wiseman, May 15 2018

nonn

The first entry 5 is optional so has offset 0.

			The list of multisets with Heinz numbers in the sequence is the following. The number of k's in row n + 1 is equal to the k-th largest term of row n.
                     5: {3}
                     8: {1,1,1}
                    30: {1,2,3}
                   360: {1,1,1,2,2,3}
               1801800: {1,1,1,2,2,3,3,4,5,6}
2746644314348614680000: {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,7,8,9,10}

Cf. A001221, A001222, A001462, A012257, A014612, A033992, A071625, A112798, A181819, A182850, A182857, A275870, A304464, A304465, A304634, A304636.

Function[m,Times@@Prime/@m]/@NestList[Join@@Table[Table[i,{Reverse[#][[i]]}],{i,Length[#]}]&,{3},6]

A082998 a(n) = card{ x <= n : omega(x) = 3 }.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 10

Offset: 1

Benoit Cloitre, May 30 2003

nonn

G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, p. 203, Publications de l'Institut Cartan, 1990.

Daniel Suteu, Table of n, a(n) for n = 1..10000

Cf. A001221, A025528, A033992, A072000, A072114, A082997.

PARI
```
a(n)=sum(i=1,n,if(omega(i)-3,0,1))
```

a(n, k = 3, m = 1, p = 2, s = sqrtnint(n\m, k), j = 1) = my(count = 0); if (k==2, while(p <= s, my(r = nextprime(p+1)); my(t = m*p); while (t <= n, my(w = n\t); if(r > w, break); count += primepi(w) - j; my(r2 = r); while(r2 <= w, my(u = t*r2*r2); if(u > n, break); while (u <= n, count += 1; u *= r2); r2 = nextprime(r2+1)); t *= p); p = r; j += 1); return(count)); while(p <= s, my(r = nextprime(p+1)); my(t = m*p); while(t <= n, my(s = sqrtnint(n\t, k-1)); if(r > s, break); count += a(n, k-1, t, r, s, j+1); t *= p); p = r; j += 1); count; \\ Daniel Suteu, Jul 21 2021

from sympy import factorint
from itertools import accumulate
def cond(n): return int(len(factorint(n))==3)
def aupto(nn): return list(accumulate(map(cond, range(1, nn+1))))
print(aupto(105)) # Michael S. Branicky, Jul 21 2021

a(n) ~ (1/2)*(n/log(n))*log(log(n))^2.

a(A033992(n)) = n. - Daniel Suteu, Jul 21 2021

A168638 Number of distinct prime divisors of n is 2 or 3.

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106

Offset: 1

Juri-Stepan Gerasimov, Dec 01 2009

nonn

			2310=2*3*5*7*11 has 5 prime factors, hence it is not here, but it is part of A064040.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Select[Range[106],MemberQ[{2,3},PrimeNu[#]]&] (* Jayanta Basu, Jun 04 2013 *)

is(n)=my(o=omega(n)); o==2 || o==3 \\ Charles R Greathouse IV, Jul 28 2016

a(n) ~ A033992(n) ~ 2n log n / (log log n)^2. - Charles R Greathouse IV, Jul 28 2016

A214195 Numbers with the number of distinct prime factors a multiple of 3.

Original entry on oeis.org

1, 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, 220, 222, 228, 230, 231, 234, 238, 240, 246, 252, 255, 258, 260, 264, 266, 270, 273, 276, 280, 282, 285

Offset: 1

Enrique Pérez Herrero, Jul 07 2012

nonn

If GCD(a(n),a(m))=1, then a(n)*a(m) is also in this sequence. - Enrique Pérez Herrero, Nov 23 2013

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000

Subsequences include A033992, A067885, A007304 and A147573.

Cf. A145784, A030230, A030231.

Select[Range[1000],Mod[PrimeNu[#],3]==0&]

is(n)=omega(n)%3==0 \\ Charles R Greathouse IV, Sep 14 2015

A010872(A001221(a(n))) = 0.

A288784 Irregular triangle read by rows: T(n,m) is the list of numbers kA002110(n) <= kt < (k + 1)A002110(n) such that A001222(kt) = n, with 1 <= k < prime(n + 1).

Original entry on oeis.org

1, 2, 3, 4, 6, 10, 12, 18, 24, 30, 42, 60, 84, 90, 120, 150, 180, 210, 330, 390, 420, 630, 840, 1050, 1260, 1470, 1680, 1890, 2100, 2310, 2730, 3570, 3990, 4290, 4620, 5460, 6930, 8190, 9240, 10920, 11550, 13650, 13860, 16170, 18480, 20790, 23100, 25410, 27720

Offset: 0

Michael De Vlieger, Jun 15 2017

A060735 and A002110 are subsets.
This sequence is a necessary but insufficient condition for A244052. Terms that are in A060735 and A002110 are also in A244052. The first terms of this sequence that are not in A244052 are {3, 4290, 881790, 903210, 1009470, 17160990, 363993630, 380570190, 406816410, 434444010, ...}.
Primorial p_n# = A002110(n) is the smallest squarefree number with n prime factors. Consider the list of squarefree numbers t with n prime factors greater than and including A002110(n) but less than 2*A002110(n). Extend the list to include products k*t of this list with 1 <= k < prime(n+1) such that k*t < (k+1)*p_n#. This list contains squarefree numbers k*t with n distinct primes and presumes that the number (k+1)*p_n# serves as a "limit" beyond which k*t > (k+1)p_n# are not in the sequence.

			Triangle begins:
    n   T(n,m)
    0:   1;
    1:   2,  3,  4;
    2:   6, 10, 12, 18, 24;
    3:  30, 42, 60, 84, 90, 120, 150, 180;
       ...

Michael De Vlieger, Table of n, a(n) for n = 0..11793 (rows 1 <= n <= 30).
Michael De Vlieger, Terms of a(n) with 1 <= n <= 653 in A002110, A060735, and A244052
Eric Weisstein's World of Mathematics, Primorial
Eric Weisstein's World of Mathematics, Squarefree

Cf. A001221, A002110, A005117, A006881, A033992, A060735, A244052.

Table[Function[P, Function[s, Flatten@ Map[Function[k, Select[k s, # < (k + 1) P &]], Range[1, Prime[n + 1] - 1]]]@ Select[Range[P, 2 P - 1], And[SquareFreeQ@ #, PrimeOmega@ # == n] &]]@ Product[Prime@ i, {i, n}], {n, 0, 5}] (* Michael De Vlieger, Jun 15 2017 *)

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

A295644 Rectangular array, by antidiagonals; row 1 is the ordered list of all k having at most 2 unitary divisors; for n > 1, row n is the ordered list of all k having 2^n unitary divisors.

Original entry on oeis.org

1, 2, 6, 3, 10, 30, 4, 12, 42, 210, 5, 14, 60, 330, 2310, 7, 15, 66, 390, 2730, 30030, 8, 18, 70, 420, 3570, 39270, 510510, 9, 20, 78, 462, 3990, 43890, 570570

Offset: 1

Clark Kimberling, Jun 26 2018

Every positive integer occurs exactly once, so that as a sequence, this is a permutation of the positive integers.
row 1: A000961
row 2: A007774
row 3: A033992
row 4: A033993
col 1: A231209

			Northwest corner:
     1    2    3    4    5    7    8    9   11
     6   10   12   14   15   18   20   21   22
    30   42   60   66   70   78   84   90  102
   210  330  390  420  462  510  546  570  630
  2310 2730 3570 3990 4290 4620 4830 5460 5610

Cf. A034444.

As an array, essentially the same as A125666.

z = 10000;
t = Table[2^PrimeNu[n], {n, 1, z}] ;(*  A035555 *)
r[n_] := Flatten[Position[t, 2^n]]; r[1] = Join[{1}, r[1]];
v[n_, k_] := r[n][[k]];
TableForm[Table[v[n, k], {n, 1, 5}, {k, 1, 15}]]  (* A295644 array *)
Table[v[n - k + 1, k], {n, 5}, {k, n, 1, -1}] // Flatten  (* A295644 sequence *)

cp's OEIS Frontend

A062509 a(n) = n^omega(n).

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A278569 Numbers of the form p^i*q^j*r^k where p,q,r are distinct odd primes and i,j,k >= 1.

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A287484 Number of squarefree k with A002110(n) <= k < A002110(n+1) such that A001221(k) = n.

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A304647 Smallest term of A304636 that requires exactly n iterations to reach a fixed point under the x -> A181819(x) map.

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A082998 a(n) = card{ x <= n : omega(x) = 3 }.

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A168638 Number of distinct prime divisors of n is 2 or 3.

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A214195 Numbers with the number of distinct prime factors a multiple of 3.

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A288784 Irregular triangle read by rows: T(n,m) is the list of numbers k*A002110(n) <= k*t < (k + 1)*A002110(n) such that A001222(k*t) = n, with 1 <= k < prime(n + 1).

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A278569 Numbers of the form p^iq^jr^k where p,q,r are distinct odd primes and i,j,k >= 1.

A288784 Irregular triangle read by rows: T(n,m) is the list of numbers kA002110(n) <= kt < (k + 1)A002110(n) such that A001222(kt) = n, with 1 <= k < prime(n + 1).