A010846 -id:A010846 - cp's OEIS frontend

A244974 Sum of numbers m <= n whose set of prime divisors is a subset of the set of prime divisors of n.

1, 3, 4, 7, 6, 16, 8, 15, 13, 30, 12, 45, 14, 36, 33, 31, 18, 79, 20, 66, 41, 64, 24, 103, 31, 70, 40, 80, 30, 235, 32, 63, 84, 114, 73, 198, 38, 120, 92, 163, 42, 310, 44, 140, 130, 132, 48, 246, 57, 213, 108, 154, 54, 300, 97, 217, 116, 150, 60, 600, 62, 156, 180, 127, 109, 540, 68, 246

Offset: 1

Michael De Vlieger, Jul 08 2014

nonn

a(n) = A000203(n) when n is prime or a perfect prime power (A000961). This is because all products of the prime divisor p in such numbers produce divisors.

a(n) > A000203(n) when n is composite and not a perfect prime power.

			For n = 4, A162306(4) = {1, 2, 4} and a(4) = 7.
For n = 5, A162306(5) = {1, 5} and a(5) = 6.
For n = 6, A162306(6) = {1, 2, 3, 4, 6} and a(6) = 16.

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

a(n) = sum of terms of n-th row of triangle A162306(n,k).

Cf. A010846, A000005, A243822.

Table[Total@ Union[{1}, Function[d, Select[Range@ n, Union[d, First /@ FactorInteger@ #] == d &]][First /@ FactorInteger@ n]], {n, 68}] (* or *)
Table[Sum[k (Floor[n^k/k] - Floor[(n^k - 1)/k]), {k, n}], {n, 68}] (* Michael De Vlieger, May 26 2016 *)

a(n) = {summ = 0; spn = factor(n)[,1]~; for (m=1, n, spm = factor(m)[,1]~; if (setintersect(spm, spn) == spm, summ += m);); summ;} \\ Michel Marcus, Jul 17 2014

a(n) = Sum_{k=1..n} k*( floor(n^k/k)-floor((n^k - 1)/k) ). - Anthony Browne, May 25 2016

a(n) = Sum_{j=1..n} Sum_{i=j..gcd(n^j,j)} i. - Wesley Ivan Hurt, Apr 05 2021

A376248 Irregular triangle where row n lists m such that rad(m) | n and bigomega(m) <= bigomega(n), where rad = A007947 and bigomega = A001222.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 2, 4, 1, 5, 1, 2, 3, 4, 6, 9, 1, 7, 1, 2, 4, 8, 1, 3, 9, 1, 2, 4, 5, 10, 25, 1, 11, 1, 2, 3, 4, 6, 8, 9, 12, 18, 27, 1, 13, 1, 2, 4, 7, 14, 49, 1, 3, 5, 9, 15, 25, 1, 2, 4, 8, 16, 1, 17, 1, 2, 3, 4, 6, 8, 9, 12, 18, 27, 1, 19, 1, 2, 4, 5, 8, 10, 20, 25, 50, 125

Offset: 1

Michael De Vlieger, Oct 09 2024

Analogous to A162306 regarding m such that rad(m) | n, but instead of taking m <= n, we take m such that bigomega(m) <= bigomega(n).
Row n is a finite set of products of prime power factors p^k (i.e., p^k | n) such that Sum_{p|n} k <= bigomega(n).
For prime power n = p^k, k >= 0 (i.e., n in A000961), row p^k of this sequence is the same as row p^k of A027750 and A162306. Therefore, for prime p, row p of this sequence is the same as row p of A027750 and A162306: {1, p}.
For n in A024619, row n of this sequence does not match row n of A162306, since the former contains gpf(n)^bigomega(n) = A006530(n)^A001222(n), which is larger than n.

			Triangle begins:
   n    row n of this sequence:
  -------------------------------------------
   1:   1;
   2:   1,  2;
   3:   1,  3;
   4:   1,  2   4;
   5:   1,  5;
   6:   1,  2,  3,  4,  6,  9;
   7:   1,  7;
   8:   1,  2,  4,  8;
   9:   1,  3,  9;
  10:   1,  2,  4,  5, 10, 25;
  11:   1, 11;
  12:   1,  2,  3,  4,  6,  8, 9, 12, 18, 27;
        ...
Row n = 10 of this sequence, presented according to 2^k, k = 0..bigomega(n) by columns, 5^i, i = 0..bigomega(n) by rows, showing terms m > n with an asterisk. The remaining m and the parenthetic 8 are in row 10 of A162306:
   1   2   4  (8)
   5  10
  25*
Row n = 12 of this sequence, presented according to 2^k, k = 0..bigomega(n) by columns, 3^i, i = 0..bigomega(n) by rows, showing terms m > n with an asterisk. The remaining m are in row 12 of A162306:
   1   2   4   8
   3   6  12
   9  18*
  27*

Michael De Vlieger, Table of n, a(n) for n = 1..17475 (rows n = 1..1000, flattened)
Michael De Vlieger, Log log scatterplot of rows n = 1..2^16 of this sequence, flattened, (3153752 terms).

Cf. A000961, A001221, A001222, A006530, A007947, A010846, A024619, A027750, A162306, A376567.

Table[Clear[p]; MapIndexed[Set[p[First[#2]], #1] &, FactorInteger[n][[All, 1]]]; k = PrimeOmega[n]; w = PrimeNu[n]; Union@ Map[Times @@ MapIndexed[p[First[#2]]^#1 &, #] &, Select[Tuples[Range[0, k], w], Total[#] <= k &] ], {n, 120}]

Row n of this sequence is { m : rad(m) | n, bigomega(m) <= bigomega(n) }.

A376567(n) = binomial(bigomega(n) + omega(n)) = Length of row n, where omega = A001221.

A138109 Positive integers k whose smallest prime factor is greater than the cube root of k and strictly less than the square root of k.

Original entry on oeis.org

6, 15, 21, 35, 55, 65, 77, 85, 91, 95, 115, 119, 133, 143, 161, 187, 203, 209, 217, 221, 247, 253, 259, 287, 299, 301, 319, 323, 329, 341, 377, 391, 403, 407, 437, 451, 473, 481, 493, 517, 527, 533, 551, 559, 583, 589, 611, 629, 649, 667, 671, 689, 697, 703

Offset: 1

David S. Newman, May 04 2008

nonn

This sequence was suggested by Moshe Shmuel Newman.
A020639(n)^2 < a(n) < A020639(n)^3. - Reinhard Zumkeller, Dec 17 2014
In other words, k = p*q with primes p, q satisfying p < q < p^2. - Charles R Greathouse IV, Apr 03 2017
If "strictly less than" in the definition were changed to "less than or equal to" then this sequence would also include the squares of primes (A001248), resulting in A251728. - Jon E. Schoenfield, Dec 27 2022

			6 is a term because the smallest prime factor of 6 is 2 and 6^(1/3) = 1.817... < 2 < 2.449... = sqrt(6).
From _Michael De Vlieger_, Apr 27 2024: (Start):
Table of p*q where p = prime(n) and q = prime(n+k):
n\k   1     2     3     4     5     6     7     8     9    10    11
-------------------------------------------------------------------
1:    6;
2:   15,   21;
3:   35,   55,   65,   85,   95,  115;
4:   77,   91,  119,  133,  161,  203,  217,  259,  287,  301,  329;
     ... (End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Subsequence of A251728 and of A006881. A006094 is a proper subset.

Cf. A020639, A010846, A079047, A162306.

a138109 n = a138109_list !! (n-1)
a138109_list = filter f [1..] where
   f x = p ^ 2 < x && x < p ^ 3 where p = a020639 x
-- Reinhard Zumkeller, Dec 17 2014

s = {}; Do[f = FactorInteger[i]; test = f[[1]][[1]]; If [test < N[i^(1/2)] && test > N[i^(1/3)], s = Union[s, {i}]], {i, 2, 2000}]; Print[s]
Select[Range[1000],Surd[#,3]Harvey P. Dale, May 10 2015 *)

is(n)=my(f=factor(n)); f[,2]==[1,1]~ && f[1,1]^3 > n \\ Charles R Greathouse IV, Mar 28 2017

list(lim)=if(lim<6, return([])); my(v=List([6])); forprime(p=3,sqrtint(1+lim\=1)-1, forprime(q=p+2, min(p^2-2,lim\p), listput(v,p*q))); Set(v) \\ Charles R Greathouse IV, Mar 28 2017

from math import isqrt
from sympy import primepi, primerange
def A138109(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(min(x//p,p**2)) for p in primerange(s+1)))
    return bisection(f,n,n) # Chai Wah Wu, Mar 05 2025

From Michael De Vlieger, Apr 27 2024: (Start)
Let k = a(n); row k of A162306 = {1, p, q, p^2, p*q}, therefore A010846(k) = 5.
A079047(n) = card({ q : p < q < p^2 }), p and q primes. (End)

A361373 Number of prime powers p^m <= n such that p | n.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 3, 2, 4, 1, 5, 1, 4, 3, 4, 1, 6, 1, 5, 3, 5, 1, 6, 2, 5, 3, 5, 1, 9, 1, 5, 4, 6, 3, 8, 1, 6, 4, 7, 1, 9, 1, 6, 5, 6, 1, 8, 2, 7, 4, 6, 1, 8, 3, 7, 4, 6, 1, 10, 1, 6, 5, 6, 3, 10, 1, 7, 4, 10, 1, 9, 1, 7, 5, 7, 3, 10, 1, 8, 4, 7, 1, 12, 3, 7

Offset: 1

Michael De Vlieger, Jun 17 2024

Let p be prime. The term "prime power" p^m, m > 0, used here is that of A246655 = A000040 U A246547, the union of primes and perfect prime powers. Essentially, 1 is not considered a prime power.

			Let S = {k <= n : rad(k) | n} = row n of A162306
a(1) = 0 since S = {1} has 0 prime powers.
a(2) = 1 since S = {1, [2]} has 1 prime power.
a(4) = 2 since S = {1, [2, 4]} has 2 prime powers.
a(6) = 3 since S = {1, [2, 3, 4], 6} has 3 prime powers.
a(10) = 4 since S = {1, [2, 4, 5, 8], 10} has 4 prime powers.
a(12) = 5 since S = {1, [2, 3, 4], 6, [8, 9], 12} has 5 prime powers, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Diagram illustrating a(1440) = 20. Terms are arranged according to prime decomposition and sorted vertically. This sequence counts primes (red) and perfect prime powers (gold).

Cf. A000005, A000040, A007947, A010846, A020639, A126706, A138109, A138511, A162306, A246655, A246547.

a := n -> add(ilog[p](n), p in NumberTheory:-PrimeFactors(n)):
seq(a(n), n = 1..92); # Peter Luschny, Jun 20 2024

{0}~Join~Table[Total@ Map[Floor@ Log[#, n] &, FactorInteger[n][[All, 1]]], {n, 2, 120}]

a(n) = if (n==1, 0, my(f=factor(n)[,1]); sum(k=1, #f, logint(n, f[k]))); \\ Michel Marcus, Jun 20 2024

from sympy import integer_log, primefactors
def A361373(n): return sum(integer_log(n,p)[0] for p in primefactors(n)) # Chai Wah Wu, Sep 20 2024

a(n) = Sum_{p | n} floor(log n / log p).
a(n) = number of prime powers in row n of A162306.
a(n) < A000005(n), since A000005 counts 1.
a(n) < A010846(n), since A010846 counts 1.
Let tau = A000005, rad = A007947, rcf = A010846, and lpf = A020639.
a(p) = tau(p) - 1 = rcf(p) - 1 = 1 since S = row p of both A027750 and A162306 = {1, p} contains the prime power p.
a(p^m) = tau(p^m) - 1 = rcf(p^m) = 1 = m since S = row p^m of both A027750 and A162306 = {1, p, p^2, ..., p^m} contains the prime powers {p, p^2, ..., p^m}.
a(k) = tau(k) - 1 = 3 for squarefree composite k = p*q, p < q < p^2 in A138109 since S = row k of A162306 = {1, p, q, p^2, p*q} contains 3 prime powers {p, q, p^2}.
a(k) < tau(k) for k in A138511 and k in A126706 since m = lpf(k)^(-1 + floor(log k / log lpf(k))) is such that m < k but m does not divide k.

A363061 Number of k <= P(n) such that rad(k) | P(n), where rad(n) = A007947(n) and P(n) = A002110(n).

Original entry on oeis.org

1, 2, 5, 18, 68, 283, 1161, 4843, 19985, 83074, 349670, 1456458, 6107257, 25547835, 106115655, 440396113, 1833079809, 7642924612, 31705433101, 131711607956, 546283729493, 2257462298234, 9339325821411, 38593708318690, 159600066415313, 661371515924516, 2736805917843710

Offset: 0

Michael De Vlieger, Jun 16 2023

			a(0) = 1 since P(0) = 1 and 1 | 1.
a(1) = 2 since P(1) = 2 and both 1 | 2 and 2 | 2.
a(2) = 5 since P(2) = 6 and rad(m) | 6 for m = {1, 2, 3, 4, 6}.
a(3) = 18 since P(3) = 30 and rad(m) | 30 for m = {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30}, etc.
Regarding a(3), we see that there are 18 terms in the tensor product of prime power ranges of 2, 3, and 5 that do not exceed 30:
5^0X | 2^0 2^1 2^2 2^3 2^4    5^1X | 2^0 2^1 2^2    5^2X | 2^0
--------------------------    ------------------    ----------
3^0  |   1   2   4   8  16    3^0  |   5  10  20    3^0  |  25
3^1  |   3   6  12  24        3^1  |  15  30
3^2  |   9  18
3^3  |  27
Hence, a(3) = 18. This approach proves handy for larger n.

Bert Dobbelaere, Python program

Cf. A002110, A007947, A010846.

f[1] = 1; f[n_] := Function[w,
ToExpression@ StringJoin["Block[{n = ", ToString@ n,
    ", k = 0}, Flatten@ Table[k++, ",
    Most@ Flatten@ Map[{#, ", "} &, #], "]; k]"] &@
      MapIndexed[
        Function[p, StringJoin["{", ToString@ Last@ p, ", 0, Log[",
          ToString@ First@ p, ", n/(",
          ToString@ InputForm[Times @@ Map[Power @@ # &, Take[w, First@ #2 - 1]]],
          ")]}"] ]@ w[[First@ #2]] &, w]]@
   Map[{#, ToExpression["p" <> ToString@ PrimePi@ #]} &,
     FactorInteger[n][[All, 1]]];
   Map[f, FoldList[Times, 1, Prime@ Range@ 9] ]

a(n) = A010846(A002110(n)).

a(n) >= 2^n.

Corrected a(15) and added a(16)-a(23) from Bert Dobbelaere, Jun 27 2023

a(24)-a(26) from Martin Ehrenstein, Jul 08 2023

A010848 Number of numbers k <= n such that at least one prime factor of n is not a prime factor of k.

Original entry on oeis.org

0, 1, 2, 2, 4, 5, 6, 4, 6, 9, 10, 10, 12, 13, 14, 8, 16, 15, 18, 18, 20, 21, 22, 20, 20, 25, 18, 26, 28, 29, 30, 16, 32, 33, 34, 30, 36, 37, 38, 36, 40, 41, 42, 42, 42, 45, 46, 40, 42, 45, 50, 50, 52, 45, 54, 52, 56, 57, 58, 58, 60, 61, 60, 32, 64, 65, 66, 66, 68, 69, 70, 60

Offset: 1

Olivier Gérard

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

Cf. A008479, A010846, A010847.

f:= n -> n - n/convert(numtheory:-factorset(n),`*`):
map(f, [$1..100]); # Robert Israel, Apr 10 2018

a(n) = n-A003557(n). - Vladeta Jovovic, Sep 15 2006

Definition corrected by Vladeta Jovovic, Sep 15 2006

A299990 a(n) = A243822(n) - A000005(n).

Original entry on oeis.org

-1, -2, -2, -3, -2, -3, -2, -4, -3, -2, -2, -4, -2, -2, -3, -5, -2, -2, -2, -4, -3, -1, -2, -5, -3, -1, -4, -4, -2, 2, -2, -6, -2, 0, -3, -4, -2, 0, -2, -5, -2, 3, -2, -3, -4, 0, -2, -5, -3, 0, -2, -3, -2, 0, -3, -5, -2, 0, -2, 2, -2, 0, -4, -7, -3, 6, -2, -2

Offset: 1

Michael De Vlieger, Feb 25 2018

sign

Since A010846(n) = A000005(n) + A243822(n), this sequence examines the balance of the two components among "regular" numbers.
Value of a(n) is generally less frequently negative as n increases.
a(1) = -1.
For primes p, a(p) = -2 since 1 | p and the cototient is restricted to the divisor p.
For perfect prime powers p^e, a(p^e) = -(e + 1), since all m < p^e in the cototient of p^e that do not have a prime factor q coprime to p^e are powers p^k with 1 < p^k <= p^e; all such p^k divide p^e.
Generally for n with A001221(n) = 1, a(n) = -1 * A000005(n), since the cototient is restricted to divisors, and in the case of p^e > 4, divisors and numbers in A272619(p^e) not counted by A010846(p^e).
For m >= 3, a(A002110(m)) is positive.
For m >= 9, a(A244052(m)) is positive.

			a(6) = -3 since 6 has 4 divisors, and 4 | 6^2; A243822(6) = 1 and A000005(6) = 4; 1 - 4 = -3. Alternatively, A010846(6) = 5; 5 - 2*4 = -3.
a(30) = 2 since 30 has 8 divisors and the numbers {4, 8, 9, 12, 16, 18, 20, 24, 25, 27} divide 30^e with e > 1; A243822(30) = 10 and A000005(30) = 8; 10 - 8 = 2. Alternatively, A010846(30) = 18; 18 - 2*8 = 2.
Some values of a(n) and related sequences:
   n  a(n) A010846(n) A243822(n) A000005(n) A272618(n)
  ----------------------------------------------------
   1   -1          1          0          1  0
   2   -2          2          0          2  0
   3   -2          2          0          2  0
   4   -3          3          0          3  0
   5   -2          2          0          2  0
   6   -3          5          1          4  {4}
   7   -2          2          0          2  0
   8   -4          4          0          4  0
   9   -3          3          0          3  0
  10   -2          6          2          4  {4,8}
  11   -2          2          0          2  0
  12   -4          8          2          6  {8,9}
  ...
  30    2         18         10          8  {4,8,9,12,16,18,20,24,25,27}
  ...
  34    0          8          4          4  {4,8,16,32}
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Examination of the relationships of the species of numbers enumerated in A010846.

Cf. A000005, A002110, A010846, A243822, A272618, A272619, A299991, A299992, A300155, A300156, A300157.

Table[Count[Range[n], _?(PowerMod[n, Floor@ Log2@ n, #] == 0 &)] - 2 DivisorSigma[0, n], {n, 68}]

a(n) = A010846(n) - 2*A000005(n).

A299991 Numbers n for which A243822(n) > A000005(n).

Original entry on oeis.org

30, 42, 60, 66, 70, 74, 78, 82, 84, 86, 90, 94, 98, 102, 106, 110, 114, 118, 120, 122, 126, 130, 132, 134, 138, 140, 142, 146, 150, 154, 156, 158, 162, 165, 166, 168, 170, 174, 178, 180, 182, 186, 190, 194, 195, 198, 202, 204, 206, 210, 214, 218, 220, 222, 226

Offset: 1

Michael De Vlieger, Feb 25 2018

nonn

Composite numbers m have nondivisors k in the cototient such that k | n^e with e > 1. These k appear in row n of A272618 and are enumerated by A243822(n). These nondivisors k are a kind of "regular" number along with divisors d of n; both are listed in row n of A162306 and are together enumerated by A010846(n).
This sequence lists numbers that have more nondivisors k in the cototient of n than divisors d.
This sequence contains all n for which A299990(n) is positive.
The smallest odd term is 165.
For m >= 3, A002110(m) is in a(n).
For m >= 9, A244052(m) is in a(n).

			30 is the first term since it is the smallest number for which A243822(n) > A000005(n), alternatively, for which A010846(n) > 2*A000005(n).

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

Cf. A000005, A002110, A010846, A162306, A243822, A272618, A299990.

Select[Range@ 226, Function[n, Count[Range[n], _?(PowerMod[n, Floor@ Log2@ n, #] == 0 &)] > 2 DivisorSigma[0, n]]]

A372720 a(n) = A000005(n) - A008479(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 3, 1, 1, 3, 1, 4, 3, 3, 1, 4, 1, 3, 1, 4, 1, 7, 1, 1, 3, 3, 3, 4, 1, 3, 3, 5, 1, 7, 1, 4, 4, 3, 1, 4, 1, 2, 3, 4, 1, 1, 3, 5, 3, 3, 1, 10, 1, 3, 4, 1, 3, 7, 1, 4, 3, 7, 1, 4, 1, 3, 3, 4, 3, 7, 1, 5, 1, 3, 1, 10, 3, 3, 3

Offset: 1

Michael De Vlieger, May 13 2024

sign

A095960(50) = 3, a(50) = 2.

a(162) = -2 is the first negative term.

			Table of a(n), b(n) = A000005(n), and c(n) = A008479(n) for n <= 12:
  n  b(n) c(n) a(n)
 ------------------
  1    1    1    0
  2    2    1    1
  3    2    1    1
  4    3    2    1
  5    2    1    1
  6    4    1    3
  7    2    1    1
  8    4    3    1
  9    3    2    1
 10    4    1    3
 11    2    1    1
 12    6    2    4
a(12) = 4 since 12 has 6 divisors {1, 2, 3, 4, 6, 12}, and row 12 of A369609 has 2 terms {6, 12}.
a(18) = 3 since 18 has 6 divisors {1, 2, 3, 6, 9, 18}, and row 18 of A369609 has 3 terms {6, 12, 18}.
a(50) = 2 since 50 has 6 divisors {1, 2, 5, 10, 25, 50}, and row 50 of A369609 has 4 terms {10, 20, 40, 50}
a(162) = -2 since 162 has 10 divisors {1,2,3,6,9,18,27,54,81,162} but row 162 of A369609 has 12 terms {6,12,18,24,36,48,54,72,96,108,144,162}.
a(500) = 0 since 500 has as many divisors {1,2,4,5,10,20,25,50,100,125,250,500} as terms in row 500 of A369609 {10,20,40,50,80,100,160,200,250,320,400,500}.

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

Cf. A000005, A003557, A008479, A095960, A119288, A162306, A183093, A303554, A355432, A360767, A369609.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[r = rad[n]; DivisorSigma[0, n] - Count[Range[n/r], _?(Divisible[r, rad[#]] &)], {n, 120}]

a(n) = my(f=factor(n)[, 1], s); forvec(v=vector(#f, i, [1, logint(n, f[i])]), if(prod(i=1, #f, f[i]^v[i])<=n, s++)); numdiv(n) - s; \\ after A008479 \\ Michel Marcus, Jun 03 2024

a(n) = A095960(n) for n in A303554, i.e., for squarefree n or prime powers n.
a(n) = A095960(n) for n in A360767, i.e., for nonsquarefree composite n such that omega(n) > 1 and A003557(n) < A119288(n), since A008479(n) is the number of terms k in row n of A010846 such that k <= A003557(n).
a(n) = A183093(n) - A355432(n).

A376567 a(n) = binomial(bigomega(n) + omega(n), omega(n)), where bigomega = A001222 and omega = A001221.

Original entry on oeis.org

1, 2, 2, 3, 2, 6, 2, 4, 3, 6, 2, 10, 2, 6, 6, 5, 2, 10, 2, 10, 6, 6, 2, 15, 3, 6, 4, 10, 2, 20, 2, 6, 6, 6, 6, 15, 2, 6, 6, 15, 2, 20, 2, 10, 10, 6, 2, 21, 3, 10, 6, 10, 2, 15, 6, 15, 6, 6, 2, 35, 2, 6, 10, 7, 6, 20, 2, 10, 6, 20, 2, 21, 2, 6, 10, 10, 6, 20, 2

Offset: 1

Michael De Vlieger, Oct 09 2024

For prime power p^k, a(p^k) = A010846(p^k) = A000005(p^k) = k+1. Therefore, for prime p, a(p) = A010846(p) = A000005(p) = 2.

For n in A024619, a(n) != A010846(n) and A010846(n) > A000005(n).

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

Cf. A000005, A001221, A001222, A007947, A010846, A024619, A068795, A376248, A376846, A376847.

with(NumberTheory):
a := n -> binomial(Omega(n) + Omega(n, distinct), Omega(n, distinct)):
seq(a(n), n = 1..79);  # Peter Luschny, Oct 25 2024

Array[Binomial[#2 + #1, #1] & @@ {PrimeNu[#], PrimeOmega[#]} &, 120]

a(n) = length of row n of A376248.

a(n) = A010846(n) - A376846(n) + A376847(n).

cp's OEIS Frontend

A244974 Sum of numbers m <= n whose set of prime divisors is a subset of the set of prime divisors of n.

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A376248 Irregular triangle where row n lists m such that rad(m) | n and bigomega(m) <= bigomega(n), where rad = A007947 and bigomega = A001222.

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A138109 Positive integers k whose smallest prime factor is greater than the cube root of k and strictly less than the square root of k.

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A361373 Number of prime powers p^m <= n such that p | n.

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A363061 Number of k <= P(n) such that rad(k) | P(n), where rad(n) = A007947(n) and P(n) = A002110(n).

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A010848 Number of numbers k <= n such that at least one prime factor of n is not a prime factor of k.

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A299990 a(n) = A243822(n) - A000005(n).

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