A334655 -id:A334655 - cp's OEIS frontend

A335097 Number of integers less than n with the same number of prime factors (counted with multiplicity) as n.

0, 0, 1, 0, 2, 1, 3, 0, 2, 3, 4, 1, 5, 4, 5, 0, 6, 2, 7, 3, 6, 7, 8, 1, 8, 9, 4, 5, 9, 6, 10, 0, 10, 11, 12, 2, 11, 13, 14, 3, 12, 7, 13, 8, 9, 15, 14, 1, 16, 10, 17, 11, 15, 4, 18, 5, 19, 20, 16, 6, 17, 21, 12, 0, 22, 13, 18, 14, 23, 15, 19, 2, 20, 24, 16, 17, 25, 18, 21, 3

Offset: 1

Ilya Gutkovskiy, Oct 31 2020

nonn

			a(10) = 3 because bigomega(10) = 2 and also bigomega(4) = bigomega(6) = bigomega(9) = 2.

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

Cf. A000079 (positions of 0's), A001222, A047983, A058933, A067004, A322838, A334655.

A:= NULL:
for n from 1 to 100 do
  t:= numtheory:-bigomega(n);
  if not assigned(R[t]) then
    A:= A,0;
    R[t]:= 1;
   else
    A:= A, R[t];
    R[t]:= R[t]+1;
   fi
od:
A; # Robert Israel, Oct 24 2021

Table[Length[Select[Range[n - 1], PrimeOmega[#] == PrimeOmega[n] &]], {n, 80}]

a(n)={my(t=bigomega(n)); sum(k=1, n-1, bigomega(k)==t)} \\ Andrew Howroyd, Oct 31 2020

from math import prod, isqrt
from sympy import isprime, primepi, primerange, integer_nthroot, primeomega
def A335097(n):
    if n==1: return 0
    if isprime(n): return primepi(n)-1
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    return int(sum(primepi(n//prod(c[1] for c in a))-a[-1][0] for a in g(n,0,1,1,primeomega(n)))-1) # Chai Wah Wu, Aug 28 2024

a(n) = |{j < n : bigomega(j) = bigomega(n)}|.

a(n) = A058933(n) - 1.

A339512 Number of subsets of {1..n} whose elements have the same number of distinct prime factors.

Original entry on oeis.org

1, 2, 3, 5, 9, 17, 18, 34, 66, 130, 132, 260, 264, 520, 528, 544, 1056, 2080, 2112, 4160, 4224, 4352, 4608, 8704, 9216, 17408, 18432, 34816, 36864, 69632, 69633, 135169, 266241, 270337, 278529, 294913, 327681, 589825, 655361, 786433, 1048577, 1572865, 1572867

Offset: 0

Ilya Gutkovskiy, Dec 07 2020

nonn

			a(5) = 17 subsets: {}, {1}, {2}, {3}, {4}, {5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 5}, {2, 3, 4}, {2, 3, 5}, {2, 4, 5}, {3, 4, 5} and {2, 3, 4, 5}.

Sebastian Karlsson, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Distinct Prime Factors

Cf. A001221, A339511, A339514.

from sympy import primefactors
def test(n):
    if n==0: return -1
    return len(primefactors(n))
def a(n):
    tests = [test(i) for i in range(n+1)]
    return sum(2**tests.count(v)-1 for v in set(tests))
print([a(n) for n in range(43)]) # Michael S. Branicky, Dec 07 2020

a(n) = 1 + Sum_{k=1..n} 2^A334655(k). - Sebastian Karlsson, Feb 18 2021

a(23)-a(42) from Michael S. Branicky, Dec 07 2020

A337557 Number of integers less than n with the same number of odd divisors as n.

Original entry on oeis.org

0, 1, 0, 2, 1, 2, 3, 3, 0, 4, 5, 6, 7, 8, 0, 4, 9, 1, 10, 11, 1, 12, 13, 14, 2, 15, 2, 16, 17, 3, 18, 5, 4, 19, 5, 3, 20, 21, 6, 22, 23, 7, 24, 25, 0, 26, 27, 28, 4, 5, 8, 29, 30, 9, 10, 31, 11, 32, 33, 12, 34, 35, 1, 6, 13, 14, 36, 37, 15, 16, 38, 6, 39, 40, 2, 41, 17, 18, 42, 43

Offset: 1

Ilya Gutkovskiy, Oct 31 2020

			a(10) = 4 because A001227(10) = 2 and also A001227(3) = A001227(5) = A001227(6) = A001227(7) = 2.

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

Cf. A001227, A047983, A334655, A335097.

Table[Length[Select[Range[n - 1], Sum[Mod[d, 2], {d, Divisors[#]}] == Sum[Mod[d, 2], {d, Divisors[n]}] &]], {n, 80}]

a(n)={my(t=numdiv(n/2^valuation(n, 2))); sum(k=1, n-1, numdiv(k/2^valuation(k, 2))==t)} \\ Andrew Howroyd, Oct 31 2020

first(n) = { my(m = Map(), res = vector(n)); for(i = 1, n, q = numdiv(i >> valuation(i, 2)); if(mapisdefined(m, q), res[i] = mapget(m, q); mapput(m, q, res[i]+1); , mapput(m, q, 1) ) ); res } \\ David A. Corneth, Oct 31 2020

a(n) = |{j < n : A001227(j) = A001227(n)}|.

A377734 Number of integers less than n that have the same smallest prime factor as n.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 3, 1, 4, 0, 5, 0, 6, 2, 7, 0, 8, 0, 9, 3, 10, 0, 11, 1, 12, 4, 13, 0, 14, 0, 15, 5, 16, 2, 17, 0, 18, 6, 19, 0, 20, 0, 21, 7, 22, 0, 23, 1, 24, 8, 25, 0, 26, 3, 27, 9, 28, 0, 29, 0, 30, 10, 31, 4, 32, 0, 33, 11, 34, 0, 35, 0, 36, 12, 37, 2, 38, 0, 39

Offset: 1

Ilya Gutkovskiy, Nov 05 2024

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Least Prime Factor.

Cf. A020639, A038110, A038111, A047983, A078898, A334655, A335097, A377730.

Table[Length[Select[Range[n - 1], If[# == 1, 1, FactorInteger[#][[1, 1]]] == If[n == 1, 1, FactorInteger[n][[1, 1]]] &]], {n, 80}]
seq[len_] := Module[{t = Table[FactorInteger[n][[1,1]], {n, 1, len}], s = Table[0, {len}]}, Do[s[[i]] = Count[t[[1;;i-1]], t[[i]]], {i, 1, len}]; s]; seq[80] (* Amiram Eldar, Nov 21 2024 *)

a(n) = if (n>1, my(p=vecmin(factor(n)[,1])); sum(k=2, n-1, p == vecmin(factor(k)[,1])), 0); \\ Michel Marcus, Nov 16 2024

a(n) = |{j < n : lpf(j) = lpf(n)}|.
a(n) = A078898(n) - 1.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Sum_{k>=1} (A038110(k)/A038111(k))^2 = 0.2847976823663... . - Amiram Eldar, Nov 21 2024

A377730 Number of integers less than n that have the same greatest prime factor as n.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 2, 2, 1, 0, 3, 0, 1, 2, 3, 0, 4, 0, 3, 2, 1, 0, 5, 4, 1, 6, 3, 0, 5, 0, 4, 2, 1, 4, 7, 0, 1, 2, 6, 0, 5, 0, 3, 7, 1, 0, 8, 6, 8, 2, 3, 0, 9, 4, 7, 2, 1, 0, 9, 0, 1, 8, 5, 4, 5, 0, 3, 2, 9, 0, 10, 0, 1, 10, 3, 6, 5, 0, 11, 11, 1, 0, 10, 4, 1, 2, 7, 0, 12

Offset: 1

Ilya Gutkovskiy, Nov 05 2024

nonn

Eric Weisstein's World of Mathematics, Greatest Prime Factor

Cf. A006530, A047983, A078899, A334655, A335097, A377734.

R:= NULL:
for n from 1 to 100 do  p:= max(numtheory:-factorset(n));  if assigned(C[p]) then C[p]:= C[p]+1 else C[p]:= 0 fi;
  R:= R, C[p]
od:R; # Robert Israel, Nov 07 2024

Table[Length[Select[Range[n - 1], FactorInteger[#][[-1, 1]] == FactorInteger[n][[-1, 1]] &]], {n, 90}]

a(n) = |{j < n : gpf(j) = gpf(n)}|.

a(n) = A078899(n) - 1.

cp's OEIS Frontend

A335097 Number of integers less than n with the same number of prime factors (counted with multiplicity) as n.

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A339512 Number of subsets of {1..n} whose elements have the same number of distinct prime factors.

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A337557 Number of integers less than n with the same number of odd divisors as n.

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A377734 Number of integers less than n that have the same smallest prime factor as n.

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A377730 Number of integers less than n that have the same greatest prime factor as n.

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