A067004 -id:A067004 - cp's OEIS frontend

A331185 a(n) = n - prime(A067004(n)), where A067004 is the ordinal transform of number of divisors of n (A000005).

-1, 0, 0, 2, 0, 4, 0, 5, 6, 5, 0, 10, 0, 7, 4, 14, 0, 15, 0, 15, 8, 5, 0, 22, 20, 7, 4, 21, 0, 27, 0, 21, 4, 3, -2, 34, 0, -3, -4, 35, 0, 35, 0, 31, 28, -1, 0, 46, 42, 31, -2, 29, 0, 43, -4, 43, -4, -9, 0, 58, 0, -9, 34, 62, -8, 49, 0, 37, -10, 51, 0, 69, 0, -9, 38, 35, -12, 55, 0, 77, 78, -15, 0, 79, -16, -17, -20, 59, 0, 83

Offset: 1

Antti Karttunen, Jan 12 2020

Antti Karttunen, Table of n, a(n) for n = 1..10201
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000005, A000040 (positions of zeros), A067004, A331186, A331187.

b[_] = 0;
c[n_] := c[n] = With[{t = DivisorSigma[0, n]}, b[t] = b[t]+1];
a[n_] := n - Prime[c[n]];
Array[a, 105] (* Jean-François Alcover, Dec 20 2021 *)

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
v067004 = ordinal_transform(vector(up_to,n,numdiv(n)));
A067004(n) = v067004[n];
A331185(n) = (n - prime(A067004(n)));

a(n) = n - A000040(A067004(n)).

A331186 Exponent of the highest power of prime(A067004(n)) which divides n, where A067004 is the ordinal transform of number of divisors of n (A000005).

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 0, 2, 1, 1, 2, 1, 1, 0, 4, 1, 2, 1, 1, 0, 0, 1, 3, 2, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 2, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 4, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 6, 0, 0, 1, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 1, 0, 4, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0

Offset: 1

Antti Karttunen, Jan 12 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000005, A067004, A286561.

Cf. also A331185, A331187.

b[_] = 0;
A067004[n_] := A067004[n] = With[{t = DivisorSigma[0, n]}, b[t] = b[t]+1];
a[n_] := IntegerExponent[n, Prime[A067004[n]]];
Array[a, 105] (* Jean-François Alcover, Dec 21 2021 *)

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
v067004 = ordinal_transform(vector(up_to,n,numdiv(n)));
A067004(n) = v067004[n];
A331186(n) = valuation(n,prime(A067004(n)));

a(n) = A286561(n, A000040(A067004(n))), where A286561(n,k) gives the k-valuation of n.

A331187 a(n) is n divided by the highest power of prime(A067004(n)) which divides it, where A067004 is the ordinal transform of number of divisors of n (A000005).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 8, 1, 2, 1, 3, 1, 2, 15, 1, 1, 2, 1, 4, 21, 22, 1, 3, 1, 26, 27, 4, 1, 10, 1, 32, 33, 34, 35, 9, 1, 38, 39, 8, 1, 6, 1, 44, 45, 46, 1, 3, 1, 50, 51, 52, 1, 54, 55, 56, 57, 58, 1, 15, 1, 62, 63, 1, 65, 66, 1, 68, 69, 70, 1, 8, 1, 74, 75, 76, 77, 78, 1, 80, 1, 82, 1, 84, 85, 86, 87, 88, 1, 90

Offset: 1

Antti Karttunen, Jan 12 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000040, A067004, A331186.

Cf. also A331185.

b[_] = 0;
A067004[n_] := A067004[n] = With[{t = DivisorSigma[0, n]}, b[t] = b[t]+1];
A331186[n_] := IntegerExponent[n, Prime[A067004[n]]];
a[n_] := n/(Prime[A067004[n]]^A331186[n]);
Array[a, 105] (* Jean-François Alcover, Dec 21 2021 *)

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
v067004 = ordinal_transform(vector(up_to,n,numdiv(n)));
A067004(n) = v067004[n];
A331187(n) = { my(p=prime(A067004(n))); n/(p^valuation(n,p)); };

a(n) = n / (A000040(A067004(n))^A331186(n)).

A081373 Number of values of k, 1 <= k <= n, with phi(k) = phi(n), where phi is Euler totient function, A000010.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Mar 24 2003

nonn

Ordinal transform of Euler totient function phi, A000010. - Antti Karttunen, Aug 26 2024

			For n = 16: phi(k) = {1,1,2,2,4,2,6,4,6,4,10,4,12,6,8,8} for k = 1,...,n; 2 numbers exist with phi(k) = phi(n) = 8: {15,16}, so a(16) = 2.
If n = p is an odd prime number, then a(p) = 1 with phi(k) = p-1.

Paul Tek, Table of n, a(n) for n = 1..100000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010, A081375 (positions of records), A210719 (of 1's).

Cf. also A067004, A303756, A303757, A303777 (ordinal transform of this sequence).

f[x_] := Count[Table[EulerPhi[j]-EulerPhi[x], {j, 1, x}], 0] Table[f[w], {w, 1, 100}]

a(n)=my(t=eulerphi(n), s); sum(k=1, n, eulerphi(k)==t) \\ Charles R Greathouse IV, Feb 21 2013, corrected by Antti Karttunen, Aug 26 2024

a(n) = #select(x -> x <= n, invphi(eulerphi(n))); \\ Amiram Eldar, Nov 08 2024, using Max Alekseyev's invphi.gp

A047983 Number of integers less than n but with the same number of divisors.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 1, 1, 2, 4, 0, 5, 3, 4, 0, 6, 1, 7, 2, 5, 6, 8, 0, 2, 7, 8, 3, 9, 1, 10, 4, 9, 10, 11, 0, 11, 12, 13, 2, 12, 3, 13, 5, 6, 14, 14, 0, 3, 7, 15, 8, 15, 4, 16, 5, 17, 18, 16, 0, 17, 19, 9, 0, 20, 6, 18, 10, 21, 7, 19, 1, 20, 22, 11, 12, 23, 8, 21, 1, 1, 24, 22, 2, 25, 26, 27

Offset: 1

Simon Colton (simonco(AT)cs.york.ac.uk)

Invented by the HR concept formation program.

			f(10) = 2 because tau(10) = 4 and also tau(6) = tau(8) = 4.

T. D. Noe, Table of n, a(n) for n = 1..10000
Simon Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2 (1999), Article 99.1.2.
Simon Colton, HR - Automatic Theory Formation in Pure Mathematics, 1998-1999. [Wayback Machine link]

Position of the 0's form A007416.

Cf. A000005, A005179, A067004.

a047983 n = length [x | x <- [1..n-1], a000005 x == a000005 n]
-- Reinhard Zumkeller, Nov 06 2011

a[n_] := With[{tau = DivisorSigma[0, n]}, Length[ Select[ Range[n-1], DivisorSigma[0, #] == tau & ]]]; Table[a[n], {n, 1, 87}] (* Jean-François Alcover, Nov 30 2011 *)
Module[{nn=90,ds},ds=DivisorSigma[0,Range[nn]];Table[Count[Take[ds,n], ds[[n]]]- 1,{n,nn}]] (* Harvey P. Dale, Feb 16 2014 *)

A047983(n) = {local(d);d=numdiv(n);sum(k=1,n-1,(numdiv(k)==d))} \\ Michael B. Porter, Mar 01 2010

from sympy import divisor_count as D
def a(n): return sum([1 for k in range(1, n) if D(k) == D(n)]) # Indranil Ghosh, Apr 30 2017

f(n) = |{k < n : tau(k) = tau(n)}|.

a(n) = A067004(n) - 1. - Amiram Eldar, Feb 04 2025

A067003 Number of numbers <= n with same number of distinct prime factors as n.

Original entry on oeis.org

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

Offset: 1

Henry Bottomley, Dec 21 2001

			a(11)=8 since 2,3,4,5,7,8,9,11 each have one distinct prime factor. a(12)=3 since 6,10,12 each have two distinct prime factors.
From _Gus Wiseman_, Dec 28 2018: (Start)
Column n lists the a(n) positive integers less than or equal to n with the same number of distinct prime factors as n:
  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20
  ---------------------------------------------------------------------
  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20
        2  3  4     5  7  8  6   9   10  11  12  14  13  16  15  17  18
           2  3     4  5  7      8   6   9   10  12  11  13  14  16  15
              2     3  4  5      7       8   6   10  9   11  12  13  14
                    2  3  4      5       7       6   8   9   10  11  12
                       2  3      4       5           7   8   6   9   10
                          2      3       4           5   7       8   6
                                 2       3           4   5       7
                                         2           3   4       5
                                                     2   3       4
                                                         2       3
                                                                 2
(End)

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Positions of 1's are A002110.
Cf. A001221, A008479, A058933, A067004. Inverse of A000961, A007774, A033992, A033993, A051270 etc.
Cf. A000010, A006049, A061142, A294277, A294278, A302242, A322837, A322841.

Table[Length[Select[Range[n],PrimeNu[#]==PrimeNu[n]&]],{n,100}] (* Gus Wiseman, Dec 28 2018 *)

a(n) = my(nb = #factor(n)~); sum(k=1, n, #factor(k)~ == nb); \\ Michel Marcus, Jul 13 2019

a(A002110(n)) = 1.

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

Original entry on oeis.org

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.

A081375 a(n) is the least number k such that A081373(k) = n.

Original entry on oeis.org

1, 2, 6, 12, 30, 42, 72, 78, 84, 90, 190, 216, 222, 228, 234, 252, 270, 540, 546, 570, 630, 738, 744, 770, 792, 858, 900, 924, 930, 990, 1050, 1638, 1710, 1890, 1980, 2100, 2310, 2418, 2442, 2508, 2562, 2574, 2604, 2700, 2772, 2790, 2850, 2970, 3150

Offset: 1

Labos Elemer, Mar 24 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010, A008479, A081373, A081374, A067004.

f[x_] := Count[Table[EulerPhi[j]-EulerPhi[x], {j, 1, x}], 0] t=Table[0, {50}]; Do[s=f[n]; If[s<51&&t[[s]]==0, t[[s]]=n], {n, 1, 4000}]; t

lista(len) = {my(v = vector(len), c = 0, k = 1, i); while(c < len, i = #select(x -> x <= k, invphi(eulerphi(k))); if(i <= len && v[i] == 0, c++; v[i] = k); k++); v;} \\ Amiram Eldar, Nov 08 2024, using Max Alekseyev's invphi.gp

A334655 Number of integers less than n with the same number of distinct prime factors as n.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Oct 31 2020

nonn

			a(12) = 2 because omega(12) = 2 and also omega(6) = omega(10) = 2.

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

Cf. A001221, A002110 (positions of 0's), A047983, A067003, A067004, A322837, A322841, A335097.

R:= NULL:
for n from 1 to 100 do
  w:= nops(numtheory:-factorset(n));
  if assigned(V[w]) then V[w]:= V[w]+1 else V[w]:= 1 fi;
  R:= R, V[w]-1
od:
R; # Robert Israel, Feb 25 2024

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

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

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

a(n) = A067003(n) - 1.

A138009 a(n) = number of positive integers k, k <= n, where d(k) >= d(n); d(n) = number of positive divisors of n.

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 6, 2, 4, 3, 10, 1, 12, 5, 6, 2, 16, 2, 18, 3, 10, 11, 22, 1, 15, 13, 14, 5, 28, 2, 30, 7, 18, 19, 20, 1, 36, 22, 23, 4, 40, 5, 42, 11, 12, 28, 46, 1, 33, 14, 31, 15, 52, 7, 34, 8, 36, 37, 58, 1, 60, 39, 19, 10, 42, 10, 66, 22, 45, 11, 70, 2, 72, 48, 25, 26, 51, 13, 78, 4

Offset: 1

Leroy Quet, Feb 27 2008

			9 has 3 positive divisors. Among the first 9 positive integers, there are four that have more than or equal the number of divisors than 9 has: 4, with 3 divisors; 6, with 4 divisors; 8, with 4 divisors; and 9, with 3 divisors. So a(9) = 4.

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

Cf. A000005, A002182, A067004, A079788.

L:= [2]: A[1]:= 1:
for n from 2 to 100 do
  v:= 2*numtheory:-tau(n);
  k:= ListTools:-BinaryPlace(L,v-1);
  A[n]:= n-k;
  L:= [op(L[1..k]),v,op(L[k+1..-1])];
od:
seq(A[i],i=1..100); # Robert Israel, Sep 26 2018

Table[Length[Select[Range[n], Length[Divisors[ # ]]>=Length[Divisors[n]]&]], {n,1,100}] (* Stefan Steinerberger, Feb 29 2008 *)

a(n) = my(dn=numdiv(n)); sum(k=1, n, numdiv(k) >= dn); \\ Michel Marcus, Sep 26 2018

From Amiram Eldar, Jun 26 2025: (Start)
a(n) = n - 1 if and only if n is prime.
a(n) = 1 if and only if n is a highly composite number (A002182). (End)

More terms from Stefan Steinerberger, Feb 29 2008

cp's OEIS Frontend

A331185 a(n) = n - prime(A067004(n)), where A067004 is the ordinal transform of number of divisors of n (A000005).

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A331186 Exponent of the highest power of prime(A067004(n)) which divides n, where A067004 is the ordinal transform of number of divisors of n (A000005).

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A331187 a(n) is n divided by the highest power of prime(A067004(n)) which divides it, where A067004 is the ordinal transform of number of divisors of n (A000005).

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A081373 Number of values of k, 1 <= k <= n, with phi(k) = phi(n), where phi is Euler totient function, A000010.

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A047983 Number of integers less than n but with the same number of divisors.

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A067003 Number of numbers <= n with same number of distinct prime factors as n.

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

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