A030513 -id:A030513 - cp's OEIS frontend

A211337 Numbers k for which the number of divisors, tau(k), is congruent to 1 modulo 3.

1, 6, 8, 10, 14, 15, 21, 22, 26, 27, 33, 34, 35, 38, 39, 46, 48, 51, 55, 57, 58, 62, 64, 65, 69, 74, 77, 80, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 112, 115, 118, 119, 120, 122, 123, 125, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 162

Offset: 1

Douglas Latimer, Apr 07 2012

nonn

Any term a(n) can be expressed as 1 term from A211484 times 1 nonzero term from A000578. - Douglas Latimer, Apr 20 2012

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 4, 36, 366, 3635, 36499, 365456, 3654240, 36538501, 365382167, 3653804173, ... . Conjecture: the asymptotic density of this sequence exists and equals 3*zeta(3)/Pi^2 = 0.3653814847007... (A346602), so, a(n) ~ k*n with k = Pi^2/(3*zeta(3)) = 2.73686555524... . This conjecture is true if this sequence and A211338 have the same density (see A059269). - Amiram Eldar, Jan 06 2024

			The divisors of 10  are: 1, 2, 5, 10 (4 divisors). 4 is congruent to 1 modulo 3. Thus 10 is a member of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Douglas Latimer)

This is an extension of A030513 (numbers with 4 divisors).
The union of A059269 and A211338 is the complementary sequence to this one.
The definition of this sequence uses A000005 (the number of divisors of n).
Cf. A000578, A074794, A211484, A346602.

Select[Range[162], Mod[DivisorSigma[0, #], 3] == 1 &] (* T. D. Noe, Apr 21 2012 *)

{plnt=1 ; mxind=100 ;for(k=1, 10^6,
if(numdiv(k) % 3 == 1, print(k); plnt++; if(mxind+1 ==  plnt, break() )))}

Conjecture: a(n) ~ k*n where k = 2/prod(1 - (p-1)/(p^(3*k))) = 2.7290077... where p ranges over the primes and k ranges over the positive integers. - Charles R Greathouse IV, Apr 13 2012

A007964 Numbers k such that product of proper divisors of k is <= k; i.e., product of divisors of k is <= k^2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 106

Offset: 1

R. Muller

Numbers which are the product of up to two primes (not necessarily distinct) or the cube of a prime. Alternatively, numbers having prime decomposition p*q, where q either is distinct from p or equals p^k for 0 <= k <= 2.
Corresponds to numbers having at most four divisors. (For numbers with exactly four divisors see A030513.) - Lekraj Beedassy, Sep 23 2003
For n > 3: numbers that can occur as fourth divisors; union of A000040, A001248, A006881 and A030078. - Reinhard Zumkeller, May 15 2006

Liu Hongyan and Zhang Wenpeng, On the simple numbers and the mean value properties, Smarandache Notions (Book Series, Vol. 14), American Research Press, 2004; pp. 171-175.

Amiram Eldar, Table of n, a(n) for n = 1..10000
F. Smarandache, Only Problems, Not Solutions! (listing the sequence as the "simple numbers" in Unsolved Problem: 23 p.23).

Cf. A007955, A058080.

Select[Range[100], DivisorSigma[0, #] < 5 &] (* Amiram Eldar, Apr 30 2020 *)

is(n)=numdiv(n)<5 \\ Charles R Greathouse IV, Sep 16 2015

Description corrected by Henry Bottomley, Nov 24 2000

A039832 Numbers k such that k and k+1 both have 4 divisors.

Original entry on oeis.org

14, 21, 26, 33, 34, 38, 57, 85, 86, 93, 94, 118, 122, 133, 141, 142, 145, 158, 177, 201, 202, 205, 213, 214, 217, 218, 253, 298, 301, 302, 326, 334, 381, 393, 394, 445, 446, 453, 481, 501, 514, 526, 537, 542, 553, 565, 622, 633, 634, 694, 697, 698, 706, 717, 745, 766, 778, 793, 802, 817

Offset: 1

N. J. A. Sloane, Felice Russo, Olivier Gérard, Dec 11 1999

nonn

			14 and 15 both have 4 as number of divisors and are consecutive.

David Wells, Curious and interesting numbers, Penguin Books, 1986, p. 91.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)

Intersection of A005237 and A030513.

Cf. A038456, A039833.

Flatten[Position[Partition[Table[DivisorSigma[0, n], {n, 1000}], 2, 1], ?(#=={4, 4}&)]] (* _Vincenzo Librandi, Oct 21 2012 *)

isA039832(n) = numdiv(n)==4 && numdiv(n+1)==4 \\ Michael B. Porter, Feb 03 2010

A073915 Triangle read by rows in which the n-th row contains the first n numbers with n divisors.

Original entry on oeis.org

1, 2, 3, 4, 9, 25, 6, 8, 10, 14, 16, 81, 625, 2401, 14641, 12, 18, 20, 28, 32, 44, 64, 729, 15625, 117649, 1771561, 4826809, 24137569, 24, 30, 40, 42, 54, 56, 66, 70, 36, 100, 196, 225, 256, 441, 484, 676, 1089, 48, 80, 112, 162, 176, 208, 272, 304, 368, 405

Offset: 1

Amarnath Murthy, Aug 18 2002

The first row contains the 1. The 2nd row contains the beginning of A000040. The 3rd contains the beginning of A001248, the 4th through 7th A030513 to A030516. The 8th through 20th rows come from A030626 to A030638. - R. J. Mathar, Mar 23 2007

			1;
2,3;
4,9,25;
6,8,10,14;
16,81,625,2401,14641;
...

T. D. Noe, Rows n = 1..100 of triangle, flattened

Cf. A073916.

d = Table[Length[Divisors[n]], {n, 2000}]; t = {}; n = 0; ok = True; While[ok, n++; If[PrimeQ[n], AppendTo[t, Prime[Range[n]]^(n - 1)], c = Flatten[Position[d, n, 1, n]]; If[Length[c] >= n, AppendTo[t, c], ok = False]]]; Flatten[t] (* T. D. Noe, Jun 23 2013 *)

Corrected and extended by Sascha Kurz, Jan 28 2003

A080256 Sum of numbers of distinct and of all prime factors of n.

Original entry on oeis.org

0, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 5, 2, 4, 4, 5, 2, 5, 2, 5, 4, 4, 2, 6, 3, 4, 4, 5, 2, 6, 2, 6, 4, 4, 4, 6, 2, 4, 4, 6, 2, 6, 2, 5, 5, 4, 2, 7, 3, 5, 4, 5, 2, 6, 4, 6, 4, 4, 2, 7, 2, 4, 5, 7, 4, 6, 2, 5, 4, 6, 2, 7, 2, 4, 5, 5, 4, 6, 2, 7, 5, 4, 2, 7, 4, 4, 4, 6, 2, 7, 4, 5, 4, 4, 4, 8, 2, 5, 5, 6, 2, 6, 2, 6, 6

Offset: 1

Reinhard Zumkeller, Feb 10 2003

a(n) = 2 iff n is prime, A000040; a(n) > 2 iff n is composite, A002808; a(n) <= 3 iff n is prime or square of prime, A000430; a(n) = 3 iff n is square of prime, A001248; a(A080257(n)) > 3;

a(n) <= 4 iff product of proper divisors <= n^2, A007964; a(n) = 4 iff n has four divisors, A030513; a(n) > 4 iff product of proper divisors > n^2, A058080; a(A064598(n)) <= 5; a(A080258(n)) = 5.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000040, A000430, A002808, A001248, A007964, A058080, A064598, A080257, A080258.

Cf. A001221, A001222, A046660, A077761, A083342.

f[n_] := Plus @@ (Last /@ FactorInteger[n] + 1); Table[ f[n], {n, 105}] (* Robert G. Wilson v, Aug 03 2005 *)

a(n) = {my(f = factor(n)); omega(f) + bigomega(f);} \\ Amiram Eldar, Sep 28 2023

a(n) = Omega(n) + omega(n) = A001221(n) + A001222(n).
Additive with a(p^e) = e + 1.
Sum_{k=1..n} a(k) = 2 * n * log(log(n)) + c * n + O(n/log(n)), where c = A077761 + A083342 = 1.29615109474508069537... . - Amiram Eldar, Sep 28 2023

A137493 Numbers with 30 divisors.

Original entry on oeis.org

720, 1008, 1200, 1584, 1620, 1872, 2268, 2352, 2448, 2592, 2736, 2800, 3312, 3564, 3888, 3920, 4050, 4176, 4212, 4400, 4464, 4608, 5200, 5328, 5508, 5808, 5904, 6156, 6192, 6768, 6800, 7452, 7500, 7600, 7632, 7938, 8112, 8496, 8624, 8784, 9200, 9396

Offset: 1

R. J. Mathar, Apr 22 2008

nonn

Maple implementation: see A030513.

Numbers of the form p^29 (subset of A122970), p*q^2*r^4 (A179669), p^4*q^5 (A179702), p^2*q^9 (like 4608) or p*q^14, where p, q and r are distinct primes. - R. J. Mathar, Mar 01 2010

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A137492 (29 divs), A139571 (31 divs).

Select[Range[10000],DivisorSigma[0,#]==30&]  (* Harvey P. Dale, Feb 18 2011 *)

is(n)=numdiv(n)==30 \\ Charles R Greathouse IV, Jun 19 2016

list(lim)=
{
  my(f=(v,s)->concat(v,listsig(lim,s,1)));
  Set(fold(f, [[], [29], [5, 4], [9, 2], [14, 1], [4, 2, 1]]));
}
listsig(lim, sig, coprime)=
{
  my(e=sig[1]);
  if(#sig<2,
    if(#sig==0 || sig[1]==0, return(if(lim<1,[],[1])));
    my(P=primes([2,sqrtnint(lim\1,e)]));
    if(coprime==1, return(if(e>1,apply(p->p^e,P),P)));
    P=select(p->gcd(p,coprime)==1, P);
    if(e>1, P=apply(p->p^e, P));
    return(P);
  );
  my(v=List(),ss=sig[2..#sig],t=leastOfSig(ss));
  forprime(p=2,sqrtnint(lim\t,e),
    if(coprime%p,
        my(u=listsig(lim\p^e,ss,coprime*p));
        for(i=1,#u, listput(v,p^e*u[i]));
    )
  );
  Vec(v);
} \\ Charles R Greathouse IV, Nov 18 2021

A000005(a(n))=30.

A036455 Numbers n such that d(d(n)) is an odd prime, where d(k) is the number of divisors of k.

Original entry on oeis.org

6, 8, 10, 14, 15, 21, 22, 26, 27, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 120, 122, 123, 125, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 168, 177, 178, 183

Offset: 1

Labos Elemer

nonn

Compare with sequence A007422 and A030513 -- the resemblance is rather strong. Still this sequence is different. For example, 36, 100, 120, and 168 are here.

			a(15) = 39 and d(39) = 4, d(d(39)) = d(4) = 3 and d(d(d(39))) = 2. After 3 iteration the equilibrium is reached.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000005, A007422, A030513, A036450, A036452, A036454, A036456, A036457, A036458.

filter:= proc(n) local r;
  r:= numtheory:-tau(numtheory:-tau(n));
  r::odd and isprime(r)
end proc:
select(filter, [$1..1000]); # Robert Israel, Feb 02 2016

fQ[n_] := Module[{d2 = DivisorSigma[0, DivisorSigma[0, n]]}, d2 > 2 && PrimeQ[d2]]; Select[Range[200], fQ] (* T. D. Noe, Jan 22 2013 *)

is(n)=isprime(n=numdiv(numdiv(n))) && n>2 \\ Charles R Greathouse IV, Jan 22 2013

d(d(d(a(n)))) = 2 for all n.

A036459(a(n)) = 3. - Ivan Neretin, Jan 25 2016

Definition clarified by R. J. Mathar and Charles R Greathouse IV, Jan 22 2013

A137484 Numbers with 21 divisors.

Original entry on oeis.org

576, 1600, 2916, 3136, 7744, 10816, 18225, 18496, 23104, 33856, 35721, 53824, 61504, 62500, 87616, 88209, 107584, 118336, 123201, 140625, 141376, 179776, 210681, 222784, 238144, 263169, 287296, 322624, 341056, 385641, 399424, 440896

Offset: 1

R. J. Mathar, Apr 22 2008

nonn

Maple implementation: see A030513.

Numbers of the form p^20 or p^2*q^6 (A189990) where p and q are distinct primes. - R. J. Mathar, Mar 01 2010

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000005, A030513, A030638 (20 divisors), A137485 (22 divisors), A189990.

Select[Range[450000],DivisorSigma[0,#]==21&] (* Vladimir Joseph Stephan Orlovsky, May 03 2011 *)

is(n)=numdiv(n)==21 \\ Charles R Greathouse IV, Jun 19 2016

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A137484(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 n+x-sum(primepi(isqrt(x//p**6)) for p in primerange(integer_nthroot(x,6)[0]+1))+primepi(integer_nthroot(x,8)[0])-primepi(integer_nthroot(x,20)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

A000005(a(n)) = 21.

Sum_{n>=1} 1/a(n) = P(2)*P(6) - P(8) + P(20) = 0.00365945..., where P is the prime zeta function. - Amiram Eldar, Jul 03 2022

A137488 Numbers with 25 divisors.

Original entry on oeis.org

1296, 10000, 38416, 50625, 194481, 234256, 456976, 1185921, 1336336, 1500625, 2085136, 2313441, 4477456, 6765201, 9150625, 10556001, 11316496, 14776336, 16777216, 17850625, 22667121, 29986576, 35153041, 45212176, 52200625

Offset: 1

R. J. Mathar, Apr 22 2008

nonn

Maple implementation: see A030513.

Numbers of the form p^24 (24th powers of A000040, subset of A010812) or p^4*q^4 (A189991), where p and q are distinct primes. - R. J. Mathar, Mar 01 2010

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000005, A010812, A030513-A030516, A030626, A030627, A030634-A030638, A005179, A003680, A096932, A061286, A061283, A135581, A175755.

a137488 n = a137488_list !! (n-1)
a137488_list = m (map (^ 24) a000040_list) (map (^ 4) a006881_list) where
   m xs'@(x:xs) ys'@(y:ys) | x < y = x : m xs ys'
                           | otherwise = y : m xs' ys
-- Reinhard Zumkeller, Nov 29 2011

lst = {}; Do[If[DivisorSigma[0, n] == 25, Print[n]; AppendTo[lst, n]], {n, 55000000}]; lst (* Vladimir Joseph Stephan Orlovsky, May 03 2011 *)
Select[Range[5221*10^4],DivisorSigma[0,#]==25&] (* Harvey P. Dale, Mar 11 2019 *)

is(n)=numdiv(n)==25 \\ Charles R Greathouse IV, Jun 19 2016

from math import isqrt
from sympy import primepi, integer_nthroot, primerange
def A137488(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(y:=integer_nthroot(x,4)[0])))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1)))-primepi(integer_nthroot(x,24)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 22 2025

A000005(a(n)) = 25.

Sum_{n>=1} 1/a(n) = (P(4)^2 - P(8))/2 + P(24) = 0.000933328..., where P is the prime zeta function. - Amiram Eldar, Jul 03 2022

A137485 Numbers with 22 divisors.

Original entry on oeis.org

3072, 5120, 7168, 11264, 13312, 17408, 19456, 23552, 29696, 31744, 37888, 41984, 44032, 48128, 54272, 60416, 62464, 68608, 72704, 74752, 80896, 84992, 91136, 99328, 103424, 105472, 109568, 111616, 115712, 118098, 130048, 134144, 140288

Offset: 1

R. J. Mathar, Apr 22 2008

nonn

Maple implementation: see A030513.

Numbers of the form p^21 or p*q^10, where p and q are distinct primes. - R. J. Mathar, Mar 01 2010

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A030513-A030516, A030626, A030627, A030634-A030638, A005179, A003680, A096932, A061286, A061283.

A137485=proc(q) local n;
for n from 1 to q do if tau(n)=22 then print(n); fi; od; end:
A137485(10^10);

Select[Range[200000],DivisorSigma[0,#]==22&] (* Vladimir Joseph Stephan Orlovsky, May 05 2011 *)

is(n)=numdiv(n)==22 \\ Charles R Greathouse IV, Jun 19 2016

from sympy import primepi, integer_nthroot, primerange
def A137485(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 n+x-sum(primepi(x//p**10) for p in primerange(integer_nthroot(x,10)[0]+1))+primepi(integer_nthroot(x,11)[0])-primepi(integer_nthroot(x,21)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

A000005(a(n))=22.

cp's OEIS Frontend

A211337 Numbers k for which the number of divisors, tau(k), is congruent to 1 modulo 3.

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A007964 Numbers k such that product of proper divisors of k is <= k; i.e., product of divisors of k is <= k^2.

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A039832 Numbers k such that k and k+1 both have 4 divisors.

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A073915 Triangle read by rows in which the n-th row contains the first n numbers with n divisors.

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A080256 Sum of numbers of distinct and of all prime factors of n.

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A137493 Numbers with 30 divisors.

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A036455 Numbers n such that d(d(n)) is an odd prime, where d(k) is the number of divisors of k.

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Maple