A030636 -id:A030636 - cp's OEIS frontend

A030513 Numbers with 4 divisors.

6, 8, 10, 14, 15, 21, 22, 26, 27, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 125, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187

Offset: 1

Jeff Burch

Essentially the same as A007422.
Numbers which are either the product of two distinct primes (A006881) or the cube of a prime (A030078).
4*a(n) are the solutions to A048272(x) = Sum_{d|x} (-1)^d = 4. - Benoit Cloitre, Apr 14 2002
Since A119479(4)=3, there are never more than 3 consecutive integers in the sequence. Triples of consecutive integers start at 33, 85, 93, 141, 201, ... (A039833). No such triple contains a term of the form p^3. - Ivan Neretin, Feb 08 2016
Numbers that are equal to the product of their proper divisors (A007956) (proof in Sierpiński). - Bernard Schott, Apr 04 2022

Wacław Sierpiński, Elementary Theory of Numbers, Ex. 2 p. 174, Warsaw, 1964.

Cf. A000005, A007422, A007956, A030515, A035533, A048272, A119479.

Equals the disjoint union of A006881 and A030078.

[n: n in [1..200] | DivisorSigma(0, n) eq 4]; // Vincenzo Librandi, Jul 16 2015

Select[Range[200], DivisorSigma[0,#]==4&] (* Harvey P. Dale, Apr 06 2011 *)

is(n)=numdiv(n)==4 \\ Charles R Greathouse IV, May 18 2015

from math import isqrt
from sympy import primepi, integer_nthroot, primerange
def A030513(n):
    def f(x): return int(n+x-primepi(integer_nthroot(x,3)[0])+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 16 2024

{n : A000005(n) = 4}. - Juri-Stepan Gerasimov, Oct 10 2009

Incorrect comments removed by Charles R Greathouse IV, Mar 18 2010

A030637 Numbers with 19 divisors.

Original entry on oeis.org

262144, 387420489, 3814697265625, 1628413597910449, 5559917313492231481, 112455406951957393129, 14063084452067724991009, 104127350297911241532841, 3244150909895248285300369, 210457284365172120330305161, 699053619999045038539170241

Offset: 1

Jeff Burch

18th powers of primes. The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime. - Omar E. Pol, May 06 2008

R. J. Mathar, Table of n, a(n) for n = 1..1000

Cf. A000040, A030634, A030635, A030636.

Prime[Range[18]]^18 (* Vladimir Joseph Stephan Orlovsky, May 05 2011 *)

is(n)=isprimepower(n)==18 \\ Charles R Greathouse IV, Jun 19 2016

a(n)=A000040(n)^(19-1)=A000040(n)^(18). - Omar E. Pol, May 06 2008

a(6) corrected, Mar 26 2007

A059269 Numbers m for which the number of divisors, tau(m), is divisible by 3.

Original entry on oeis.org

4, 9, 12, 18, 20, 25, 28, 32, 36, 44, 45, 49, 50, 52, 60, 63, 68, 72, 75, 76, 84, 90, 92, 96, 98, 99, 100, 108, 116, 117, 121, 124, 126, 132, 140, 144, 147, 148, 150, 153, 156, 160, 164, 169, 171, 172, 175, 180, 188, 196, 198, 200, 204, 207, 212, 220, 224, 225, 228

Offset: 1

Avi Peretz (njk(AT)netvision.net.il), Jan 24 2001

tau(n) is divisible by 3 iff at least one prime in the prime factorization of n has exponent of the form 3*m + 2. This sequence is an extension of the sequence A038109 in which the numbers has at least one prime with exponent 2 (the case of m = 0 here ) in their prime factorization.
The union of A211337 and A211338 is the complementary sequence to this one. - Douglas Latimer, Apr 12 2012
Numbers whose cubefree part (A050985) is not squarefree (A005117). - Amiram Eldar, Mar 09 2021

			a(7) = 28 is a term because the number of divisors of 28, d(28) = 6, is divisible by 3.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical Notes, XIII. A Sequal to Note IV, Elemente der Mathematik, Vol. 18 (1963), pp. 8-11.
S. S. Pillai, On a congruence property of the divisor function, J. Indian Math. Soc. (N. S.), Vol. 6, (1942), pp. 118-119.
L. G. Sathe, On a congruence property of the divisor function, American Journal of Mathematics, Vol. 67, No. 3 (1945), pp. 397-406.

Cf. A000005, A005117, A038109, A050985, A211337, A211338, A253905.

Characteristic function: A353470.

with(numtheory): for n from 1 to 1000 do if tau(n) mod 3 = 0 then printf(`%d,`,n) fi: od:

Select[Range[230], Divisible[DivisorSigma[0, #], 3] &] (* Amiram Eldar, Jul 26 2020 *)

is(n)=vecmax(factor(n)[,2]%3)==2 \\ Charles R Greathouse IV, Apr 10 2012

is(n)=numdiv(n)%3==0 \\ Charles R Greathouse IV, Sep 18 2015

Conjecture: a(n) ~ k*n where k = 1/(1 - Product(1 - (p-1)/(p^(3*i)))) = 3.743455... where p ranges over the primes and i ranges over the positive integers. - Charles R Greathouse IV, Apr 13 2012
The asymptotic density of this sequence is 1 - zeta(3)/zeta(2) = 1 - 6*zeta(3)/Pi^2 = 0.2692370305... (Sathe, 1945). Therefore, the above conjecture, a(n) ~ k*n, is true, but k = 1/(1-6*zeta(3)/Pi^2) = 3.7141993349... - Amiram Eldar, Jul 26 2020
A001248 UNION A030515 UNION A030627 UNION A030630 UNION A030633 UNION A030636 UNION ... - R. J. Mathar, May 05 2023

More terms from James Sellers, Jan 24 2001

A030638 Numbers with 20 divisors.

Original entry on oeis.org

240, 336, 432, 528, 560, 624, 648, 810, 816, 880, 912, 1040, 1104, 1134, 1232, 1360, 1392, 1456, 1488, 1520, 1536, 1776, 1782, 1840, 1904, 1968, 2000, 2064, 2106, 2128, 2256, 2288, 2320, 2480, 2544, 2560, 2576, 2754, 2832, 2835, 2928

Offset: 1

Jeff Burch

nonn

Numbers of the form p^19, p*q^9 (A179692), p*q*r^4 (A179644) or p^3*q^4 (A179666), where p, q and r are distinct primes. - R. J. Mathar, Mar 01 2010

R. J. Mathar, Table of n, a(n) for n = 1..1000

Cf. A030636, A030637, A137484.

Select[Range[20000],DivisorSigma[0,#]==20&] (* Vladimir Joseph Stephan Orlovsky, May 05 2011 *)

is(n)=numdiv(n)==20 \\ Charles R Greathouse IV, Jun 19 2016

A179646 Product of the 5th power of a prime and different distinct prime of the 2nd power (p^5*q^2).

Original entry on oeis.org

288, 800, 972, 1568, 3872, 5408, 6075, 9248, 11552, 11907, 12500, 16928, 26912, 28125, 29403, 30752, 41067, 43808, 53792, 59168, 67228, 70227, 70688, 87723, 89888, 111392, 119072, 128547, 143648, 151263, 153125, 161312, 170528, 199712

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 21 2010

nonn

288=2^5*3^2, 800=2^5*5^2,..

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, Prime Signatures
Index to sequences related to prime signature

Cf. A030636, A046308, A007774.

Cf. A085548, A085965, A085967.

f[n_]:=Sort[Last/@FactorInteger[n]]=={2,5}; Select[Range[200000], f]

list(lim)=my(v=List(),t);forprime(p=2,(lim\4)^(1/5),t=p^5;forprime(q=2,sqrt(lim\t),if(p==q,next);listput(v,t*q^2)));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A189988(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**4)) for p in primerange(integer_nthroot(x,4)[0]+1))+primepi(integer_nthroot(x,6)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

Sum_{n>=1} 1/a(n) = P(2)*P(5) - P(7) = A085548 * A085965 - A085967 = 0.007886..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020

A067004 Number of numbers <= n with same number of divisors as n.

Original entry on oeis.org

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

Offset: 1

Henry Bottomley, Dec 21 2001

nonn

			a(10)=3 since 6,8,10 each have four divisors. a(11)=5 since 2,3,5,7,11 each have two divisors.

Paul Tek, Table of n, a(n) for n = 1..10000

Cf. A000005, A008479, A058933, A067003. Inverse of A000040, A001248, A030513, A030514, A030515, A030516, A030626, A030627, A030628, A030629, A030630, A030631, A030632, A030633, A030634, A030635, A030636, A030637, A030638 etc.
Cf. also A079788, A138009.
A047983(n) = a(n)-1.

N:= 1000: # to get a(1) to a(N)
R:= Vector(N):
for n from 1 to N do
  v:= numtheory:-tau(n);
  R[v]:= R[v]+1;
  A[n]:= R[v];
od:
seq(A[n],n=1..N); # Robert Israel, May 04 2015

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

a(n)=my(d=numdiv(n)); sum(k=1,n,numdiv(k)==d) \\ Charles R Greathouse IV, Sep 02 2015

Ordinal transform of A000005. - Franklin T. Adams-Watters, Aug 28 2006

a(A000040(n)^(p-1)) = n if p is prime. - Robert Israel, May 04 2015

A166546 Natural numbers n such that d(n) + 1 is prime.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 79, 80, 82, 83, 84, 85, 86, 87, 89, 90

Offset: 1

Giovanni Teofilatto, Oct 16 2009

nonn

Natural numbers n such that d(d(n)+1)= 2. - Giovanni Teofilatto, Oct 26 2009

The complement is the union of A001248, A030514, A030516, A030626, A030627, A030629, A030631, A030632, A030633 etc. - R. J. Mathar, Oct 26 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..6955

Cf. A000005.

Cf. A073915. - R. J. Mathar, Oct 26 2009

[n: n in [1..100] | IsPrime(NumberOfDivisors(n)+1)]; // Vincenzo Librandi, Jan 20 2019

Select[Range@90, PrimeQ[DivisorSigma[0, #] + 1] &] (* Vincenzo Librandi, Jan 20 2019 *)

isok(n) = isprime(numdiv(n)+1); \\ Michel Marcus, Jan 20 2019

{1} U A000040 U A030513 U A030515 U A030628 U A030630 U A030634 U A030636 U A137485 U A137491 U A137493 U ... . - R. J. Mathar, Oct 26 2009

A036457 Numbers k for which exactly 5 applications of A000005 are needed to reach 2.

Original entry on oeis.org

60, 72, 84, 90, 96, 108, 126, 132, 140, 150, 156, 160, 180, 198, 200, 204, 220, 224, 228, 234, 240, 252, 260, 276, 288, 294, 300, 306, 308, 315, 336, 340, 342, 348, 350, 352, 360, 364, 372, 380, 392, 396, 414, 416, 420, 432, 444, 450, 460, 468, 476, 480

Offset: 1

Labos Elemer

nonn

Subsequences include A030630 (numbers with 12 divisors), A030636 (numbers with 18 divisors), A030638 (numbers with 20 divisors), A137491 (numbers with 28 divisors), etc. [edited by Jon E. Schoenfield, May 12 2018]

			a(13)=180; the successive iterates are 18, 6, 4, 3, and finally the 5th is 2;
a(3)=84; divisor numbers are 12, 6, 4, 3, and 2.

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

Cf. A000005, A007624, A036450, A036452, A036453, A036455, A030630.

A036459:= proc(n) option remember;
  if n <= 2 then 0 else 1 + procname(numtheory:-tau(n)) fi
end proc:
select(A036459 = 5, [$1..1000]); # Robert Israel, Jan 25 2016

Select[Range@ 480, Last@ # == 2 && #[[5]] != 2 &@ NestList[DivisorSigma[0, #] &, #, 5] &] (* Michael De Vlieger, Jan 26 2016 *)

is(n)=for(i=1,4,n=numdiv(n); if(n<3, return(0))); numdiv(n)==2 \\ Charles R Greathouse IV, Sep 17 2015

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

A036459(a(n)) = 5. - Ivan Neretin, Jan 25 2016

New name from Robert Israel, Jan 25 2016

A274360 Numbers n such that n and n+1 both have 18 divisors.

Original entry on oeis.org

6075, 23275, 25856, 26900, 33524, 45324, 46475, 61299, 61347, 77076, 82075, 93924, 96236, 107775, 111924, 117324, 118700, 133524, 137924, 155771, 209524, 210176, 219275, 229275, 230643, 234099, 257724, 258475, 272924, 275300, 278271, 312987, 325899, 332667, 348524

Offset: 1

Charles R Greathouse IV, Jun 19 2016

nonn

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

Intersection of A005237 and A030636.

Cf. A000005, A274357.

PARI
```
is(n)=numdiv(n)==18 && numdiv(n+1)==18
```

list(lim)=my(v=List()); forprime(p=2,sqrtint(lim\12), forprime(q=2,sqrtint(lim\p^2\2), if(p==q,next); my(pq2=(p*q)^2); forprime(r=2,lim\pq2, if(p==r || q==r, next); t=pq2*r; if(numdiv(t-1)==18, listput(v,t-1)); if(numdiv(t+1)==18, listput(v,t))))); Set(v)

cp's OEIS Frontend

A030513 Numbers with 4 divisors.

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A030637 Numbers with 19 divisors.

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A059269 Numbers m for which the number of divisors, tau(m), is divisible by 3.

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A030638 Numbers with 20 divisors.

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A179646 Product of the 5th power of a prime and different distinct prime of the 2nd power (p^5*q^2).

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A067004 Number of numbers <= n with same number of divisors as n.

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A166546 Natural numbers n such that d(n) + 1 is prime.

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