A030627 -id:A030627 - 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

A137492 Numbers with 29 divisors.

Original entry on oeis.org

268435456, 22876792454961, 37252902984619140625, 459986536544739960976801, 144209936106499234037676064081, 15502932802662396215269535105521, 28351092476867700887730107366063041

Offset: 1

R. J. Mathar, Apr 22 2008

Maple implementation: see A030513.

28th 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

T. D. Noe, Table of n, a(n) for n = 1..1000
OEIS Wiki, Index entries for number of divisors

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

Cf. A000040.

Prime[Range[13]]^28 (* Vladimir Joseph Stephan Orlovsky, May 05 2011 *)

a(n)=prime(n)^28 \\ Charles R Greathouse IV, Jun 19 2016

A000005(a(n))=29.

a(n)=A000040(n)^(29-1)=A000040(n)^(28). - Omar E. Pol, May 06 2008

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

A030628 1 together with numbers of the form p*q^4 and p^9, where p and q are distinct primes.

Original entry on oeis.org

1, 48, 80, 112, 162, 176, 208, 272, 304, 368, 405, 464, 496, 512, 567, 592, 656, 688, 752, 848, 891, 944, 976, 1053, 1072, 1136, 1168, 1250, 1264, 1328, 1377, 1424, 1539, 1552, 1616, 1648, 1712, 1744, 1808, 1863, 1875, 2032, 2096, 2192, 2224, 2349, 2384

Offset: 1

Jeff Burch

Also 1 together with numbers with 10 divisors. Also numbers n such that product of all proper divisors of n equals n^4.

If M(n) denotes the product of all divisors of n, then n is said to be k-multiplicatively perfect if M(n)=n^k. All such numbers are of the form p*q^(k-1) or p^(2k-1). The sequence A030628 is therefore 5-multiplicatively perfect. See the Links for A007422. - Walter Kehowski, Sep 13 2005

D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston, MA, 1976. p. 119.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, entry for 48, page 106, 1997.

R. J. Mathar, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Divisor Product
Index to sequences related to prime signature

Cf. A030515, A030627, A030629.

with(numtheory): k:=5: MPL:=[]: for z from 1 to 1 do for n from 1 to 5000 do if convert(divisors(n),`*`) = n^k then MPL:=[op(MPL),n] fi od; od; MPL; # Walter Kehowski, Sep 13 2005

Join[{1},Select[Range[6000],DivisorSigma[0,#]==10&]] (* Vladimir Joseph Stephan Orlovsky, May 05 2011 *)
Select[Range[2500],Times@@Most[Divisors[#]]==#^4&] (* Harvey P. Dale, Nov 04 2024 *)

{v=[]; for(n=1,500,v=concat(v, if(numdiv(n)==10,n,",")); ); v} \\ Jason Earls, Jun 18 2001

list(lim)=my(v=List([1]), t); forprime(p=2, (lim\2+.5)^(1/4), t=p^4; forprime(q=2, lim\t, if(p==q, next); listput(v, t*q))); forprime(p=2,(lim+.5)^(1/9),listput(v,p^9)); vecsort(Vec(v)) \\ Charles R Greathouse IV, Apr 26 2012

from sympy import primepi, primerange, integer_nthroot
def A030628(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-1+x-sum(primepi(x//p**4) for p in primerange(integer_nthroot(x,4)[0]+1))+primepi(integer_nthroot(x,5)[0])-primepi(integer_nthroot(x,9)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

Union A178739 U A179665 {1}. - R. J. Mathar, Apr 03 2011

Better description from Sharon Sela (sharonsela(AT)hotmail.com), Dec 23 2001

More terms from Walter Kehowski, Sep 13 2005

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

A267892 Numbers with 9 odd divisors.

Original entry on oeis.org

225, 441, 450, 882, 900, 1089, 1225, 1521, 1764, 1800, 2178, 2450, 2601, 3025, 3042, 3249, 3528, 3600, 4225, 4356, 4761, 4900, 5202, 5929, 6050, 6084, 6498, 6561, 7056, 7200, 7225, 7569, 8281, 8450, 8649, 8712, 9025, 9522, 9800, 10404, 11858, 12100, 12168, 12321, 12996, 13122, 13225, 14112, 14161, 14400, 14450, 15129

Offset: 1

Omar E. Pol, Apr 03 2016

nonn

Positive integers that have exactly nine odd divisors.
Numbers k such that the symmetric representation of sigma(k) has 9 subparts. - Omar E. Pol, Dec 29 2016
From Robert Israel, Dec 29 2016: (Start)
Numbers k such that A000265(k) is in A030627.
Numbers of the form 2^j*p^8 or 2^j*p^2*q^2 where p and q are distinct odd primes. (End)
Numbers that can be formed in exactly 8 ways by summing sequences of 2 or more consecutive positive integers. - Julie Jones, Aug 13 2018

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

Column 9 of A266531.
Cf. A001227, A038547, A237593, A279387.
Numbers with exactly k odd divisors (k = 1..10): A000079, A038550, A072502, apparently A131651, A267696, A230577, A267697, A267891, this sequence, A267893.
Cf. A000265, A030627.

A:=List([1..16000],n->DivisorsInt(n));; B:=List([1..Length(A)],i->Filtered(A[i],IsOddInt));;
a:=Filtered([1..Length(B)],i->Length(B[i])=9); # Muniru A Asiru, Aug 14 2018

N:= 10^5: # to get all terms <= N
P:= select(isprime, [seq(i,i=3..floor(sqrt(N)/2),2)]);
Aodd:= select(`<=`,map(t -> t^8, P) union {seq(seq(P[i]^2*P[j]^2,i=1..j-1),j=1..nops(P))}, N):
A:= map(t -> seq(2^j*t,j=0..ilog2(N/t)), Aodd):
sort(convert(A,list)); # Robert Israel, Dec 29 2016

Select[Range[5^6], Length[Divisors@ # /. d_ /; EvenQ@ d -> Nothing] == 9 &] (* Michael De Vlieger, Apr 04 2016 *)
Select[Range[16000],Total[Boole[OddQ[Divisors[#]]]]==9&] (* Harvey P. Dale, May 12 2019 *)

isok(n) = sumdiv(n, d, (d%2)) == 9; \\ after Michel Marcus.

A001227(a(n)) = 9.

Sum_{n>=1} 1/a(n) = (P(2)-1/4)^2 - P(4) + 2*P(8) + 7/128 = 0.026721189882055998428..., where P(s) is the prime zeta function. - Amiram Eldar, Sep 16 2024

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.

A137491 Numbers with 28 divisors.

Original entry on oeis.org

960, 1344, 1728, 2112, 2240, 2496, 3264, 3520, 3648, 4160, 4416, 4928, 5440, 5568, 5824, 5832, 5952, 6080, 7104, 7290, 7360, 7616, 7872, 8000, 8256, 8512, 9024, 9152, 9280, 9920, 10176, 10206, 10304, 11328, 11712, 11840, 11968, 12864, 12992, 13120

Offset: 1

R. J. Mathar, Apr 22 2008

nonn

Maple implementation: see A030513.

Numbers of the form p^27 (subset of A122968), p*q^13, p*q*r^6 (A179672) or p^3*q^6 (A179694), 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. A030513-A030516, A030626, A030627, A030634-A030638, A005179, A003680, A096932, A061286, A061283.

with(numtheory): A137491:=n->`if`(tau(n)=28,n,NULL): seq(A137491(n), n=1..15000); # Wesley Ivan Hurt, Sep 18 2014

Select[Range[30000],DivisorSigma[0,#]==28&] (* Vladimir Joseph Stephan Orlovsky, May 06 2011 *)

is(n)=numdiv(n)==28 \\ Charles R Greathouse IV, Jun 19 2016

A000005(a(n)) = 28.

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

cp's OEIS Frontend

A030513 Numbers with 4 divisors.

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A137492 Numbers with 29 divisors.

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

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A030628 1 together with numbers of the form p*q^4 and p^9, where p and q are distinct primes.

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

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A267892 Numbers with 9 odd divisors.

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