A030633 -id:A030633 - 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

A030635 Numbers with 17 divisors.

Original entry on oeis.org

65536, 43046721, 152587890625, 33232930569601, 45949729863572161, 665416609183179841, 48661191875666868481, 288441413567621167681, 6132610415680998648961, 250246473680347348787521, 727423121747185263828481, 12337511914217166362274241

Offset: 1

Jeff Burch

16th 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, A030632, A030633, A030634.

Prime[Range[16]]^16 (* Vladimir Joseph Stephan Orlovsky, May 05 2011 *)

is(n)=isprimepower(n)==16 \\ Charles R Greathouse IV, Jun 19 2016

a(n)=A000040(n)^16. - Omar E. Pol, May 06 2008

a(n) = A179645(n)^2. - R. J. Mathar, May 26 2017

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

A030634 Numbers with 16 divisors.

Original entry on oeis.org

120, 168, 210, 216, 264, 270, 280, 312, 330, 378, 384, 390, 408, 440, 456, 462, 510, 520, 546, 552, 570, 594, 616, 640, 680, 690, 696, 702, 714, 728, 744, 750, 760, 770, 798, 858, 870, 888, 896, 910, 918, 920, 930, 945, 952, 966, 984, 1000

Offset: 1

Jeff Burch

nonn

Numbers of the form p^15 (subset of A010803), p*q^7, p*q*r^3 or p^3*q^3, or p*q*r*s, where p, q, r and s are distinct primes. - R. J. Mathar, Mar 01 2010

R. J. Mathar, Table of n, a(n) for n = 1..1000
Jérôme Germoni, Nombres à huit diviseurs, Images des Mathématiques, CNRS, 2017 (in French).

Cf. A030626, A030631, A030632, A030633.

[n: n in [1..1000] | DivisorSigma(0, n) eq 16]; // Vincenzo Librandi, Oct 05 2017

Select[Range[3000],DivisorSigma[0,#]==16&] (* Vladimir Joseph Stephan Orlovsky, May 05 2011 *)

is(n)=numdiv(n)==16 \\ Charles R Greathouse IV, Jun 19 2016

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

A275387 Numbers of ordered pairs of divisors d < e of n such that gcd(d, e) > 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 3, 1, 2, 0, 8, 0, 2, 2, 6, 0, 8, 0, 8, 2, 2, 0, 18, 1, 2, 3, 8, 0, 15, 0, 10, 2, 2, 2, 24, 0, 2, 2, 18, 0, 15, 0, 8, 8, 2, 0, 32, 1, 8, 2, 8, 0, 18, 2, 18, 2, 2, 0, 44, 0, 2, 8, 15, 2, 15, 0, 8, 2, 15, 0, 49, 0, 2, 8, 8, 2, 15, 0, 32, 6, 2

Offset: 1

Michel Lagneau, Aug 03 2016

nonn

Number of elements in the set {(x, y): x|n, y|n, x < y, gcd(x, y) > 1}.
Every element of the sequence is repeated indefinitely, for instance:
a(n)=0 if n prime;
a(n)=1 if n = p^2 for p prime (A001248);
a(n)=2 if n is a squarefree semiprime (A006881);
a(n)=3 if n = p^3 for p prime (A030078);
a(n)=6 if n = p^4 for p prime (A030514);
a(n)=8 if n is a number which is the product of a prime and the square of a different prime (A054753);
a(n)=10 if n = p^5 for p prime (A050997);
a(n)=15 if n is in the set {A007304} union {64} = {30, 42, 64, 66, 70,...} = {Sphenic numbers} union {64};
a(n)=18 if n is the product of the cube of a prime (A030078) and a different prime (see A065036);
a(n)=21 if n = p^7 for p prime (A092759);
a(n)=24 if n is square of a squarefree semiprime (A085986);
a(n)=32 if n is the product of the 4th power of a prime (A030514) and a different prime (see A178739);
a(n)=36 if n = p^9 for p prime (A179665);
a(n)=44 if n is the product of exactly four primes, three of which are distinct (A085987);
a(n)=45 if n is a number with 11 divisors (A030629);
a(n)=49 if n is of the form p^2*q^3, where p,q are distinct primes (A143610);
a(n)=50 if n is the product of the 5th power of a prime (A050997) and a different prime (see A178740);
a(n)=55 if n if n = p^11 for p prime(A079395);
a(n)=72 if n is a number with 14 divisors (A030632);
a(n)=80 if n is the product of four distinct primes (A046386);
a(n)=83 if n is a number with 15 divisors (A030633);
a(n)=89 if n is a number with prime factorization pqr^3 (A189975);
a(n)=96 if n is a number that are the cube of a product of two distinct primes (A162142);
a(n)=98 if n is the product of the 7th power of a prime and a distinct prime (p^7*q) (A179664);
a(n)=116 if n is the product of exactly 2 distinct squares of primes and a different prime (p^2*q^2*r) (A179643);
a(n)=126 if n is the product of the 5th power of a prime and different distinct prime of the 2nd power (p^5*q^2) (A179646);
a(n)=128 if n is the product of the 8th power of a prime and a distinct prime (p^8*q) (A179668);
a(n)=150 if n is the product of the 4th power of a prime and 2 different distinct primes (p^4*q*r) (A179644);
a(n)=159 if n is the product of the 4th power of a prime and a distinct prime of power 3 (p^4*q^3) (A179666).
It is possible to continue with a(n) = 162, 178, 209, 224, 227, 238, 239, 260, 289, 309, 320, 333,...

			a(12) = 8 because the divisors of 12 are {1, 2, 3, 4, 6, 12} and GCD(d_i, d_j)>1 for the 8 following pairs of divisors: (2,4), (2,6), (2,12), (3,6), (3,12), (4,6), (4,12) and (6,12).

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

Cf. A001248, A006881, A007304, A030078, A030514, A030632, A046386, A050997, A054753, A063647, A065036, A066446, A079395, A085986, A085987, A092759, A143610, A162142, A178739, A178740, A179644, A179646, A179664, A189975.

Cf. A333976 (same with d <= e).

with(numtheory):nn:=100:
for n from 1 to nn do:
x:=divisors(n):n0:=nops(x):it:=0:
for i from 1 to n0 do:
  for j from i+1 to n0 do:
   if gcd(x[i],x[j])>1
    then
    it:=it+1:
    else
   fi:
  od:
od:
  printf(`%d, `,it):
od:

Table[Sum[Sum[(1 - KroneckerDelta[GCD[i, k], 1]) (1 - Ceiling[n/k] + Floor[n/k]) (1 - Ceiling[n/i] + Floor[n/i]), {i, k - 1}], {k, n}], {n, 100}] (* Wesley Ivan Hurt, Jan 01 2021 *)

a(n)=my(d=divisors(n)); sum(i=2,#d, sum(j=1,i-1, gcd(d[i],d[j])>1)) \\ Charles R Greathouse IV, Aug 03 2016

a(n)=my(f=factor(n)[,2],t=prod(i=1,#f,f[i]+1)); t*(t-1)/2 - (prod(i=1,#f,2*f[i]+1)+1)/2 \\ Charles R Greathouse IV, Aug 03 2016

a(n) = A066446(n) - A063647(n).

a(n) = Sum_{d1|n, d2|n, d1Wesley Ivan Hurt, Jan 01 2021

A336595 Numbers whose number of divisors is divisible by 5.

Original entry on oeis.org

16, 48, 80, 81, 112, 144, 162, 176, 208, 240, 272, 304, 324, 336, 368, 400, 405, 432, 464, 496, 512, 528, 560, 567, 592, 624, 625, 648, 656, 688, 720, 752, 784, 810, 816, 848, 880, 891, 912, 944, 976, 1008, 1040, 1053, 1072, 1104, 1134, 1136, 1168, 1200, 1232

Offset: 1

Amiram Eldar, Jul 26 2020

nonn

The asymptotic density of this sequence is 1 - zeta(5)/zeta(4) = 0.0419426259... (Sathe, 1945).

			16 is a term since A000005(16) = 5 is divisible by 5.

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 63.

Amiram Eldar, 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, A008587, A059269, A336596.

Cf. A030514, A030628, A113849, A178739, A179665, A030633, A030638, A137488, A137493, A175745.

q:= n-> is(irem(numtheory[tau](n), 5)=0):
select(q, [$1..1300])[];  # Alois P. Heinz, Jul 26 2020

Select[Range[1300], Divisible[DivisorSigma[0, #], 5] &]

A030514 UNION A030628 \ {1} UNION A030633 UNION A030638 UNION A137488 UNION A137493 UNION A175745 UNION A175749 UNION A175752 UNION A175756 UNION ... - R. J. Mathar, May 05 2023

A350767 a(1)=1. Thereafter, a(n+1) is the least unused number k such that either d(j(n)) properly divides d(k) or d(k) properly divides d(j(n)), where j(n) = a(n)+1 and d is the divisor counting function A000005.

Original entry on oeis.org

1, 6, 8, 12, 10, 14, 2, 15, 48, 18, 20, 3, 28, 21, 5, 7, 11, 4, 22, 24, 32, 13, 17, 9, 19, 23, 26, 29, 27, 25, 30, 33, 31, 37, 40, 34, 41, 35, 49, 43, 47, 16, 38, 42, 39, 46, 44, 53, 51, 59, 45, 54, 56, 60, 50, 61, 66, 52, 55, 57, 67, 71, 58, 62, 72, 63, 192, 65

Offset: 1

David James Sycamore, Jan 14 2022

nonn

If d(j(n)) is prime p then d(a(n+1)) must be properly divisible by p. In practice the proper divisor for computation of a(n+1) toggles between d(j(n)) and d(k).
Conjecture: This is a permutation of the positive integers. Numbers with the same number (tau) of divisors appear in their natural orders (e.g., primes, semiprimes, squares).
The plot, after the first few terms, resolves itself into points tightly packed on and around a straight line of slope 1, with exceptional points appearing as significant upward or downward "spikes".
When d(j(n)) is prime p appearing for the first time in the sequence J = {d(j(a(n)), n>=1}, then a(n+1) is the smallest number with 2p divisors, which produces a significantly large upward spike above the straight line (6, 12, 48, 192, 3072, 12288, ...).
When d(j(a(n)) is 2p, seen for the first time in J, then a(n+1) is the smallest number with p divisors, which produces a large downward spike, below the straight line (2, 4, 16, 64, 1024, 4096, ...).
The sequence of fixed points starts: 1, 46, 69, 74, 110, 140, 142, 152, 154, 178, ... apparently becoming denser as n increases.

			a(1)=1, so j(1)=2, d(j(1))=2, a prime, so we need the smallest unused k such that d(k) is properly divisible by 2, hence a(2)=6.
a(2)=6, j(2)=4, d(j(2))=3, a prime so we need the smallest unused k such that d(k) is properly divisible by 3, hence a(3)=8.

Michael De Vlieger, Table of n, a(n) for n = 1..10001
Michael De Vlieger, Scatterplot of a(n), n = 1..256, color coded to show primes in red, evens in blue, 2 in magenta, odd composites in black. A gold highlight indicates terms such that (a(n)+1) | a(n+1). Outlying points are annotated.
Michael De Vlieger, Log-log scatterplot of a(n), n = 1..2^16. Terms in A3680(p) such that p is prime appear annotated in red, and those in A061286 appear in blue.

Cf. A000040, A001248, A030514, A054753, A030516, A030626, A085986, A178739, A030629, A030630, A030631, A030632, A030633, A030634, A005179, A061286, A003680.

Block[{c, d, j, k, u, nn}, nn = 120; j = u = 2; c[] = 0; c[1] = 1; Do[d[i] = DivisorSigma[0, i], {i, 2^(Ceiling@ Log2[nn] + 3)}]; {1}~Join~Reap[Do[k = u; While[Nand[Or[Divisible[d[j], d[k]], Divisible[d[k], d[j]]], d[j] != d[k], c[k] == 0], k++]; Sow[k]; c[k] = i; j = k + 1; If[k == u, While[c[u] != 0, u++]], {i, nn}] ][[-1, -1]] ] (* _Michael De Vlieger, Jan 14 2022 *)

isok(k, last, set) = if (!setsearch(set, k), my(ndk=numdiv(k), ndl=numdiv(last+1)); (ndl != ndk) && ((!(ndk % ndl)) || (!(ndl % ndk))));
lista(nn) = {my(last = 1, list = List(last), set = Set(list)); for (n=2, nn, my(k=1); while (!isok(k, last, set), k++); listput(list, k); set = Set(list); last = k;); Vec(list);} \\ Michel Marcus, Jan 15 2022

More terms from Michael De Vlieger, Jan 14 2022

cp's OEIS Frontend

A030513 Numbers with 4 divisors.

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A030635 Numbers with 17 divisors.

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

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A030634 Numbers with 16 divisors.

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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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A275387 Numbers of ordered pairs of divisors d < e of n such that gcd(d, e) > 1.

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