A030513 -id:A030513 - cp's OEIS frontend

A137491 Numbers with 28 divisors.

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

A137487 Numbers with 24 divisors.

Original entry on oeis.org

360, 420, 480, 504, 540, 600, 630, 660, 672, 756, 780, 792, 864, 924, 936, 990, 1020, 1050, 1056, 1092, 1120, 1140, 1152, 1170, 1176, 1188, 1224, 1248, 1350, 1368, 1380, 1386, 1400, 1404, 1428, 1470, 1500, 1530, 1540, 1596, 1632, 1638, 1650, 1656, 1710

Offset: 1

R. J. Mathar, Apr 22 2008

nonn

Maple implementation: see A030513.

Numbers of the form p^23, p^2*q^7, p*q^2*r^3 (like 360, 504), p*q*r^5 (like 480, 672), p*q*r*s^2 (like 420, 660), p^3*q^5 (like 864) or p*q^11, where p, q, r and s 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.

Select[Range[3000],DivisorSigma[0,#]==24&] (* Vladimir Joseph Stephan Orlovsky, May 05 2011 *)

is(n)=numdiv(n)==24 \\ Charles R Greathouse IV, Jun 19 2016

A000005(a(n))=24.

A332269 Numbers m with only one divisor d such that sqrt(m) < d < m.

Original entry on oeis.org

6, 8, 10, 14, 15, 16, 21, 22, 26, 27, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 81, 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

Bernard Schott, May 04 2020

nonn

Equivalently: numbers with only one proper divisor > sqrt(n).
Also: numbers with only one nontrivial divisor d with 1 < d < sqrt(n).
Four subsequences (see examples):
1) Squarefree semiprimes (A006881) p*q with p < q, then this unique divisor is q.
2) Cube of primes p^3 (A030078), then this unique divisor is p^2.
3) Primes^4 (A030514), then this unique divisor is p^3.
4) Numbers with 4 divisors: A030513 = A006881 Union A030078.
For n = 1 to n = 21, we have a(n) = A319238(n) = A331231(n) but a(22) = 65 <> A319238(22) = A331231(22) = 64.
From Marius A. Burtea, May 07 2020: (Start)
The sequence contains terms that are consecutive numbers.
If the numbers 4*k + 1 and 6*k + 1, k >= 1, are prime numbers, then the numbers 12*k + 2 and 12*k + 3 are terms. Examples: (14, 15), (38, 39), (86, 87), (122, 123), (158, 159), (218, 219), (302, 303), ...
If the numbers 6*m + 1, 10*m + 1 and 15*m + 2, m >= 1, are prime numbers, then the numbers 30*m + 3, 30*m + 4 and 30*m + 5 are terms. Examples: (33, 34, 35), (93, 94, 95), (213, 214, 215), (393, 394, 395), (633, 634, 635), ... (End)
There are never more than 3 consecutive terms because one of them would be divisible by 4, and neither 8 nor 16 belong to such a string of 4 consecutive terms.

			The divisors of 15 are {1, 3, 5, 15} and only 5 satisfies sqrt(15) < 5 < 15, hence 15 is a term.
The divisors of 27 are {1, 3, 9, 27} and only 9 satisfies sqrt(27) < 9 < 27, hence 27 is a term.
The divisors of 16 are {1, 2, 4, 8, 16} and only 8 satisfies sqrt(16) < 8 < 16, hence 16 is a term.
The divisors of 28 are {1, 2, 4, 7, 14, 28} but 7 and 14 satisfy sqrt(28) < 7 < 14 < 28, hence 28 is not a term.

Disjoint union of A006881, A030078, and A030514.
Disjoint union of A030513 and A030514.
Cf. A000005, A038548.

[k:k in [1..200]|#[d:d in Divisors(k)|d gt Sqrt(k) and d lt k] eq 1]; // Marius A. Burtea, May 07 2020

Select[Range[200], MemberQ[{4, 5}, DivisorSigma[0, #]] &] (* Amiram Eldar, May 04 2020 *)

isok(m) = #select(x->(x^2 > m), divisors(m)) == 2; \\ Michel Marcus, May 05 2020

m is a term iff tau(m) - A038548(m) = 2 where tau = A000005.

A137489 Numbers with 26 divisors.

Original entry on oeis.org

12288, 20480, 28672, 45056, 53248, 69632, 77824, 94208, 118784, 126976, 151552, 167936, 176128, 192512, 217088, 241664, 249856, 274432, 290816, 299008, 323584, 339968, 364544, 397312, 413696, 421888, 438272, 446464, 462848, 520192, 536576

Offset: 1

R. J. Mathar, Apr 22 2008

nonn

Maple implementation: see A030513.

Numbers of the form p^25 (5th powers of A050997, subset of A010813) or p*q^12, 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.

Select[Range[900000],DivisorSigma[0,#]==26&] (* Vladimir Joseph Stephan Orlovsky, May 05 2011 *)

is(n)=numdiv(n)==26 \\ Charles R Greathouse IV, Jun 19 2016

A000005(a(n))=26.

A210994 Numbers n such that A000005(n) <> 4.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 28, 29, 30, 31, 32, 36, 37, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 60, 61, 63, 64, 66, 67, 68, 70, 71, 72, 73, 75, 76, 78, 79, 80, 81, 83, 84, 88, 89, 90, 92, 96

Offset: 1

Omar E. Pol, Aug 11 2012

nonn

			6 is not in the sequence because 6 has four divisors: 1, 2, 3, 6.

Complement of A030513. Column 4 of A210976.

Cf. A018252, A213367.

Select[Range[100],DivisorSigma[0,#]!=4&] (* Harvey P. Dale, Jan 07 2013 *)

A276045 Primes p such that d(p*(2p+1)) = 8 where d(n) is the number of divisors of n (A000005).

Original entry on oeis.org

7, 13, 17, 19, 43, 47, 59, 61, 71, 79, 101, 107, 109, 149, 151, 163, 167, 197, 223, 257, 263, 271, 311, 317, 347, 349, 353, 383, 389, 401, 421, 439, 449, 461, 479, 503, 521, 523, 557, 569, 599, 601, 613, 631, 673, 677, 691, 701, 811, 821, 827, 839, 853, 863, 881, 919

Offset: 1

Anthony Hernandez, Aug 17 2016

Primes p such that 2p+1 is in A030513. - Robert Israel, Aug 17 2016
From Anthony Hernandez, Aug 29 2016: (Start)
Conjecture: this sequence is infinite.
It appears that the prime numbers in this sequence which have 7 for as final digit form the sequence A104164.
Conjecture: this sequence contains infinitely many twin primes. The first few twin primes in this sequence are 17,19,59,61,107,109,521,523,599,601,... (End)
From Bernard Schott, Apr 28 2020: (Start)
This sequence equals the union of {13} and A234095; proof by double inclusion:
-> 1st inclusion: {13} Union A234095 is included in A276045.
1) if p = 13, then 13*27 = 351 = 3^3 * 13, hence d(351) = 8 and 13 belongs to A276045.
2) if p is in A234095, then p*(2*p+1) = p*r*s (p,r,s primes) and d(p*r*s) = 8, hence p is in 276045.
-> 2nd inclusion: A276045 is included in {13} Union A234095.
If p is in A276095, then m=p*(2*p+1) has 8 divisors and there are only three possibilities: m = u*v*w, or m = u^3*v or m = u^7 with u, v, w are distinct primes.
1st case: if p*(2*p+1) = u*v*w then u=p, and 2p+1=v*w is semiprime; hence, p is in A234095 Union {13}.
2nd case: if p*(2p+1) = u^3*v then  p=v and 2*p+1=u^3 ==> 2*p = u^3-1 = (u-1)*(u^2+u+1) with 2 and p are primes; then (u-1=2, u^2+u+1=p) so u=3, and p=3^2+3+1=13; hence p = 13 belongs to {13} Union A234095.
3rd case: p*(2p+1) = u^7 is impossible.
Conclusion: this sequence = {13} Union A234095. (End)

			d(7*(2*7+1))=d(105)=8 so 7 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A030513, A030626.

Equals {13} Union A234095.

select(n -> isprime(n) and numtheory:-tau(n*(2*n+1))=8,
[seq(i, i=3..1000, 2)]); # Robert Israel, Aug 17 2016

Select[Prime@ Range@ 160, DivisorSigma[0, # (2 # + 1)] == 8 &] (* Michael De Vlieger, Aug 28 2016 *)

lista(nn) = forprime(p=2, nn, if (numdiv(p*(2*p+1))==8, print1(p, ", "))); \\ Michel Marcus, Aug 17 2016

A323644 Numbers with 3 or 4 divisors.

Original entry on oeis.org

4, 6, 8, 9, 10, 14, 15, 21, 22, 25, 26, 27, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 121, 122, 123, 125, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 169, 177, 178, 183, 185, 187, 194, 201, 202, 203, 205

Offset: 1

Omar E. Pol, Feb 26 2019

Also numbers k such that the noncentral divisors of k are 1 and k.

Also numbers which are either semiprimes (A001358) or the cube of a prime (A030078). In other words: numbers which are either the product of two distinct primes (A006881) or the square of a prime (A001248) or the cube of a prime (A030078).

			4 is in the sequence because 4 has three divisors, they are 1, 2, 4. On the other hand, the noncentral divisors of 4 are 1 and 4, in accordance with the first comment.
6 is in the sequence because 6 has four divisors, they are 1, 2, 3, 6. On the other hand, the noncentral divisors of 6 are 1 and 6, in accordance with the first comment.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Union of A001248 and A030513.

Cf. A001358, A006881, A030078, A161840, A323643.

Select[Range[200], MemberQ[{3, 4}, DivisorSigma[0, #]] &] (* Amiram Eldar, Dec 03 2020 *)

isok(n) = my(nd=numdiv(n)); (nd==3) || (nd==4); \\ Michel Marcus, Feb 26 2019

A174322 a(n) is the smallest n-digit number with exactly 4 divisors.

Original entry on oeis.org

6, 10, 106, 1003, 10001, 100001, 1000001, 10000001, 100000001, 1000000006, 10000000003, 100000000007, 1000000000007, 10000000000015, 100000000000013, 1000000000000003, 10000000000000003, 100000000000000015, 1000000000000000007, 10000000000000000001

Offset: 1

Jaroslav Krizek, Nov 27 2010

a(n) = the smallest n-digit number of the form p^3 or p^1*q^1, (p, q = distinct primes).

Michael S. Branicky, Table of n, a(n) for n = 1..70

Subsequence of A030513.

Cf. A182648 (largest n-digit numbers with exactly 4 divisors).

Table[k=10^(n-1); While[DivisorSigma[0, k] != 4, k++]; k, {n, 10}]

from sympy import divisors
def a(n):
    k = 10**(n-1)
    while len(divisors(k)) != 4: k += 1
    return k
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Jun 10 2021

# faster alternative for larger terms
from sympy import divisors
def a(n):
    k = 10**(n-1) - 1
    divs = -1
    while divs != 4:
      k += 1
      divs = 0
      for d in divisors(k, generator=True):
        divs += 1
        if divs > 4: break
    return k
print([a(n) for n in range(1, 22)]) # Michael S. Branicky, Jun 10 2021

A000005(a(n)) = 4.

A182648 a(n) is the largest n-digit number with exactly 4 divisors.

Original entry on oeis.org

8, 95, 998, 9998, 99998, 999997, 9999998, 99999997, 999999991, 9999999997, 99999999997, 999999999997, 9999999999989, 99999999999997, 999999999999998, 9999999999999994, 99999999999999989, 999999999999999993, 9999999999999999991, 99999999999999999983

Offset: 1

Jaroslav Krizek, Nov 27 2010

a(n) is the largest n-digit number of the form p^3 or p^1*q^1, (p, q = distinct primes).

Large overlap with A098450 which considers p^2 and p*q with n digits. - R. J. Mathar, Apr 23 2024

Michael S. Branicky, Table of n, a(n) for n = 1..62

Subsequence of A030513.

Cf. A174322, A098450.

Table[k=10^n-1; While[DivisorSigma[0,k] != 4, k--]; k, {n,10}]
lnd4[n_]:=Module[{k=10^n-1},While[DivisorSigma[0,k]!=4,k--];k]; Array[lnd4,20] (* Harvey P. Dale, Aug 20 2024 *)

from sympy import divisors
def a(n):
    k = 10**n - 1
    divs = -1
    while divs != 4:
      k -= 1
      divs = 0
      for d in divisors(k, generator=True):
        divs += 1
        if divs > 4: break
    return k
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Jun 10 2021

A000005(a(n)) = 4.

a(19) and beyond from Michael S. Branicky, Jun 10 2021

cp's OEIS Frontend

A137491 Numbers with 28 divisors.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A137487 Numbers with 24 divisors.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A332269 Numbers m with only one divisor d such that sqrt(m) < d < m.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A137489 Numbers with 26 divisors.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A210994 Numbers n such that A000005(n) <> 4.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A276045 Primes p such that d(p*(2p+1)) = 8 where d(n) is the number of divisors of n (A000005).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A323644 Numbers with 3 or 4 divisors.