A061286 -id:A061286 - 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.

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.

A299795 Numbers of the form p*2^(p-1) where p is prime.

Original entry on oeis.org

4, 12, 80, 448, 11264, 53248, 1114112, 4980736, 96468992, 7784628224, 33285996544, 2542620639232, 45079976738816, 189115999977472, 3307330976350208, 238690780250636288, 17005592192950992896, 70328211781017665536, 4943727411754159833088, 83822005070936202543104

Offset: 1

Peter Luschny, Mar 03 2018

nonn

G. C. Greubel, Table of n, a(n) for n = 1..460

A subsequence of A001787 and A300332.

[NthPrime(n)*2^(NthPrime(n) -1): n in [1..30]]; // G. C. Greubel, Mar 07 2018

Primes := select(isprime, [$1..71]):
seq(p*2^(p-1), p in Primes);

Table[Prime[n]*2^(Prime[n] -1), {n,1,30}] (* G. C. Greubel, Mar 07 2018 *)

a(n) = my(p = prime(n)); p*2^(p-1); \\ Michel Marcus, Mar 07 2018

From Michel Marcus, Mar 07 2018: (Start)
a(n) = prime(n)*2^(prime(n)-1).
a(n) = A000040(n)*A061286(n).
a(n) = A001787(A000040(n)).
(End)

A358820 a(n) is the least novel k such that d(k)|n, where d is the divisor counting function A000005.

Original entry on oeis.org

1, 2, 4, 3, 16, 5, 64, 6, 9, 7, 1024, 8, 4096, 11, 25, 10, 65536, 12, 262144, 13, 49, 17, 4194304, 14, 81, 19, 36, 15, 268435456, 18, 1073741824, 21, 121, 23, 625, 20, 68719476736, 29, 169, 22, 1099511627776, 28, 4398046511104, 26, 100, 31, 70368744177664, 24

Offset: 1

David James Sycamore, Dec 02 2022

nonn

In other words, a(n) = Min{k_j; 1 <= j <= d(n), such that d(k_j) = m_j}, where m_j|n, and k_j has not appeared earlier.
a(n) is composite iff n is odd, and prime p (the least that has not occurred earlier) iff 2|n, and if for any other m|n, and k such that d(k) = m; k > p.
The primes appear in natural order, and records > 1 are 2^(prime(k)-1); k = 1,2,...
Conjectured to be a permutation of the positive integers.
For each n, there is some k <= n such that a(k*d(n)) = n, so (1) a((log 2 + o(1))*n log n/log log n) > n by Wigert's theorem and (2) this sequence is a permutation of the positive integers. - Charles R Greathouse IV, Dec 03 2022

			a(1)=1 since d(1)=1 and 1 has no other divisors.
a(2)=2 since 2 is the smallest number having just 2 divisors.
a(5)=16 since 5 is prime and 16 is the smallest number having 5 divisors.
a(15)=25 since 15 has divisors 25 is the least novel number having 3 divisors, 81 is the least having 5 divisors and 144 is the least having 15 divisors.

Michael De Vlieger, Table of n, a(n) for n = 1..1024
Michael De Vlieger, Log log scatterplot of Log_10(a(n)), n = 1..256.
Index entries for sequences that are permutations of the natural numbers

Cf. A000005, A005179, A061286, A128555 (inverse).

kk = 2^32; nn = 60; c[] = False; s = Union[Flatten@ Monitor[Table[a^2*b^3, {b, kk^(1/3)}, {a, Sqrt[kk/b^3]}], b]]^2; u = 1; v = 1; w = 1; Do[Which[PrimeQ[n], k = 2^(n - 1), CoprimeQ[n, 6], k = w; While[Nand[! c[#], Divisible[n, DivisorSigma[0, #]]] &[s[[k]]], k++]; If[k == w, While[c[s[[k]]], w++]]; k = s[[k]], OddQ[n], k = v; While[Nand[! c[#], Divisible[n, DivisorSigma[0, #]]] &[k^2], k++]; If[k == v, While[c[v^2], v++]]; k *= k, True, k = u; While[Nand[! c[k], Divisible[n, DivisorSigma[0, k]]], k++]]; Set[{a[n], c[k]}, {k, True}]; If[k == u, While[c[u], u++]], {n, nn}]; Array[a, nn] (* _Michael De Vlieger, Dec 03 2022 *)

from functools import lru_cache
from itertools import count, islice
from sympy import divisor_count, isprime
@lru_cache(maxsize=None)
def d(n): return divisor_count(n)
def agen():
    mink, seen = 1, set()
    for n in count(1):
        k = mink if not isprime(n) else 2**(n-1)
        dk = d(k)
        while k in seen or n%dk != 0: k += 1; dk = d(k)
        while mink in seen: mink += 1
        yield k
        seen.add(k)
print(list(islice(agen(), 22))) # Michael S. Branicky, Dec 02 2022

a(prime(k)) = 2^(prime(k) - 1) (see A061286).

n log log n/log n << a(n) <= 2^(n-1), see comments. - Charles R Greathouse IV, Dec 03 2022

a(26) and beyond from Michael S. Branicky, Dec 02 2022

a(24) corrected by Michael De Vlieger, Dec 05 2022

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.

A200815 Number of iterations of k -> d(k) until n reaches an odd prime.

Original entry on oeis.org

0, 1, 0, 2, 0, 2, 1, 2, 0, 3, 0, 2, 2, 1, 0, 3, 0, 3, 2, 2, 0, 3, 1, 2, 2, 3, 0, 3, 0, 3, 2, 2, 2, 2, 0, 2, 2, 3, 0, 3, 0, 3, 3, 2, 0, 3, 1, 3, 2, 3, 0, 3, 2, 3, 2, 2, 0, 4, 0, 2, 3, 1, 2, 3, 0, 3, 2, 3, 0, 4, 0, 2, 3, 3, 2, 3, 0, 3, 1, 2, 0, 4, 2, 2, 2, 3, 0

Offset: 3

Charles R Greathouse IV, Nov 23 2011

nonn

Csajbók and Kasza call this the tau-iteration length.

			d(10) = 4 and d(4) = 3, an odd prime, so a(10) = 2.

Antti Karttunen, Table of n, a(n) for n = 3..10000
Tímea Csajbók and János Kasza, Iterating the tau-function, Annales Univ. Sci. Budapest., Sec. Math. 35 (2011), pp. 83-93.

Cf. A036459, A060937, A061286, A200810.

nop[n_]:=Length[NestWhileList[DivisorSigma[0,#]&,n,#<3 || CompositeQ[ #]&]]-1; Array[ nop,100,3] (* Harvey P. Dale, Nov 14 2020 *)

a(n)=my(i);while(!isprime(n),i++;n=numdiv(n));i

a(n) <= pi(log_2(n)) = A000720(A000523(n)).

a(n) = A036459(n)-1 = A060937(n)-2, for n >= 3. - Antti Karttunen, Oct 06 2017

A269781 a(n) is the smallest k different from n such that (n, k) is an amicably refactorable pair (see the comments).

Original entry on oeis.org

4, 3, 16, 4, 64, 24, 36, 16, 1024, 6, 4096, 64, 4, 5, 65536, 12, 262144, 6, 4, 1024, 4194304, 8, 81, 4096, 4, 6, 268435456, 16, 1073741824, 6, 4, 65536, 16, 9, 68719476736, 262144, 4, 8, 1099511627776, 32, 4398046511104, 6, 36, 4194304, 70368744177664, 10, 729, 48, 4, 6, 4503599627370496, 32, 16, 8

Offset: 3

Waldemar Puszkarz, May 01 2016

nonn

Let m and k be distinct integers and numdiv(n) be the number of divisors of n (A000005(n)). We call m and k amicably refactorable if numdiv(m) divides k and numdiv(k) divides m.
For any n with no amicably refactorable partner, a(n) = 0.
Conjecture: the sequence contains no zeros.
1 does not have an amicable partner as all other numbers have more than one divisor and 2 does not have an amicable partner as all other numbers with two divisors are odd primes and cannot be divided by the number of divisors of 2, also 2. All other numbers may have an amicably refactorable partner, though for some, primes, semiprimes and squares of primes in particular, this number can be quite large.
For primes and semiprimes, a(n) = 2^(f(n) - 1), (see A061286), where f(n) is their largest prime factor. For squares of primes, a(n) = 3^(|sqrt(n)| - 1), except for n = 9 for which this formula yields 9; this forces us to choose the next best candidate: 36.

			For n=5, a(5)=16 as the number of divisors of n (2) divides a(n) while the number of divisors of a(n) (5) divides 5 and 16 is the smallest number for which this happens.

Cf. A000005 (number of divisors), A033950 (refactorable numbers), A061286 (subsequence for odd prime indices and semiprime indices), A268037, A272353 (related sequences).

A269781 = {}; Do[k = 1; If[PrimeQ[n] || PrimeNu[n] == 2 && PrimeOmega[n] == 2, AppendTo[A269781, 2^(First[Last[FactorInteger[n]]] - 1)], If[PrimeQ @ Sqrt @ n && (n > 9), AppendTo[A269781, 3^(Sqrt[n] - 1)],While[k != n && !(Divisible[n, DivisorSigma[0, k]] && Divisible[k, DivisorSigma[0, n]]), k++]; If[k == n, k = n + 1; While[!(Divisible[n, DivisorSigma[0, k]] && Divisible[k, DivisorSigma[0, n]]), k++]]; AppendTo[A269781, k]]], {n, 3, 56}]; A269781

A278741 Odd numbers k such that tau(k-1) is a prime.

Original entry on oeis.org

3, 5, 17, 65, 1025, 4097, 65537, 262145, 4194305, 268435457, 1073741825, 68719476737, 1099511627777, 4398046511105, 70368744177665, 4503599627370497, 288230376151711745, 1152921504606846977, 73786976294838206465, 1180591620717411303425, 4722366482869645213697

Offset: 1

Jaroslav Krizek, Nov 27 2016

nonn

tau(k) = A000005(k) = the number of divisors of k.
Conjecture: prime terms are in A249759: 3, 5, 17, 65537, ...
Supersequence of A256438 and A249759. Subsequence of {A009087(n) + 1}.

			Odd number 65 is in the sequence because tau(64) = 7 (prime).

Cf. A000005, A000040, A001348, A009087, A061286, A249759, A256438.

[n: n in[2..10000000] |  IsOdd(n) and IsPrime(NumberOfDivisors(n-1))];

isok(n) = (n % 2) && isprime(numdiv(n-1)); \\ Michel Marcus, Nov 27 2016

a(n) = A061286(n) + 1.
sigma(a(n)-1) = A001348(n), i.e., Mersenne numbers.
tau(a(n)-1) = A000040(n), i.e., all primes; a(n) = the smallest odd number k such that tau(a(n)-1) = prime(n) = A000040(n).

A135620 a(n) = 2^(prime(n) - 2).

Original entry on oeis.org

1, 2, 8, 32, 512, 2048, 32768, 131072, 2097152, 134217728, 536870912, 34359738368, 549755813888, 2199023255552, 35184372088832, 2251799813685248, 144115188075855872, 576460752303423488, 36893488147419103232, 590295810358705651712, 2361183241434822606848, 151115727451828646838272

Offset: 1

Omar E. Pol, Mar 01 2008

G. C. Greubel, Table of n, a(n) for n = 1..465

Cf. A000040, A034785, A040976, A061286.

Partial differences of A135482.

[2^(NthPrime(n)-2): n in [1..25]]; // Vincenzo Librandi, Oct 23 2016

Table[2^(Prime[n] - 2), {n,1,25}] (* G. C. Greubel, Oct 23 2016 *)

a(n) = 2^(prime(n)-2); \\ Michel Marcus, Oct 23 2016

a(n) = 2^(A000040(n)-2) = 2^(A040976(n)) = 2^A000040(n)/4 = A061286(n)/2.

a(n) = A034785(n)/4. - Alois P. Heinz, Jun 08 2025

A187678 The smallest integer for which the number of divisors is the n-th nonprime.

Original entry on oeis.org

1, 6, 12, 24, 36, 48, 60, 192, 144, 120, 180, 240, 576, 3072, 360, 1296, 12288, 900, 960, 720, 840, 9216, 196608, 5184, 1260, 786432, 36864, 1680, 2880, 15360, 3600, 12582912, 2520, 46656, 6480, 589824, 61440, 6300, 82944, 6720, 2359296, 805306368, 5040, 3221225472, 14400, 7560

Offset: 1

Juri-Stepan Gerasimov, Mar 12 2011

nonn

Amiram Eldar, Table of n, a(n) for n = 1..5789

Cf. A005179, A018252, A061286.

a(n) = A005179(A018252(n)).

cp's OEIS Frontend

A137491 Numbers with 28 divisors.

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A137487 Numbers with 24 divisors.

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A299795 Numbers of the form p*2^(p-1) where p is prime.

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A358820 a(n) is the least novel k such that d(k)|n, where d is the divisor counting function A000005.

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A137489 Numbers with 26 divisors.

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A200815 Number of iterations of k -> d(k) until n reaches an odd prime.

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A269781 a(n) is the smallest k different from n such that (n, k) is an amicably refactorable pair (see the comments).

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