A075027 -id:A075027 - cp's OEIS frontend

A075025 Numbers k such that d(k) < d(k-1) and d(k) < d(k+1), where d(k) is the number of divisors of k.

5, 7, 9, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 51, 53, 55, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 89, 91, 97, 101, 103, 107, 109, 111, 113, 115, 121, 125, 127, 129, 131, 137, 139, 149, 151, 153, 155, 157, 161, 163, 167, 169, 173, 175, 179, 181, 183

Offset: 1

Amarnath Murthy, Sep 02 2002

nonn

All primes > 3 are members.
Is this sequence of positive density?  I expect a(n) ~ 4n but can only prove n (log log n)^k/ log n << a(n) << n for arbitrary k. - Charles R Greathouse IV, May 01 2011
Number of terms < 10^k: 3, 32, 324, 3222, 32026, 318583, 3181133, 31766404, ..., . - Robert G. Wilson v, May 01 2011

			17 is in the sequence because d(16) = 5, d(17) = 2, d(18) = 6 and 5 > 2 < 6.

Cf. A000005, A361797 (even terms).

Cf. A075026, A075027.

fQ[n_] := DivisorSigma[0, n - 1] > DivisorSigma[0, n] < DivisorSigma[0, n + 1]; Select[ Range@ 200, fQ] (* Robert G. Wilson v, May 01 2011 *)

isok(k) = if (k>1, numdiv(k) < min(numdiv(k-1), numdiv(k+1))); \\ Michel Marcus, Mar 26 2023

Corrected and extended by Jason Earls, Sep 04 2002

A323379 Odd k such that d(k-1) < d(k) and d(k) > d(k+1), d = A000005.

Original entry on oeis.org

165, 315, 357, 405, 495, 525, 555, 567, 585, 627, 675, 693, 765, 795, 825, 855, 891, 915, 945, 957, 975, 1005, 1053, 1071, 1125, 1155, 1173, 1305, 1323, 1365, 1395, 1425, 1485, 1515, 1575, 1617, 1677, 1683, 1725, 1755, 1785, 1815, 1827, 1845, 1911, 1965, 1995

Offset: 1

Jianing Song, Jan 12 2019

nonn

Numbers k such that k is in A138171 and that k-1 is in A138172.
It's often the case that an odd number has fewer divisors than at least one of its adjacent even numbers. This sequence lists the exceptions.
Most terms are congruent to 3 modulo 6. The smallest term congruent to 1 modulo 6 is 2275, and the smallest term congruent to 5 modulo 6 is 6125.

			d(314) = 4, d(315) = 12, d(316) = 6, so 315 is a term.

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

Intersection of A075027 and A005408.
Cf. A000005, A138171, A138172.
Similar sequences: A076773, A323380.

q:= k-> k::odd and (d-> d(k-1)d(k+1))(numtheory[tau]):
select(q, [$1..2000])[];  # Alois P. Heinz, Sep 28 2021

Select[Range[3, 2001, 2], (d = DivisorSigma[0, #] & /@ (# + Range[-1,1]))[[2]] > d[[1]] && d[[2]] > d[[3]] &] (* Amiram Eldar, Jul 22 2019 *)

forstep(n=3,2000,2,if(numdiv(n)>numdiv(n-1)&&numdiv(n)>numdiv(n+1), print1(n, ", ")))

A075026 Define a number k to occupy a divisor cavity if d(k-1) > d(k) < d(k+1) where d(k) is the number of divisors of k. Sequence gives composite numbers occupying a divisor cavity.

Original entry on oeis.org

9, 25, 49, 51, 55, 65, 69, 77, 91, 111, 115, 121, 125, 129, 153, 155, 161, 169, 175, 183, 185, 187, 209, 221, 235, 237, 247, 249, 259, 265, 267, 274, 287, 289, 291, 295, 305, 309, 319, 321, 323, 329, 339, 341, 343, 351, 355, 361, 365, 369, 371, 377, 386, 391

Offset: 0

Amarnath Murthy, Sep 02 2002

nonn

Cf. A000005, A075025, A075027.

q:= k-> not isprime(k) and (d->
        d(k-1)>d(k) and d(k)Alois P. Heinz, Sep 28 2021

Select[Flatten[Position[Partition[DivisorSigma[0,Range[400]],3,1],?(#[[1]]> #[[2]]<#[[3]]&),1,Heads->False]]+1,CompositeQ] (* _Harvey P. Dale, Oct 23 2019 *)

Corrected and extended by Jason Earls, Sep 04 2002

A347719 a(n) is the smallest prime(n)-rough number k that has more divisors than k-1 and more divisors than k+1.

Original entry on oeis.org

4, 165, 2275, 18473, 45617, 71383, 257393, 257393, 1239907, 1275797, 1851847, 4411843, 6865337, 6865337, 8312467, 15763207, 24157963, 33684317, 33684317, 60428597, 61182103, 61694813, 73803517, 104622971, 128397967, 128397967, 173805187, 214820797, 284708981

Offset: 1

Jon E. Schoenfield, Sep 26 2021

nonn

Equivalently, a(n) is the smallest number k that is not divisible by any of the first n-1 primes such that d(k-1) < d(k) > d(k+1), where d(k) = A000005(k) is the number of divisors of k.
a(1) = 4 is the initial term of A075027: 4 is the smallest k such that d(k-1) < d(k) > d(k+1).
a(2) = 165 = A323379(1) is the smallest odd k such that d(k-1) < d(k) > d(k+1).
For n > 2, since neither 2 nor 3 divides k, one of k's two nearest neighbors, k-1 and k+1, is a multiple of 3, and both of those neighbors are even numbers, so one of those neighbors is a multiple of 6, and thus (since that neighbor is not 6, 12, or 18) that neighbor has at least 8 divisors, so k must have at least 9 divisors, so k must be the product of at least 4 primes (counted with multiplicity). For distinct primes p, q, r, and s, the product k cannot be p^4 (only 5 divisors) or p^3 * q (only 8 divisors), nor can it be p^2 * q^2 (which would have 9 divisors, but since k would be an odd square not divisible by 3, k-1 would be a proper multiple of 24 and would thus have more than 9 divisors), so k must be p^2 * q * r (12 divisors) or p*q*r*s (16 divisors).
For n > 3, k cannot simultaneously be congruent to +-1 (mod 10) and be of the form p^2 * q * r: one of its two neighbors would be divisible by 4 and the other twice an odd number, one would be divisible by 3, and one would be divisible by 5, so at least one of the two would have at least 12 divisors.
It appears that the least prime factor of a(n) is usually the n-th prime, but there are 299 exceptions among the first 1000 terms, beginning with the terms for n = 7, 13, and 18 (see the Example section).
Conjecture: no term is the product of more than 4 prime factors, counted with multiplicity.

			In the table below, an asterisk after the number in the "n-th prime" column appears in the row for each number n such that the least prime factor of a(n) is not the n-th prime.
.
                number of divisors of
                ======================  n-th   prime factorization
   n      a(n)  a(n)-1    a(n)  a(n)+1  prime        of a(n)
  --  --------  ------  ------  ------  -----  -------------------
   1         4       2       3       2      2     2 *  2
   2       165       6       8       4      3     3 *  5 * 11
   3      2275       8      12       6      5     5 *  5 *  7 * 13
   4     18473       8      12       8      7     7 *  7 * 13 * 29
   5     45617      10      12       8     11    11 * 11 * 13 * 29
   6     71383       8      12       8     13    13 * 17 * 17 * 19
   7    257393      10      12       8     17*   19 * 19 * 23 * 31
   8    257393      10      12       8     19    19 * 19 * 23 * 31
   9   1239907       8      16       6     23    23 * 31 * 37 * 47
  10   1275797       6      12       8     29    29 * 29 * 37 * 41
  11   1851847       8      12       8     31    31 * 31 * 41 * 47
  12   4411843       8      16      12     37    37 * 43 * 47 * 59
  13   6865337       8      12       8     41*   43 * 43 * 47 * 79
  14   6865337       8      12       8     43    43 * 43 * 47 * 79
  15   8312467       8      12       6     47    47 * 47 * 53 * 71
  16  15763207       8      12       8     53    53 * 59 * 71 * 71
  17  24157963      12      16       6     59    59 * 71 * 73 * 79
  18  33684317       6      16      12     61*   67 * 71 * 73 * 97

Cf. A000005, A075027, A323379.

A369182 Numbers k such that d(k) is greater than d(k-2), d(k-1), d(k+1) and d(k+2) where d(k) is the number of divisors of k.

Original entry on oeis.org

12, 24, 30, 36, 48, 60, 66, 72, 80, 84, 90, 96, 100, 108, 112, 120, 126, 132, 140, 144, 150, 156, 160, 165, 168, 176, 180, 192, 204, 210, 216, 220, 224, 228, 234, 240, 252, 256, 260, 264, 270, 276, 280, 288, 294, 300, 312, 315, 320, 324, 330, 336, 357, 360, 364, 368, 372

Offset: 1

Zhicheng Wei, Jan 15 2024

nonn

			30 is a term since d(28) = 6, d(29) = 2, d(30) = 8, d(31) = 2, d(32) = 6 and 6 < 8 > 6, 2 < 8 > 2.

Cf. A000005.

Subsequence of A075027.

SequencePosition[Table[DivisorSigma[0, n], {n, 1, 400}], {a_, b_, c_, d_, e_} /; Max[a, b, d, e] < c][[;; , 1]] + 2 (* Amiram Eldar, Jan 16 2024 *)

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A075025 Numbers k such that d(k) < d(k-1) and d(k) < d(k+1), where d(k) is the number of divisors of k.

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A323379 Odd k such that d(k-1) < d(k) and d(k) > d(k+1), d = A000005.

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A075026 Define a number k to occupy a divisor cavity if d(k-1) > d(k) < d(k+1) where d(k) is the number of divisors of k. Sequence gives composite numbers occupying a divisor cavity.

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A347719 a(n) is the smallest prime(n)-rough number k that has more divisors than k-1 and more divisors than k+1.

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A369182 Numbers k such that d(k) is greater than d(k-2), d(k-1), d(k+1) and d(k+2) where d(k) is the number of divisors of k.

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