A323379 -id:A323379 - cp's OEIS frontend

A075027 Numbers k such that d(k-1) < d(k) > d(k+1), d = A000005.

4, 6, 8, 10, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 50, 52, 54, 56, 60, 64, 66, 68, 70, 72, 78, 80, 84, 88, 90, 92, 96, 100, 102, 108, 110, 112, 114, 120, 124, 126, 128, 130, 132, 138, 140, 144, 150, 152, 154, 156, 160, 162, 165, 168, 170, 174, 176, 180

Offset: 1

Amarnath Murthy, Sep 02 2002

nonn

Obviously every term is composite.
The average of each twin prime pair is a term.
a(55) = 165 is the first odd term; A323379 lists all odd terms. - Jon E. Schoenfield, Sep 26 2021

			10 is a term since d(9) = 3, d(10) = 4, d(11) = 2 and 3 < 4 > 2.

Michael De Vlieger, Table of n, a(n) for n = 1..3403 (first 1001 terms from Harvey P. Dale)

Cf. A000005, A014574, A075025, A075026, A323379.

q:= k-> (d-> d(k-1)d(k+1))(numtheory[tau]):
select(q, [$1..200])[];  # Alois P. Heinz, Sep 28 2021

#[[2,1]]&/@Select[Partition[Table[{n,DivisorSigma[0,n]},{n,200}],3,1], #[[1,-1]]<#[[2,-1]]>#[[3,-1]]&] (* Harvey P. Dale, Oct 09 2011 *)

More terms from Jason Earls, Sep 04 2002

A323380 Odd n such that sigma(n) > sigma(n+1) and sigma(n) > sigma(n-1), sigma = A000203.

Original entry on oeis.org

315, 405, 525, 693, 765, 945, 1125, 1155, 1395, 1575, 1755, 1785, 1845, 1995, 2205, 2475, 2565, 2805, 2835, 3003, 3045, 3285, 3315, 3465, 3645, 3675, 3885, 4095, 4125, 4275, 4347, 4455, 4515, 4725, 4995, 5115, 5355, 5445, 5733, 5775, 5805, 6045, 6195, 6237, 6405, 6435

Offset: 1

Jianing Song, Jan 12 2019

nonn

Numbers k such that k is in A067828 and that k - 1 is in A067825.
It's often the case that the sum of divisors for an odd number is less than at least one of its adjacent even numbers. This sequence lists the exceptions.
Most terms are congruent to 3 modulo 6. It seems that the smallest term not congruent to 3 modulo 6 is greater than 10^12.

			sigma(314) = 474, sigma(315) = 624, sigma(316) = 560, so 315 is a term.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A000203, A067825, A067828;

Similar sequences: A076773, A323379.

Select[Range[1,8000,2],DivisorSigma[1,#] > DivisorSigma[1,(#+1)] && DivisorSigma[1,#] > DivisorSigma[1,(#-1)] &] (* K. D. Bajpai, Nov 19 2019 *)

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

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.

cp's OEIS Frontend

A075027 Numbers k such that d(k-1) < d(k) > d(k+1), d = A000005.

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A323380 Odd n such that sigma(n) > sigma(n+1) and sigma(n) > sigma(n-1), sigma = A000203.

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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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