A075025 -id:A075025 - 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

A067345 Square array read by antidiagonals: T(n,k)=(T(n,k-1)*n^2-Catalan(k-1))/(n-1) with a(n,1)=1 and a(1,k)=Catalan(k) where Catalan(k)=C(2k,k)/(k+1)=A000108(k).

Original entry on oeis.org

1, 2, 1, 5, 3, 1, 14, 10, 4, 1, 42, 35, 17, 5, 1, 132, 126, 74, 26, 6, 1, 429, 462, 326, 137, 37, 7, 1, 1430, 1716, 1446, 726, 230, 50, 8, 1, 4862, 6435, 6441, 3858, 1434, 359, 65, 9, 1, 16796, 24310, 28770, 20532, 8952, 2582, 530, 82, 10, 1, 58786, 92378, 128750

Offset: 1

Henry Bottomley, Jan 16 2002

Also table given by Sum_{k, 0<=k<=n}A039598(n,k)*x^k ; table begins : x=0 : 1, 2, 5, 42, 132, ...(see A000108); x=1 : 1, 3, 10, 35, 126, ...(see A001700); x=2 : 1, 4, 17, 74, 326, ...(see A049027); x=3 : 1, 5, 26, 137, 726, ...(see A075025); x=4 : 1, 6, 37, 230, 1434, ...(see A075026); x=5 : 1, 7, 50, 359, 2582, ... - Philippe Deléham, Mar 21 2007

Rows include A000108, A001700, A049027. Columns essentially include A000012, A000027, A002522.

T(n, k) =A067346(n, k)/(n-1) =A067347(n, k)/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

A361797 Even numbers k which have fewer divisors than both neighboring odd numbers, i.e., tau(k) < min{tau(k-1), tau(k+1)}.

Original entry on oeis.org

274, 386, 626, 926, 1126, 1174, 1234, 1546, 1574, 1594, 1646, 1774, 1814, 1954, 2036, 2066, 2092, 2186, 2234, 2276, 2302, 2374, 2386, 2402, 2404, 2554, 2638, 2738, 2876, 2906, 3158, 3244, 3334, 3394, 3446, 3554, 3566, 3574, 3758, 3814, 3994, 4124, 4166, 4174

Offset: 1

Steven Lu, Mar 25 2023

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Even terms of A075025. Cf. A000005.

Tau:= map(numtheory:-tau, [$1..10001]):
select(t -> Tau[t] < Tau[t-1] and Tau[t] < Tau[t+1], [seq(i,i=2..10000,2)]); # Robert Israel, Mar 28 2023

Select[2 Range[10000],
 DivisorSigma[0, #] < DivisorSigma[0, # + 1] &&
   DivisorSigma[0, #] < DivisorSigma[0, # - 1] &]

isok(k) = !(k%2) && (numdiv(k) < min(numdiv(k-1), numdiv(k+1))); \\ Michel Marcus, Mar 26 2023

A369155 Numbers k such that d(k) < d(k - 1) and d(k) < d(k + 1), and d(k) is also a record for this type of number where d(k) is the number of divisors of k.

Original entry on oeis.org

5, 9, 51, 153, 351, 3249, 6579, 19551, 47151, 122451, 246975, 393471, 3292289, 10792495, 15270849, 25770879, 58967271, 60642945, 242340175, 481701375, 5122147185, 6644739375, 6971026699, 21061868751, 92330654625, 213089528575, 1159484186575, 1305664357375

Offset: 1

Zhicheng Wei, Jan 14 2024

Numbers in A075025 with record number of divisors.

			351 is a term in this sequence because d(351) = 8, d(350) = 12, and d(352) = 12, so 351 is a number that has fewer divisors than each of its neighbors, but no number below 351 has that property and has at least 8 divisors.

Cf. A000005. Subsequence of A075025.

lista(kmax) = my(d1 = numdiv(1), d2 = numdiv(2), d3, dm = 0); for(k = 3, kmax, d3 = numdiv(k); if(d2 < d1 && d2 < d3 && d2 > dm, print1(k-1, ", "); dm = d2); d1 = d2; d2 = d3); \\ Amiram Eldar, Jan 16 2024

a(9)-a(20) from Michel Marcus, Jan 15 2024
a(21)-a(22) from Amiram Eldar, Jan 16 2024
a(23)-a(28) from Martin Ehrenstein, Feb 08 2024

cp's OEIS Frontend

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

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A067345 Square array read by antidiagonals: T(n,k)=(T(n,k-1)*n^2-Catalan(k-1))/(n-1) with a(n,1)=1 and a(1,k)=Catalan(k) where Catalan(k)=C(2k,k)/(k+1)=A000108(k).

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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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Extensions

A361797 Even numbers k which have fewer divisors than both neighboring odd numbers, i.e., tau(k) < min{tau(k-1), tau(k+1)}.

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A369155 Numbers k such that d(k) < d(k - 1) and d(k) < d(k + 1), and d(k) is also a record for this type of number where d(k) is the number of divisors of k.

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