A075026 -id:A075026 - 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

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.

Original entry on oeis.org

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

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

cp's OEIS Frontend

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

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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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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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