A138171 -id:A138171 - cp's OEIS frontend

A112275 Smallest number greater than n having at least as many divisors as n.

2, 3, 4, 6, 6, 8, 8, 10, 10, 12, 12, 18, 14, 15, 16, 18, 18, 20, 20, 24, 22, 24, 24, 30, 26, 27, 28, 30, 30, 36, 32, 36, 34, 35, 36, 48, 38, 39, 40, 42, 42, 48, 44, 45, 48, 48, 48, 60, 50, 52, 52, 54, 54, 56, 56, 60, 58, 60, 60, 72, 62, 63, 64, 66, 66, 70, 68, 70, 70, 72, 72, 84

Offset: 1

Reinhard Zumkeller, Sep 01 2005

nonn

A000005(n) <= A000005(a(n)) and A000005(k) < A000005(n) for n

A000005(2*k-1) <= A000005(2*k) for 1<=k<=22. - Corrected by Robert Israel, Jul 23 2019

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

Cf. A079427, A112276.

Cf. A138171 (odd n for which a(n) > n+1).

N:= 1000: # for all terms before the first term > N
taus:= map(numtheory:-tau,[$1..N]):
for n from 1 to N do
found:= false:
for k from n+1 to N while not found do
   if taus[k]>=taus[n] then found:= true; A[n]:= k fi
od;
if not found then break fi
od:
seq(A[i],i=1..n-1); # Robert Israel, Jul 23 2019

kmax[n_] := 2 n;
a[n_] := Module[{tau = DivisorSigma[0, n], k},
     For[k = n + 1, k <= kmax[n], k++,
          If[DivisorSigma[0, k] >= tau, Return[k]]];
     Print["a(n) = k not found for n = ", n]];
Array[a, 100] (* Jean-François Alcover, Dec 15 2021 *)

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, ", ")))

A138172 Even n where d(n) < d(n+1), where d(n) = number of positive divisors of n.

Original entry on oeis.org

62, 74, 134, 146, 164, 188, 194, 206, 254, 274, 278, 284, 314, 356, 362, 386, 398, 404, 422, 428, 454, 458, 482, 494, 524, 538, 554, 566, 584, 614, 626, 662, 674, 692, 734, 746, 758, 764, 794, 818, 824, 854, 866, 890, 914, 926, 934, 944, 956, 974, 998, 1004, 1028

Offset: 1

Leroy Quet, Mar 03 2008

nonn

Terms calculated by M. F. Hasler.

First term == 0 (mod 6) is a(133) = 2274. - Jianing Song, Apr 03 2018

Jianing Song and Muniru A Asiru, Table of n, a(n) for n = 1..10000(Terms 1 through 1019 from Jianing Song)

Cf. A000005, A138046, A138171.

Filtered([2,4..1300],n->Tau(n)Muniru A Asiru, Apr 05 2018

with(numtheory): A138172:=n->`if`(n mod 2 = 0 and tau(n) < tau(n+1), n, NULL): seq(A138172(n), n=1..1000); # Wesley Ivan Hurt, Apr 10 2015

2 Position[Partition[Array[DivisorSigma[0, #] &, 10^3, 2], 2, 2], ?(#1 < #2 & @@ # &)][[All, 1]] (* _Michael De Vlieger, Apr 07 2018 *)

lista(nn) = {forstep(n=2, nn, 2, if (numdiv(n) < numdiv(n+1), print1(n, ", ")););} \\ Michel Marcus, Apr 10 2015

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cp's OEIS Frontend

A112275 Smallest number greater than n having at least as many divisors as n.

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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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A138172 Even n where d(n) < d(n+1), where d(n) = number of positive divisors of n.

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