A055927 -id:A055927 - cp's OEIS frontend

A188629 Numbers k such that k^2 has one more divisor than k^2 - 1.

2, 4, 8, 14, 16, 22, 38, 58, 135, 158, 178, 256, 297, 382, 502, 542, 568, 676, 718, 878, 1202, 1215, 1312, 1318, 1382, 1438, 1593, 1622, 1822, 2018, 2144, 2336, 2558, 2578, 2744, 2858, 2902, 3062, 3118, 3296, 3375, 3778, 3993, 4023, 4064, 4192, 4282

Offset: 1

Juri-Stepan Gerasimov, Apr 06 2011

nonn

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

Cf. A000005, A000290, A055927, A172438.

isA188629 := proc(n) if numtheory[tau](n^2) = numtheory[tau](n^2-1)+1 then true; else false; end if; end proc:
for n from 1 do if isA188629(n) then print(n) ; end if; end do: # R. J. Mathar, Apr 14 2011

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

is(k) = k > 1 && numdiv(k^2-1) + 1 == numdiv(k^2); \\ Amiram Eldar, Apr 17 2024

A339776 Numbers m such that tau(m) = tau(m + 1) - 1 = tau(m + 2), where tau(k) = the number of divisors of k (A000005).

Original entry on oeis.org

3, 252003, 293763, 770883, 1444803, 2630883, 6543363, 8421603, 9375843, 18992163, 19731363, 21883683, 22108803, 25786083, 25989603, 32512803, 35259843, 48972003, 98049603, 101566083, 132204003, 155201763, 160224963, 162766563, 187197123, 208455843, 291658083

Offset: 1

Jaroslav Krizek, Dec 16 2020

nonn

Corresponding values of tau(a(n)): 2, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, ...
Triplets of [tau(a(n)), tau(a(n) + 1), tau(a(n) + 2)] = [tau(a(n)), tau(a(n)) + 1, tau(a(n))]: [2, 3, 2], [8, 9, 8], [8, 9, 8], [8, 9, 8], [8, 9, 8], [8, 9, 8], [8, 9, 8], [8, 9, 8], [8, 9, 8], ...
a(n) is one less than a perfect square. - David A. Corneth, Dec 29 2020

			tau(3) = 2, tau(4) = 3, tau(5) = 2.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000005, A339777.
Subsequence of A005563.
Intersection of A062832 and A055927.

[m: m in [2..10^6] | #Divisors(m + 1) - 1 eq #Divisors(m) and #Divisors(m + 2) eq #Divisors(m)]

d1 = 1; d2 = 2; s = {}; Do[d3 = DivisorSigma[0, n]; If[Equal @@ {d1, d2 - 1, d3}, AppendTo[s, n - 2]]; d1 = d2; d2 = d3, {n, 3, 10^7}]; s (* Amiram Eldar, Dec 17 2020 *)

isok(m) =  my(nb = numdiv(m)); (numdiv(m+2) == nb) && (numdiv(m+1) == nb+1); \\ Michel Marcus, Dec 18 2020

More terms from Amiram Eldar, Dec 16 2020

A350142 Numbers m of the form 2^k + 1 such that tau(m-2) = tau(m-1) - 1.

Original entry on oeis.org

3, 5, 17, 65, 257, 65537, 4294967297

Offset: 1

Jaroslav Krizek, Dec 16 2021

Corresponding pairs of values [tau(m-2), tau(m-1)]: [1, 2], [2, 3], [4, 5], [6, 7], [8, 9], [16, 17], [32, 33], ...
There are no other terms <= 2^1206 + 1 (from A046801 data).
The first 5 known Fermat primes from A019434 are in this sequence. Corresponding values of tau(A019434(n - 2)): 1, 2, 4, 8, 16, ...
Conjecture 1: Also numbers m of the form 2^k + 1 such that tau(m - 2) = k.
Conjecture 2: If 6th Fermat prime F_p6 exists, then tau(F_p6 - 2) is a power of 2 and tau(F_p6 - 1) = tau(F_p6 - 2) + 1.
Conjecture 3: Sequence is finite with 7 terms; supersequence of A262534.

			For number 257 holds: tau(255) = 8, tau(256) = 9.

Cf. A000005, A019434, A262534, A343144, A347078.

Intersection of (A055927+2) and A000051.

[2^k + 1: k in [1..50] | #Divisors(2^k) - #Divisors(2^k-1) eq 1];

cp's OEIS Frontend

A188629 Numbers k such that k^2 has one more divisor than k^2 - 1.

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A339776 Numbers m such that tau(m) = tau(m + 1) - 1 = tau(m + 2), where tau(k) = the number of divisors of k (A000005).

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A350142 Numbers m of the form 2^k + 1 such that tau(m-2) = tau(m-1) - 1.

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