A062832 -id:A062832 - cp's OEIS frontend

A067888 Numbers k such that tau(k+1) = tau(k-1) where tau(k) = A000005(k).

4, 6, 7, 9, 12, 18, 19, 30, 34, 41, 42, 51, 55, 56, 60, 72, 86, 92, 94, 102, 103, 108, 124, 129, 137, 138, 142, 144, 150, 153, 160, 180, 183, 184, 185, 186, 192, 198, 199, 202, 204, 214, 216, 218, 220, 228, 231, 236, 240, 243, 244, 247, 248, 249, 266, 270, 282

Offset: 1

Benoit Cloitre, Mar 02 2002

If (p,p+2) are twin primes, then the composite number p+1 is in this sequence. The primes occurring in this sequence are listed in A067889. See A055574 for the analog with sigma instead of tau. - M. F. Hasler, Aug 06 2015

Giovanni Resta, Table of n, a(n) for n = 1..10000

Equals A062832 + 1. - Michel Marcus, Feb 11 2018

Cf. A000005, A055574, A067889.

Select[Range[300], Equal @@ DivisorSigma[0, # + {-1, 1}] &] (* Amiram Eldar, Jan 23 2025 *)

is_A067888(n)=n>1&&numdiv(n-1)==numdiv(n+1) \\ M. F. Hasler, Aug 06 2015

A276542 Numbers k such that the k-th and (k+1)st triangular numbers have the same number of divisors.

Original entry on oeis.org

3, 4, 5, 11, 17, 28, 29, 33, 41, 42, 52, 55, 59, 66, 68, 71, 76, 85, 88, 91, 93, 101, 107, 114, 123, 137, 141, 143, 149, 150, 159, 170, 172, 179, 183, 185, 186, 188, 191, 196, 197, 201, 203, 208, 213, 215, 217, 219, 227, 232, 235, 236, 239, 243, 244, 247, 265

Offset: 1

K. D. Bajpai, Apr 09 2017

nonn

The k-th triangular number T(k) = k*(k+1)/2.

The lesser member of each twin-prime pair appears in this sequence. Hence, A001359 is a subset of this sequence.

			a(3) = 5; T(5) = 5*(5+1)/2 = 15; T(5+1) = 6*(6+1)/2 = 21; 15 and 21 have 4 divisors each.
a(6) = 28; T(28) = 28*(28+1)/2 = 406; T(28+1) = 29*(29+1)/2 = 435; 406 and 435 have 8 divisors each

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A000217, A001359, A062832.

Cf. A319035 (the corresponding triangular numbers).

T:=List([1..270],n->n*(n+1)/2);;  a:=Filtered([1..Length(T)-1],i->Tau(T[i])=Tau(T[i+1])); # Muniru A Asiru, Dec 06 2018

[n: n in [1..300] | DivisorSigma(0, n*(n + 1) div 2) eq DivisorSigma(0, (n + 1)*(n + 1 + 1) div 2)]; // Vincenzo Librandi, Dec 06 2018

T:= seq(numtheory:-tau(n*(n+1)/2), n=1..1000):
select(t -> T[t]=T[t+1], [$1..999]); # Robert Israel, Apr 09 2017

Select[Range[1000], DivisorSigma[0, #*(# + 1)/2] == DivisorSigma[0, (# + 1)*(# + 1 + 1)/2] &]
SequencePosition[DivisorSigma[0,#]&/@Accumulate[Range[300]],{x_,x_}][[All, 1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 02 2018 *)

k=[]; for(n=1, 1000, a=numdiv(n*(n+1)/2); b=numdiv((n+1)*(n+1+1)/2); if(a==b, k=concat(k, n))); k

from sympy import divisor_count
for n in range(1,20):
    if divisor_count(n*(n+1)/2)==divisor_count((n+1)*(n+2)/2):
        print(n, end=', ') # Stefano Spezia, Dec 06 2018

A229254 Numbers k such that k and k+2 have the same number (A000005) and sum of divisors (A000203).

Original entry on oeis.org

33, 54, 918, 1240, 3304, 4148, 4187, 7169, 12565, 15085, 19688, 24881, 25019, 26609, 38982, 51835, 53963, 59987, 76360, 77057, 96728, 143369, 150419, 167560, 170561, 205727, 215069, 220817, 278920, 418307, 564857, 731320, 785270, 907254, 910315, 986153

Offset: 1

Jaroslav Krizek, Sep 20 2013

nonn

Also numbers k such that A229335(k) = A229335(k+2).

Intersection of A007373 and A062832.

			Divisors of 54 = {1, 2, 3, 6, 9, 18, 27, 54}, divisors of 56 = {1, 2, 4, 7, 8, 14, 28, 56}, both have 8 divisors and sum = 120.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A007373, A062832, A000005, A000203, A229335.

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

isok(n) = (numdiv(n) == numdiv(n+2)) && (sigma(n) == sigma(n+2)); \\ Michel Marcus, Sep 20 2013

More terms from Michel Marcus, Sep 20 2013

A276553 Numbers n such that n^2 and (n + 1)^2 have the same number of divisors.

Original entry on oeis.org

2, 14, 15, 21, 33, 34, 38, 44, 57, 75, 81, 85, 86, 93, 94, 98, 116, 118, 122, 133, 135, 141, 142, 145, 147, 158, 171, 177, 201, 202, 205, 213, 214, 217, 218, 230, 244, 253, 272, 285, 296, 298, 301, 302, 326, 332, 334, 375, 381, 387, 393, 394, 405, 429, 434, 445

Offset: 1

K. D. Bajpai, Apr 10 2017

nonn

Except for a(1), all the terms are composite.

			We see that 14^2 = 196, the divisors of which are 1, 2, 4, 7, 14, 28, 49, 98, 196, and there are nine of them. And we see that 15^2 = 225, the divisors of which are 1, 3, 5, 9, 15, 25, 45, 75, 225, and there are nine of them. Both 14^2 and 15^2 have the same number of divisors, hence 14 is in the sequence.
And we see that 16^2 = 256, the divisors of which are the powers of 2 from 2^0 to 2^8, that's nine divisors. Both 15^2 and 16^2 have the same number of divisors, hence 15 is also in the sequence.
But 16 is not in the sequence, since 17 is prime and 17^2 consequently only has three divisors.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000290, A048691, A062832, A276542, A284378.
Cf. A052213 (a subsequence).
Positions of zeros in A284570.

N:= 1000: # to get all terms <= N
T:= map(t -> numtheory:-tau(t^2), [$1..N+1]):
select(t -> T[t]=T[t+1], [$1..N]); # Robert Israel, Apr 10 2017

Select[Range[1000], DivisorSigma[0, #^2] == DivisorSigma[0, (# + 1)^2] &]

k=[]; for(n=1, 1000, a=numdiv(n^2); b=numdiv((n+1)^2); if(a==b, k=concat(k, n))); k

from sympy.ntheory import divisor_count
print([n for n in range(1, 501) if divisor_count(n**2) == divisor_count((n + 1)**2)]) # Indranil Ghosh, Apr 10 2017
(Scheme, with Antti Karttunen's IntSeq-library) (define A276553 (ZERO-POS 1 1 A284570)) ;; Antti Karttunen, Apr 15 2017

A276713 Numbers n such that n and n+3 have the same number of divisors (A000005).

Original entry on oeis.org

2, 35, 55, 62, 74, 82, 91, 102, 115, 119, 122, 135, 142, 143, 155, 158, 172, 186, 202, 203, 206, 214, 215, 218, 242, 255, 259, 262, 282, 295, 298, 299, 302, 323, 326, 343, 351, 354, 355, 362, 391, 395, 399, 425, 426, 435, 451, 466, 478, 482, 492, 502, 511, 514

Offset: 1

Jaroslav Krizek, Sep 16 2016

nonn

			35 is in sequence because tau(35) = tau(38) = 4.

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Cf. A065559 (smallest k such that tau(k) = tau(k+n)), A015861 (sigma(n) = sigma(n+3)), A276714.

Cf. Similar sequences with numbers n such that n and n+k have the same number of divisors for k = 1: A005237, for k = 2: A062832.

[n: n in [1..10000] | NumberOfDivisors(n) eq  NumberOfDivisors(n+3)]

with(numtheory): A276713:=n->`if`(tau(n) = tau(n+3), n, NULL): seq(A276713(n), n=1..10^3); # Wesley Ivan Hurt, May 02 2017

SequencePosition[DivisorSigma[0,Range[600]],{x_,,,x_}][[All,1]] (* Harvey P. Dale, Nov 12 2022 *)

isok(n) = numdiv(n) == numdiv(n+3); \\ Michel Marcus, May 03 2017

A330703 Numbers k such that psi(k) = psi(k + 2) where psi(k) is the Dedekind psi function (A001615).

Original entry on oeis.org

6, 9, 12, 14, 18, 20, 33, 44, 62, 70, 92, 108, 116, 138, 164, 175, 212, 254, 280, 308, 320, 332, 348, 356, 452, 490, 524, 558, 572, 692, 716, 764, 833, 932, 956, 1004, 1105, 1124, 1172, 1188, 1436, 1496, 1562, 1593, 1676, 1724, 1772, 1964, 2002, 2036, 2088, 2132

Offset: 1

Amiram Eldar, Dec 26 2019

nonn

			6 is in the sequence since psi(6) = psi(8) = 12.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jozsef Sandor, On the composition of some arithmetic functions, II, Journal of Inequalities in Pure and Applied Mathematics, Vol. 6, No. 3 (2005), Article 73.

Cf. A001494, A001615, A007373, A062832, A291043.

psi[1] = 1; psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); Select[Range[10^3], psi[#] == psi[# + 2] &]

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

A339777 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

8, 110888, 149768, 1119363, 1172888, 2676495, 3143528, 4782968, 5895183, 8596623, 9168783, 15896168, 19114383, 28174863, 48052623, 50523663, 58186383, 72641528, 82664463, 98168463, 113465103, 139523343, 178810383, 208860303, 223681935, 230675343, 248755983, 249260943

Offset: 1

Jaroslav Krizek, Dec 16 2020

nonn

Corresponding values of tau(a(n)): 4, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, ...
Triplets of [tau(a(n)), tau(a(n) + 1), tau(a(n) + 2)] = [tau(a(n)), tau(a(n)) - 1, tau(a(n))]: [4, 3, 4], [16, 15, 16], [16, 15, 16], [16, 15, 16], [16, 15, 16], [16, 15, 16], [16, 15, 16], [16, 15, 16], [16, 15, 16], ...
a(n) is one less than a perfect square. - David A. Corneth, Dec 29 2020

			tau(8) = 4, tau(9) = 3, tau(10) = 4.

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

Cf. A000005, A339776.
Subsequence of A005563.
Intersection of A062832 and A068208.

[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 *)
Position[Partition[DivisorSigma[0,Range[59*10^5]],3,1],?(#[[1]]==#[[2]]+1==#[[3]]&),1,Heads->False]//Flatten (* _Harvey P. Dale, May 25 2023 *)

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

A356766 Least number k such that k and k+2 both have exactly 2n divisors, or -1 if no such number exists.

Original entry on oeis.org

3, 6, 18, 40, 127251, 198, 26890623, 918, 17298, 6640, 25269208984375, 3400, 3900566650390623, 640062, 8418573, 18088, 1164385682220458984373, 41650, 69528379848480224609373, 128464, 34084859373, 12164094, 150509919493198394775390625, 90270, 418514293125, 64505245696

Offset: 1

Jean-Marc Rebert, Aug 26 2022

nonn

			For n=1, numdiv(3) = numdiv(5) = 2 = 2*1, and no number < 3 satisfies this, hence a(1) = 3.

Numbers k such that k and k+2 both have exactly m divisors: A001359 (m=2), A356742 (m=4), A356743 (m=6), A356744 (m=8).

Cf. A000005 (d(n)), A003680, A005238, A006558, A006601, A062832, A067888, A067889, A075036.

a={}; n=1; nmax=10; For[k=1, n<=nmax, k++, If[DivisorSigma[0, k] == DivisorSigma[0, k+2] == 2n, AppendTo[a, k]; k=1; n++]]; a (* Stefano Spezia, Aug 26 2022 *)
Flatten[Table[SequencePosition[DivisorSigma[0,Range[27*10^6]],{2n,,2n},1],{n,10}],1][[;;,1]] (* The program generates the first 10 terms of the sequence. To generate more, increase the Range constant but the program will take a long time to run. *) (* _Harvey P. Dale, Jul 01 2023 *)

a(n)=for(k=1,+oo,if(numdiv(k)==2*n&&numdiv(k+2)==2*n,return(k)))

More terms from Jinyuan Wang, Aug 28 2022

cp's OEIS Frontend

A067888 Numbers k such that tau(k+1) = tau(k-1) where tau(k) = A000005(k).

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A276542 Numbers k such that the k-th and (k+1)st triangular numbers have the same number of divisors.

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A229254 Numbers k such that k and k+2 have the same number (A000005) and sum of divisors (A000203).

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A276553 Numbers n such that n^2 and (n + 1)^2 have the same number of divisors.

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A276713 Numbers n such that n and n+3 have the same number of divisors (A000005).

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A330703 Numbers k such that psi(k) = psi(k + 2) where psi(k) is the Dedekind psi function (A001615).

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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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