A067888 -id:A067888 - cp's OEIS frontend

A067889 Primes sandwiched between two numbers having same number of divisors.

7, 19, 41, 103, 137, 199, 307, 349, 491, 739, 823, 919, 1013, 1061, 1193, 1277, 1289, 1409, 1433, 1447, 1481, 1543, 1609, 1667, 1721, 1747, 2153, 2357, 2441, 2617, 2683, 2777, 3259, 3319, 3463, 3581, 3593, 3769, 3797, 3911, 3943, 4013, 4217, 4423, 4457

Offset: 1

Benoit Cloitre, Mar 02 2002

Primes p such that tau(p+1) = tau(p-1) where tau(k) = A000005(k).

These are the primes in sequence A067888 of numbers n such that tau(n+1) = tau(n-1). - M. F. Hasler, Aug 06 2015

			7 is a member as 6 and 8 both have 4 divisors; 19 is a member as 18 and 20 both have 6 divisors each.

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A067888, A067891 (analog with sigma).

with(numtheory):j := 0:for i from 1 to 10000 do b := ithprime(i): if nops(divisors(b-1))=nops(divisors(b+1)) then j := j+1:a[j] := b:fi:od:seq(a[k],k=1..j);

Prime[ Select[ Range[ 700 ], Length[ Divisors[ Prime[ #1 ] - 1 ]] == Length[ Divisors[ Prime[ #1 ] + 1 ]] & ]]
Select[Prime[Range[1000]],DivisorSigma[0,#-1]==DivisorSigma[0,#+1]&] (* Harvey P. Dale, Jun 08 2018 *)

is_A067889(p)=numdiv(p-1)==numdiv(p+1)&&isprime(p) \\ M. F. Hasler, Jul 31 2015

a(n) seems curiously to be asymptotic to 25*n*log(n). [From the number of terms up to 10^8, 10^9, 10^10 and 10^11, i.e., 306147, 2616930, 22835324 and 202105198, this constant can be estimated by 25.858..., 25.858..., 25.845... and 25.872..., respectively. - Amiram Eldar, Jun 28 2022]

A062832 Numbers k such that k and k+2 have the same number of divisors.

Original entry on oeis.org

3, 5, 6, 8, 11, 17, 18, 29, 33, 40, 41, 50, 54, 55, 59, 71, 85, 91, 93, 101, 102, 107, 123, 128, 136, 137, 141, 143, 149, 152, 159, 179, 182, 183, 184, 185, 191, 197, 198, 201, 203, 213, 215, 217, 219, 227, 230, 235, 239, 242, 243, 246, 247, 248, 265, 269, 281

Offset: 1

Jason Earls, Jul 20 2001

nonn

The lesser member of every twin-prime pair occurs in this sequence. Hence A001359 is a subsequence. - T. D. Noe, Sep 17 2007

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Equals A067888 - 1. - Michel Marcus, Feb 11 2018

Select[Range[300],DivisorSigma[0,#]==DivisorSigma[0,#+2]&] (* Harvey P. Dale, Mar 01 2012 *)
Position[Partition[DivisorSigma[0,Range[300]],3,1],?(#[[1]]==#[[3]]&)]// Flatten//Quiet (* _Harvey P. Dale, Mar 17 2017 *)

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

A190646 Least number k such that d(k-1) = d(k+1) = 2n or 0 if no such k exists, where d(n)=A000005(n).

Original entry on oeis.org

4, 7, 19, 41, 127252, 199, 26890624, 919, 17299, 6641, 25269208984376, 3401, 3900566650390624, 640063, 8418574, 18089, 1164385682220458984374, 41651, 69528379848480224609374, 128465, 34084859374, 12164095, 150509919493198394775390626, 90271

Offset: 1

Juri-Stepan Gerasimov, May 15 2011, May 20 2011

nonn

a(28) = 2319679. a(30) = 3568049.
From Chai Wah Wu, Mar 13 2019: (Start)
a(26) = 64505245697, a(27) = 3959299, a(29) = 237828698392557762563228607177734374, a(31) = 26711406049549496732652187347412109374, a(32) = 441559, a(34) = 12535291248641, a(36) = 352351, a(37) = 1749348542212388688829378224909305572509765626, a(38) = 193405731995647.
Conjecture: if p is an odd prime, then a(p) is even.
(End)

			a(16)=18089 because d(18088)=d(18090)=2*16.

Hugo van der Sanden, Table of n, a(n) for n = 1..40

Cf. A067888, A075036, A190355.

a(7), a(11), a(13), and a(15) from T. D. Noe, May 25 2011
a(17), a(19), a(21)-a(23) from Chai Wah Wu, Mar 13 2019
b-file extending to a(40) from Hugo van der Sanden, Mar 04 2022

A275418 Numbers n such that n - 1 has exactly as many odd divisors as n + 1.

Original entry on oeis.org

3, 4, 6, 11, 12, 13, 18, 21, 23, 25, 27, 30, 34, 39, 42, 45, 47, 56, 57, 60, 72, 75, 81, 86, 87, 92, 93, 94, 95, 99, 102, 105, 108, 109, 117, 123, 124, 131, 135, 138, 139, 142, 144, 147, 150, 155, 159, 160, 165, 169, 177, 180, 184, 186, 192, 193, 198, 202, 204, 207, 213, 214, 216

Offset: 1

Juri-Stepan Gerasimov, Jul 27 2016

Numbers n > 1 such that d(2n - 2) + d(n + 1) = d(2n + 2) + d(n - 1) where d = A000005.
Conjectures:
(1) There are only finitely many terms n such that A001227(n - 1) = A001227(n + 1) is odd: 3, 99, 577, 3363, ... (see A276188).
(2) There are only finitely many terms n such that A001227(n - 1) = A001227(n) = A001227(n + 1) = 2: 6, 11, 12, 13, 23, 47, 192, 193, 383, 786432, ... (see also A181490-A181493, A276136).
(3) There are only finitely many prime terms p such that A001227(p - 1) = A001227(p + 1) is prime: 11, 13, 23, 47, 193, 383, 577, ... (see also A275598).
I don't find any more for conjecture #3 up to 10^10. - Charles R Greathouse IV,  Aug 22 2016

			3 is in this sequence because 2 and 4 both have only one odd divisor, 1.
4 is in this sequence because 3 and 5 both have exactly two odd divisors each (1 and 3 for the former, 1 and 5 for the latter).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Supersequence of A014574.

Cf. A000005, A001227, A067888, A181490-A181493, A273401, A275598, A276136, A276188.

[n: n in [2..216] | NumberOfDivisors(2*(n-1))+ NumberOfDivisors(n+1) eq NumberOfDivisors(2*(n+1))+ NumberOfDivisors(n-1)];

N:= 1000: # to get all terms < N
nod:= proc(n) numtheory:-tau(n/2^padic:-ordp(n,2)) end proc:
X:= map(nod,[$1..N]):
select(t -> X[t+1]=X[t-1], [$2..N-1]); # Robert Israel, Aug 04 2016

f[n_] := Count[Divisors@ n, k_ /; OddQ@ k]; Select[Range[2, 240], f[# - 1] == f[# + 1] &] (* Michael De Vlieger, Jul 28 2016 *)
Flatten[Position[Partition[Table[Count[Divisors[n],?OddQ],{n,300}],3,1],?(#[[1]]==#[[3]]&),{1},Heads->False]]+1 (* Harvey P. Dale, Nov 02 2016 *)

a001227(n) = sumdiv(n, d, d%2);
is(n) = a001227(n-1)==a001227(n+1) \\ Felix Fröhlich, Jul 27 2016

is(n)=numdiv((n-1)>>valuation(n-1,2)) == numdiv((n+1)>>valuation(n+1,2)) \\ Charles R Greathouse IV, Jul 29 2016

Name edited by Alonso del Arte, Aug 23 2016

A377319 a(n) is the smallest positive integer k such that n + k and n - k have the same number of divisors.

Original entry on oeis.org

1, 2, 1, 1, 2, 1, 3, 3, 1, 6, 3, 2, 3, 6, 1, 1, 3, 2, 9, 5, 2, 6, 3, 3, 6, 12, 1, 4, 6, 4, 1, 5, 2, 2, 6, 2, 3, 1, 1, 8, 3, 2, 11, 3, 4, 7, 3, 1, 6, 2, 3, 1, 1, 4, 7, 9, 1, 4, 7, 4, 3, 6, 5, 2, 2, 2, 3, 6, 1, 4, 4, 4, 3, 6, 4, 9, 6, 2, 5, 5, 2, 8, 1, 3, 3, 2, 3

Offset: 4

Felix Huber, Nov 17 2024

nonn

If the strong Goldbach conjecture is true, that every even number >= 8 is the sum of two distinct primes, then a positive integer k <= A082467(n) exists for n >= 4.

			a(8) = 2 because 10 and 6 have both four divisors. 9 and 7 have a different number of divisors.

Felix Huber, Table of n, a(n) for n = 4..10000

Cf. A000005, A067888 , A082467, A171937, A377320, A377321.

A377319:=proc(n)
   local k;
   for k to n-1 do
      if NumberTheory:-tau(n+k)=NumberTheory:-tau(n-k) then
         return k
      fi
   od;
end proc;
seq(A377319(n),n=4..90);

A377319[n_] := Module[{k = 0}, While[DivisorSigma[0, ++k + n] != DivisorSigma[0, n - k]]; k];
Array[A377319, 100, 4] (* Paolo Xausa, Dec 03 2024 *)

a(n) = my(k=1); while (numdiv(n+k) != numdiv(n-k), k++); k; \\ Michel Marcus, Nov 17 2024

1 <= a(n) <= A082467(n).

A260256 Numbers n such that tau(n + 2) = tau(n - 2) where tau(k) = A000005(k).

Original entry on oeis.org

5, 8, 9, 12, 15, 21, 24, 30, 36, 37, 39, 45, 53, 60, 67, 68, 69, 81, 84, 89, 93, 99, 105, 111, 112, 113, 117, 120, 121, 127, 129, 131, 143, 144, 157, 158, 165, 172, 173, 184, 185, 188, 195, 202, 203, 204, 207, 211, 215, 216, 217, 219, 222, 225, 226, 231, 248, 251, 276, 277, 279, 284, 288

Offset: 1

Juri-Stepan Gerasimov, Jul 21 2015

nonn

Pinner proves that this sequence is infinite, and in particular a(n) << n (log n)^7. The correct order is conjectured to be around n sqrt(log n). - Charles R Greathouse IV, Jul 21 2015

			8 is a member as 10 and 6 both have 4 divisors.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Christopher G. Pinner, Repeated values of the divisor function, Quart. J. Math. Oxford Ser. (2) 48:192 (1997), pp. 499-502.

Cf. A067888, A067889.

[ n : n in [3..300] | Denominator((NumberOfDivisors(n-2))/(NumberOfDivisors(n+2))) eq 1 and Denominator((NumberOfDivisors(n+2))/(NumberOfDivisors(n-2))) eq 1];

Select[ Range@ 290, DivisorSigma[0, # - 2] == DivisorSigma[0, # + 2] &] (* Robert G. Wilson v, Jul 21 2015 *)

is(n)=n>4&&numdiv(n-2)==numdiv(n+2) \\ Charles R Greathouse IV, Jul 21 2015

A000005(a(n) + 2) = A000005(a(n) - 2).

A192220 Semiprimes s such that tau(s-1) = tau(s+1) where tau = A000005.

Original entry on oeis.org

4, 6, 9, 34, 51, 55, 86, 94, 129, 142, 183, 185, 202, 214, 218, 247, 249, 302, 341, 394, 415, 446, 471, 473, 535, 583, 634, 698, 723, 737, 807, 851, 905, 922, 926, 949, 1042, 1138, 1149, 1205, 1211, 1241, 1257, 1262, 1313, 1315, 1337, 1346, 1402, 1527, 1546, 1577, 1594, 1642, 1646, 1673, 1687

Offset: 1

Juri-Stepan Gerasimov, Jun 26 2011

nonn

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

Cf. A000005, A001358, A067888, A067889.

Select[Range[2000], PrimeOmega[#] == 2 && Equal @@ DivisorSigma[0, # + {-1, 1}] &] (* Amiram Eldar, Jan 23 2025 *)

isok(k) = bigomega(k) == 2 && numdiv(k-1) == numdiv(k+1); \\ Amiram Eldar, Jan 23 2025

A001358 INTERSECT A067888.

A294173 Numbers k whose nearest neighbors have the same number of divisors, the same number of distinct prime factors, and the same sum of divisors.

Original entry on oeis.org

34, 55, 919, 1241, 4149, 4188, 7170, 12566, 15086, 24882, 25020, 26610, 51836, 53964, 59988, 77058, 143370, 150420, 167561, 170562, 205728, 215070, 220818, 418308, 564858, 731321, 907255, 910316, 986154, 1239870, 1569336, 1622914, 1841861, 1887240, 1979307, 2229012, 2262108

Offset: 1

Torlach Rush, Feb 10 2018

nonn

mu(k-1) = mu(k+1), where mu(k) = A008683(k), since k-1 and k+1 have the same number of distinct prime factors.
tau(k-1) = tau(k+1) = abs(phi(k-1) - phi(k+1)) iff abs(phi(k-1) - phi(k+1)) = 4, where phi(j) is A000010. When tau(j) = 4 omega(j) = 2 and phi(j), the product of two even numbers is divisible by 4.
For known elements:
- sigma(k +- 1) and tau(k +- 1) the greatest common divisor is 4.
- sigma(k +- 1) is divisible by tau(k +- 1).
- the digital root of sigma(k +- 1) is either 3 or 9.
- the prime signature of k +- 1 is the same (see question below).
The first prime terms are 919, 110495719, 2587274227, 3908452759, 4020447619, and 9314901619. - Giovanni Resta, Feb 12 2018
Are the prime signatures of k +- 1 always the same? - Andrey Zabolotskiy, Feb 14 2018

			34 is in the sequence because tau(33)=tau(35)=4, omega(33)=omega(35)=2, and sigma(33)=sigma(35)=48.
919 is in the sequence because tau(918)=tau(920)=16, omega(918)=omega(920)=3, and sigma(918)=sigma(920)=2160.

Charles R Greathouse IV, Table of n, a(n) for n = 1..1478

Intersection of A067888, A088070, and A055574.

Cf. A229254, A008683, A000010.

Filtered([2..2000000],k->Sigma(k-1)=Sigma(k+1) and Number(FactorsInt(k-1))=Number(FactorsInt(k+1)) and Tau(k-1)=Tau(k+1)); # Muniru A Asiru, Feb 17 2018

with(numtheory):
select(k->sigma(k-1)=sigma(k+1) and mobius(k-1)=mobius(k+1) and tau(k-1)=tau(k+1), [$2..2000000]); # Muniru A Asiru, Feb 17 2018

1 + Position[Partition[Array[{DivisorSigma[0, #], DivisorSigma[1, #], PrimeOmega[#]} &, 10^6], 3, 1], ?(#[[1]] == +#[[-1]] &), {1}, Heads -> False][[All, 1]] (* _Michael De Vlieger, Feb 17 2018 *)

list(lim)=my(v=List(),k2=7,s2=sigma(k2),k1=8,s1=sigma(k1),s); forfactored(k=9,1+lim\1, s=sigma(k); if(s==s2 && numdiv(k)==numdiv(k2) && omega(k)==omega(k2), listput(v,k1[1])); k2=k1; k1=k; s2=s1; s1=s); Vec(v) \\ Charles R Greathouse IV, Feb 20 2018

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

A384195 a(n) = tau(n+1) - tau(n-1), where tau(n) = A000005(n), the number of divisors of n.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, -1, 2, 0, -2, 2, 1, -2, 1, 0, 0, 2, -2, -2, 4, 1, -4, 1, 2, -2, 2, 0, -2, 2, -2, 0, 5, -2, -5, 2, 4, -2, 0, 0, -2, 4, -2, -4, 6, 1, -4, 1, 0, -2, 2, 2, 0, 0, -4, -2, 8, 0, -8, 4, 3, -2, 1, -2, -2, 2, 2, -2, 4, 0, -8, 4, 2, -2, 2, -2, 2, 3, -6, -3, 8, 2, -8

Offset: 2

Dan Dart, May 21 2025

sign

Conjecture: a(n)=0 for an infinite number of values.

			a(10) = tau(11) - tau(9) = 2 - 3 = -1.

Cf. A000005, A051950, A051951, A067888.

divides ∷ (Integral a) ⇒ a → a → Bool
divides a b = b `mod` a == 0
properFactors ∷ Integral a ⇒ a → [a]
properFactors n = 1 : filter (`divides` n) [2..(n `div` 2)]
factors ∷ Integral a ⇒ a → [a]
factors 1 = [1]
factors n = properFactors n <> [n]
tauSuccMinusTauPred :: Integer -> Integer
tauSuccMinusTauPred n = fromIntegral (length (factors (n + 1))) - fromIntegral (length (factors (n - 1)))

a(n) = numdiv(n+1) - numdiv(n-1); \\ Michel Marcus, May 21 2025

a(n) = tau(n+1) - tau(n-1).

cp's OEIS Frontend

A067889 Primes sandwiched between two numbers having same number of divisors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A062832 Numbers k such that k and k+2 have the same number of divisors.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A190646 Least number k such that d(k-1) = d(k+1) = 2n or 0 if no such k exists, where d(n)=A000005(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A275418 Numbers n such that n - 1 has exactly as many odd divisors as n + 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Extensions

A377319 a(n) is the smallest positive integer k such that n + k and n - k have the same number of divisors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A260256 Numbers n such that tau(n + 2) = tau(n - 2) where tau(k) = A000005(k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A192220 Semiprimes s such that tau(s-1) = tau(s+1) where tau = A000005.

Views

Author

Keywords

Links

Crossrefs