A275598 -id:A275598 - cp's OEIS frontend

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

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

A284037 Primes p such that p-1 and p+1 have two distinct prime factors.

Original entry on oeis.org

11, 13, 19, 23, 37, 47, 53, 73, 97, 107, 163, 193, 383, 487, 577, 863, 1153, 2593, 2917, 4373, 8747, 995327, 1492993, 1990657, 5308417, 28311553, 86093443, 6879707137, 1761205026817, 2348273369087, 5566277615617, 7421703487487, 21422803359743, 79164837199873

Offset: 1

Giuseppe Coppoletta, Mar 28 2017

nonn

Either p-1 or p+1 must be of the form 2^i * 3^j, since among three consecutive numbers exactly one is a multiple of 3. - Giovanni Resta, Mar 29 2017

Subsequence of A219528. See the previous comment. - Jason Yuen, Mar 08 2025

			7 is not a term because n + 1 = 8 has only one prime factor.
23 is a term because it is prime and n - 1 = 22 has two distinct prime factors (2, 11) and n + 1 = 24 has two distinct prime factors (2, 3).
43 is not a term because n - 1 = 42 has three distinct prime factors (2, 3, 7).

Robert Israel, Table of n, a(n) for n = 1..51 (all terms < 10^75)

Cf. A000668, A001221, A005105, A005109, A033845, A058383, A067386, A092506, A215504, A219528, A275598.

N:= 10^20: # To get all terms <= N
Res:= {}:
for i from 1 to ilog2(N) do
  for j from 1 to floor(log[3](N/2^i)) do
    q:= 2^i*3^j;
    if isprime(q-1) and nops(numtheory:-factorset((q-2)/2^padic:-ordp(q-2,2)))=1 then Res:= Res union {q-1} fi;
    if isprime(q+1) and nops(numtheory:-factorset((q+2)/2^padic:-ordp(q+2,2)))=1 then Res:= Res union {q+1} fi
od od:
sort(convert(Res,list)); # Robert Israel, Apr 16 2017

mx = 10^30; ok[t_] := PrimeQ[t] && PrimeNu[t-1]==2==PrimeNu[t+1]; Sort@ Reap[Do[ w = 2^i 3^j; Sow /@ Select[ w+ {1,-1}, ok], {i, Log2@ mx}, {j, 1, Log[3, mx/2^i]}]][[2, 1]] (* terms up to mx, Giovanni Resta, Mar 29 2017 *)

isok(n) = isprime(n) && (omega(n-1)==2) && (omega(n+1)==2); \\ Michel Marcus, Apr 17 2017

omega=sloane.A001221; [n for n in prime_range(10^6) if 2==omega(n-1)==omega(n+1)]

sorted([2^i*3^j+k for i in (1..40) for j in (1..20) for k in (-1,1) if is_prime(2^i*3^j+k) and sloane.A001221(2^i*3^j+2*k)==2])

A001221(a(n)) = 1 and A001221(a(n) - 1) = A001221(a(n) + 1) = 2.

a(33)-a(34) from Giovanni Resta, Mar 29 2017

A276188 Numbers k > 1 such that the number of odd divisors of k-1 is odd and is equal to the number of odd divisors of k+1.

Original entry on oeis.org

3, 99, 577, 3363

Offset: 1

Juri-Stepan Gerasimov, Aug 23 2016

Conjecture: this sequence is finite.

Any further terms are greater than 10^10. - Charles R Greathouse IV, Aug 22 2016

			99 is in this sequence because there are 3 odd divisors 1, 7 and 49 of 98 and there are 3 odd divisors 1, 5 and 25 of 100, and 3 is odd.

Cf. A275418, A275598, A276136.

[n: n in [2..100000] | NumberOfDivisors(2*(n-1))- NumberOfDivisors(n-1) eq NumberOfDivisors(2*(n+1))-NumberOfDivisors(n+1) and ((NumberOfDivisors(2*(n+1))- NumberOfDivisors(n+1)) mod 2) eq 1 ];

odo[n_]:=Module[{c=Select[Divisors[n],OddQ]},If[OddQ[Length[c]],Length[c],0]]; Flatten[ Position[ Partition[Array[odo,3500],3,1],?(AllTrue[{#[[1]],#[[3]]},OddQ]&&#[[1]]==#[[3]]&),1,Heads->False]]+1 (* _Harvey P. Dale, Apr 07 2023 *)

A369615 Powers of primes (A000961) whose neighbors have a prime number as their greatest odd divisor.

Original entry on oeis.org

4, 11, 13, 23, 25, 27, 47, 81, 193, 383, 2187, 1594323

Offset: 1

Juri-Stepan Gerasimov, Jan 27 2024

From Jon E. Schoenfield, Jan 28 2024: (Start)
If it exists, a(13) > 10^2000.
Conjecture: a(12) = 1594323 is the final term of the sequence.
(End)

			(prime = greatest odd divisor of a(n)-1; a(n); prime = greatest odd divisor of a(n)+1): (3; 4; 5), (5; 11; 3), (3; 13; 7), (11; 23; 3), (3; 25; 13), (13; 27; 7), (23; 47; 3), (5; 81; 41), (3; 193; 97), (191; 383; 3), (1093; 2187; 547), (797161; 1594323; 398581).

Intersection of A000961 and A369329.
Cf. A038550.
Comparable sequences: A275598, A343973.

[k: k in [2..1600000] | #Divisors(2*k-2)-#Divisors(k-1) eq 2 and
 #PrimeDivisors(k) eq 1 and #Divisors(2*k+2)-#Divisors(k+1) eq 2];

q[n_] := PrimeQ[n/2^IntegerExponent[n, 2]]; Select[Range[2*10^6], PrimePowerQ[#] && And @@ q /@ {# - 1, # + 1} &] (* Amiram Eldar, Jan 28 2024 *)

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

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A284037 Primes p such that p-1 and p+1 have two distinct prime factors.

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A276188 Numbers k > 1 such that the number of odd divisors of k-1 is odd and is equal to the number of odd divisors of k+1.

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A369615 Powers of primes (A000961) whose neighbors have a prime number as their greatest odd divisor.

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Magma

Mathematica