A067889 -id:A067889 - 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

A067891 Primes p such that sigma(p+1) = sigma(p-1).

Original entry on oeis.org

367, 919, 30593, 95393, 117571, 124759, 147341, 197261, 334541, 344417, 463219, 732257, 755081, 931757, 982759, 1996759, 2401219, 2962697, 3013447, 4722941, 7892827, 13333097, 13358407, 17946259, 19828483, 19855471, 19904981

Offset: 1

Benoit Cloitre, Mar 02 2002

Giovanni Resta, Table of n, a(n) for n = 1..769 (terms < 10^12)

Cf. A067889 (analog with tau).

fQ[p_] := DivisorSigma[1, p - 1] == DivisorSigma[1, p + 1]; p = 2; lst = {}; While[p < 100000000, If[fQ@ p, AppendTo[lst, p]]; p = NextPrime@ p]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 03 2009 and modified by Robert G. Wilson v, May 08 2016 *)

for(n=2,10^8,if(isprime(n) && sigma(n+1)==sigma(n-1),print1(n,","))) \\ It is more efficient to use forprime(...).

is_A067891(p)=sigma(p-1)==sigma(p+1)&&isprime(p) \\ M. F. Hasler, Jul 31 2015

More terms from Rick L. Shepherd, Apr 19 2002

A171667 Lesser of a pair of twin primes (p,p+2) sandwiched between two numbers (p-1,p+3) having the same number of divisors.

Original entry on oeis.org

11, 29, 59, 431, 599, 827, 1031, 1319, 1619, 1787, 2111, 2141, 2267, 2687, 2711, 3299, 3329, 3371, 3527, 3671, 4001, 4091, 4229, 4259, 5021, 5099, 5519, 5867, 6299, 6659, 6779, 7331, 7457, 8087, 8231, 8387, 8627, 8861, 8999, 9419, 9461, 9767, 10139

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 14 2009

nonn

			First term 11: 10={1,2,5,10},14={1,2,7,14} Second term 29: 28={1,2,4,7,14,28},32={1,2,4,8,16,32}

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

Cf. A005237, A067889

f[n_]:=Length[Divisors[n]]; lst={};Do[p=Prime[n];If[PrimeQ[p+2]&&f[p-1]==f[p+3],AppendTo[lst,p]],{n,7!}];lst
Select[Range[11000],DivisorSigma[0,#-1]==DivisorSigma[0,#+3]&&AllTrue[{#, #+2},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 04 2019 *)

forprime(p=o=0,1e4,(2+o==o=p)&&numdiv(p-3)==numdiv(p+1)&&print1(p-2",")) \\ M. F. Hasler, Jul 31 2015

Name edited by M. F. Hasler, Jul 31 2015

A074460 Primes which are sandwiched between two numbers having the same unordered canonical form.

Original entry on oeis.org

19, 307, 349, 491, 739, 919, 1013, 1061, 1277, 1667, 1747, 2357, 2683, 3259, 3581, 3797, 3943, 4013, 4597, 4877, 4987, 5051, 5741, 6067, 7757, 9349, 9413, 9739, 9851, 9923, 9949, 10133, 10243, 10949, 11093, 11149, 12619, 12941, 12979, 13879, 14051

Offset: 1

Robert G. Wilson v, Aug 22 2002

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

Cf. A074497, A074498, A061715.

Subsequence of A067889.

f[n_] := Flatten[Table[{ # [[2]]}] & /@ FactorInteger[n]]; Prime[ Select[ Range[1700], Sort[ f[ Prime[ # ] - 1]] == Sort[ f[ Prime[ # ] + 1]] & ]]

A145337 a(n) = d(p(n)+1) - d(p(n)-1), where d(m) = the number of divisors of m, p(n) = the n-th prime.

Original entry on oeis.org

1, 1, 1, 0, 2, -2, 1, 0, 4, 2, -2, -5, 0, -2, 6, 2, 8, -8, -2, 4, -8, 2, 8, 4, -6, -1, 0, 8, -4, -2, -4, 4, 0, 4, 6, -4, -8, -4, 12, 2, 14, -10, 6, -10, 3, 0, -10, 4, 8, -4, 4, 12, -14, 10, -1, 12, 10, -6, -8, -8, -2, 6, 0, 8, -12, 2, -10, -14, 8, 0, -4, 20, 2, -4, -4, 12, 10, -14, -7, -8

Offset: 0

Leroy Quet, Oct 08 2008

sign

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A008328, A008329, A067889, A103664, A103665, A145338.

A000005 := proc(n) numtheory[tau](n) ; end:
A008328 := proc(n) A000005(ithprime(n)-1) ; end:
A008329 := proc(n) A000005(ithprime(n)+1) ; end:
A145337 := proc(n) A008329(n)-A008328(n) ; end: # R. J. Mathar, Oct 10 2008

DivisorSigma[0,#+1]-DivisorSigma[0,#-1]&/@Prime[Range[80]] (* Harvey P. Dale, Nov 01 2011 *)

a(n) = A008329(n) - A008328(n). - R. J. Mathar, Oct 10 2008

More terms from R. J. Mathar and Ray Chandler, Oct 10 2008

A145339 a(n) = the minimum of d(p(n)-1) and d(p(n)+1), where d(m) is the number of divisors of m and p(n) is the n-th prime.

Original entry on oeis.org

1, 2, 3, 4, 4, 4, 5, 6, 4, 6, 6, 4, 8, 6, 4, 6, 4, 4, 6, 8, 4, 8, 4, 8, 6, 8, 8, 4, 8, 8, 8, 8, 8, 8, 6, 8, 4, 6, 4, 6, 4, 8, 8, 4, 9, 12, 6, 8, 4, 8, 8, 8, 6, 8, 8, 4, 6, 10, 4, 8, 6, 6, 12, 8, 4, 6, 6, 6, 4, 12, 8, 4, 8, 8, 12, 4, 6, 4, 8, 8, 8, 4, 8, 8, 8, 8, 14, 4, 12, 10, 4, 4, 8, 12, 8, 4, 6, 12, 6, 4, 6

Offset: 1

Leroy Quet, Oct 08 2008

nonn

Cf. A145340, A008328, A008329, A067889, A103664, A103665.

Table[Min[DivisorSigma[0, Prime[n]-1], DivisorSigma[0, Prime[n]+1]], {n, 1, 100}] (* Stefan Steinerberger, Oct 11 2008 *)

a(n) = my(p = prime(n)); min(numdiv(p-1), numdiv(p+1)); \\ Michel Marcus, Sep 28 2018

More terms from Stefan Steinerberger and Ray Chandler, Oct 11 2008

A171668 Fibonacci numbers sandwiched between two numbers having same number of divisors.

Original entry on oeis.org

34, 55, 144, 10946, 46368, 196418, 9227465, 1134903170, 4052739537881, 117669030460994, 420196140727489673, 12200160415121876738, 3928413764606871165730, 22698374052006863956975682, 68330027629092351019822533679447, 13598018856492162040239554477268290

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 14 2009

nonn

Amiram Eldar, Table of n, a(n) for n = 1..70

Cf. A000005, A000045, A005237, A005238, A067889, A171666, A171667.

f[n_]:=Length[Divisors[n]]; lst={};Do[fi=Fibonacci[n];If[f[fi-1]==f[fi+1],AppendTo[lst,fi]],{n,170}];lst

a(15)-a(16) from Amiram Eldar, Aug 08 2024

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).

A103886 Rearrangement of prime numbers p according to number of divisors of p-/+1.

Original entry on oeis.org

2, 7, 13, 3, 19, 31, 5, 41, 37, 11, 103, 43, 17, 137, 61, 23, 199, 67, 29, 307, 73, 47, 349, 97, 53, 491, 101, 59, 739, 109

Offset: 1

Zak Seidov, Feb 20 2005

nonn

Number of divisors of p-1 (less than, equals and larger than) number of divisors of p+1. A103887 - position of p(n) in the sequence.

Cf. A067889, A103664, A103665, A103887.

Table[{A103664[[n]], A067889[[n]], A103665[[n]]}, {n, 10}]//Flatten

a(3k-2)=A103664(k); a(3k-1)=A067889(k); a(3k)=A103665(k); (k=1, 2, ...)

A103887 Position of p(n) in rearrangement of prime numbers A103886.

Original entry on oeis.org

1, 4, 7, 2, 10, 3, 13, 5, 16, 19, 6, 9, 8, 12, 22, 25, 28, 15, 18

Offset: 1

Zak Seidov, Feb 20 2005

nonn

Cf. A067889, A103664, A103665, A103886.

Table[Position[A103886, Prime[n]], {n, 19}]//Flatten

cp's OEIS Frontend

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

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A067891 Primes p such that sigma(p+1) = sigma(p-1).

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A171667 Lesser of a pair of twin primes (p,p+2) sandwiched between two numbers (p-1,p+3) having the same number of divisors.

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A074460 Primes which are sandwiched between two numbers having the same unordered canonical form.

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A145337 a(n) = d(p(n)+1) - d(p(n)-1), where d(m) = the number of divisors of m, p(n) = the n-th prime.

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A145339 a(n) = the minimum of d(p(n)-1) and d(p(n)+1), where d(m) is the number of divisors of m and p(n) is the n-th prime.

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A171668 Fibonacci numbers sandwiched between two numbers having same number of divisors.

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

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