A008864 -id:A008864 - cp's OEIS frontend

A008329 Number of divisors of p+1, p prime.

2, 3, 4, 4, 6, 4, 6, 6, 8, 8, 6, 4, 8, 6, 10, 8, 12, 4, 6, 12, 4, 10, 12, 12, 6, 8, 8, 12, 8, 8, 8, 12, 8, 12, 12, 8, 4, 6, 16, 8, 18, 8, 14, 4, 12, 12, 6, 12, 12, 8, 12, 20, 6, 18, 8, 16, 16, 10, 4, 8, 6, 12, 12, 16, 4, 8, 6, 6, 12, 12, 8, 24, 10, 8, 12, 16, 16, 4, 8, 8, 24, 4, 20, 8, 16

Offset: 1

N. J. A. Sloane

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

Cf. A000005, A008864.

[NumberOfDivisors(NthPrime(n)+1): n in [1..100]]; // Vincenzo Librandi, Mar 25 2018

for i from 1 to 500 do if isprime(i) then print(tau(i+1)); fi; od;
A008329 := proc(n)
        numtheory[tau](ithprime(n)+1) ;
end proc: # R. J. Mathar, Oct 30 2015

DivisorSigma[0,#]&/@(Prime[Range[100]]+1)  (* Harvey P. Dale, Apr 12 2011 *)

a(n) = numdiv(prime(n)+1); \\ Michel Marcus, Mar 25 2018

a(n) = A000005(A008864(n)). - Sean A. Irvine, Mar 25 2018

Offset corrected by Leroy Quet, Oct 08 2008

A084921 a(n) = lcm(p-1, p+1) where p is the n-th prime.

Original entry on oeis.org

3, 4, 12, 24, 60, 84, 144, 180, 264, 420, 480, 684, 840, 924, 1104, 1404, 1740, 1860, 2244, 2520, 2664, 3120, 3444, 3960, 4704, 5100, 5304, 5724, 5940, 6384, 8064, 8580, 9384, 9660, 11100, 11400, 12324, 13284, 13944, 14964, 16020, 16380

Offset: 1

Reinhard Zumkeller, Jun 11 2003

nonn

This sequence consists of terms of sequences A055523 and A055527 for prime n > 2. - Toni Lassila (tlassila(AT)cc.hut.fi), Feb 02 2004

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Christian Krause, LODA program for A084921
Index entries for sequences related to lcm's

Cf. A000040, A006093, A008864, A009286, A024700, A055523, A055527, A084920, A084922.

a084921 n = lcm (p - 1) (p + 1)  where p = a000040 n
-- Reinhard Zumkeller, Jun 01 2013

[3] cat [(p^2-1)/2: p in PrimesInInterval(3,300)]; // G. C. Greubel, May 03 2024

LCM[#-1,#+1]&/@Prime[Range[50]] (* Harvey P. Dale, Oct 09 2018 *)

a(n)=if(n<2,3,(prime(n)^2-1)/2) \\ Charles R Greathouse IV, May 15 2013

[3]+[(n^2-1)/2 for n in prime_range(3,301)] # G. C. Greubel, May 03 2024

a(n) = A084920(n)/2 for n > 1.
a(n) = 3*A084922(n) for n > 2.
a(n) = A009286(A000040(n)). - Enrique Pérez Herrero, May 17 2012
a(n) ~ 0.5 n^2 log^2 n. - Charles R Greathouse IV, May 15 2013
Product_{n>=1} (1 + 1/a(n)) = 2. - Amiram Eldar, Jan 23 2021
a(n) = (A000040(n)^2 - 1) / 2 for n > 1. - Christian Krause, Mar 27 2021
a(n) = (3/2)*A024700(n-2), for n > 1. - G. C. Greubel, May 03 2024

A377703 First differences of the sequence A345531(k) = least prime-power greater than the k-th prime.

Original entry on oeis.org

1, 3, 1, 5, 3, 3, 4, 2, 6, 1, 9, 2, 4, 2, 10, 2, 3, 7, 2, 6, 2, 8, 8, 4, 2, 4, 2, 4, 8, 7, 9, 2, 10, 2, 6, 6, 4, 2, 10, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 2, 13, 7, 6, 2, 6, 4, 2, 6, 18, 4, 2, 4, 14, 6, 6, 6, 4, 6, 2, 12, 6, 4, 6, 8, 4, 8, 10, 2, 10, 2, 6

Offset: 1

Gus Wiseman, Nov 07 2024

nonn

What is the union of this sequence? In particular, does it contain 17?

First differences of A345531.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect-powers, differences A053289, seconds A376559.
A007916 lists the non-perfect-powers, differences A375706, seconds A376562.
A024619 lists the non-prime-powers, differences A375735, seconds A376599.
A080101 counts prime-powers between primes (exclusive).
A246655 lists the prime-powers, differences A057820 without first term.
A361102 lists the non-powers of primes, differences A375708.
A366833 counts prime-powers between primes, see A053607, A304521, A377057 (positive), A377286 (zero), A377287 (one), A377288 (two).
A377432 counts perfect-powers between primes, see A377434 (one), A377436 (zero), A377466 (multiple).
Cf. A000040, A008864, A031218, A377281, A377282, A377286, A377289, A377468.

Differences[Table[NestWhile[#+1&, Prime[n]+1,!PrimePowerQ[#]&],{n,100}]]

from sympy import factorint, prime, nextprime
def A377703(n): return -next(filter(lambda m:len(factorint(m))<=1, count((p:=prime(n))+1)))+next(filter(lambda m:len(factorint(m))<=1, count(nextprime(p)+1))) # Chai Wah Wu, Nov 14 2024

A373821 Run-lengths of run-lengths of first differences of odd primes.

Original entry on oeis.org

1, 11, 1, 19, 1, 1, 1, 5, 1, 6, 1, 16, 1, 27, 1, 3, 1, 1, 1, 6, 1, 9, 1, 29, 1, 2, 1, 18, 1, 1, 1, 5, 1, 3, 1, 17, 1, 19, 1, 30, 1, 17, 1, 46, 1, 17, 1, 27, 1, 30, 1, 5, 1, 36, 1, 41, 1, 10, 1, 31, 1, 44, 1, 4, 1, 14, 1, 6, 1, 2, 1, 32, 1, 13, 1, 17, 1, 5

Offset: 1

Gus Wiseman, Jun 22 2024

nonn

Run-lengths of A333254.
The first term other than 1 at an odd positions is at a(101) = 2.
Also run-lengths (differing by 0) of run-lengths (differing by 0) of run-lengths (differing by 1) of composite numbers.

			The odd primes are:
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, ...
with first differences:
2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, ...
with run-lengths:
2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, ...
with run-lengths a(n).

Run-lengths of run-lengths of A046933(n) = A001223(n) - 1.
Run-lengths of A333254.
A000040 lists the primes.
A001223 gives differences of consecutive primes.
A027833 gives antirun lengths of odd primes (partial sums A029707).
A065855 counts composite numbers up to n.
A071148 gives partial sums of odd primes.
A373820 gives run-lengths of antirun-lengths of odd primes.
For prime runs: A001359, A006512, A025584, A067774, A373406.
For composite runs: A005381, A008864, A054265, A176246, A251092, A373403.
Cf. A006560, A037201, A038664, A069010, A373824, A373825.

Length/@Split[Length /@ Split[Differences[Select[Range[3,1000],PrimeQ]]]//Most]//Most

A377288 Numbers k such that there are exactly two prime-powers between prime(k)+1 and prime(k+1)-1.

Original entry on oeis.org

4, 9, 30, 327, 3512

Offset: 1

Gus Wiseman, Oct 25 2024

Is this sequence finite? For this conjecture see A053706, A080101, A366833.

Any further terms are > 10^12. - Lucas A. Brown, Nov 08 2024

			Primes 9 and 10 are 23 and 29, and the interval (24, 25, 26, 27, 28) contains the prime-powers 25 and 27, so 9 is in the sequence.

Lucas A. Brown, Python program.

The interval from A008864(n) to A006093(n+1) has A046933 elements.
For powers of 2 instead of primes see A013597, A014210, A014234, A244508, A304521.
The corresponding primes are A053706.
The nearest prime-power before prime(n)-1 is A065514, difference A377289.
The nearest prime-power after prime(n)+1 is A345531, difference A377281.
These are the positions of 2 in A080101, or 3 in A366833.
For at least one prime-power we have A377057, primes A053607.
For no prime-powers we have A377286.
For exactly one prime-power we have A377287.
For squarefree instead of prime-power see A377430, A061398, A377431, A068360.
A000015 gives the least prime-power >= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A031218 gives the greatest prime-power <= n.
A246655 lists the prime-powers not including 1, complement A361102.
Cf. A001597, A002808, A024619, A053707, A064113, A065890, A075526, A095195, A224363, A276781, A376596, A376597, A377282.

Select[Range[100], Length[Select[Range[Prime[#]+1,Prime[#+1]-1],PrimePowerQ]]==2&]

prime(a(n)) = A053706(n).

A023509 Greatest prime divisor of prime(n) + 1.

Original entry on oeis.org

3, 2, 3, 2, 3, 7, 3, 5, 3, 5, 2, 19, 7, 11, 3, 3, 5, 31, 17, 3, 37, 5, 7, 5, 7, 17, 13, 3, 11, 19, 2, 11, 23, 7, 5, 19, 79, 41, 7, 29, 5, 13, 3, 97, 11, 5, 53, 7, 19, 23, 13, 5, 11, 7, 43, 11, 5, 17, 139, 47, 71, 7, 11, 13, 157, 53, 83, 13, 29, 7, 59, 5, 23, 17, 19, 3, 13

Offset: 1

Clark Kimberling

nonn

A005382 is the records subsequence of this sequence. - David James Sycamore, May 05 2025

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

Cf. A006530, A008864, A005382.

Table[FactorInteger[Prime[n] + 1][[-1, 1]], {n, 100}]

A023509(n) = {local(f);f=factor(prime(n)+1);f[matsize(f)[1],1]} \\ Michael B. Porter, Feb 02 2010

a(n) = A006530(A008864(n)). - R. J. Mathar, Feb 06 2019

A028815 a(n) = prime(n) + 1 (starting with 1).

Original entry on oeis.org

2, 3, 4, 6, 8, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 54, 60, 62, 68, 72, 74, 80, 84, 90, 98, 102, 104, 108, 110, 114, 128, 132, 138, 140, 150, 152, 158, 164, 168, 174, 180, 182, 192, 194, 198, 200, 212, 224, 228, 230, 234, 240, 242, 252, 258, 264, 270, 272, 278, 282, 284

Offset: 0

N. J. A. Sloane

n-th noncomposite (unit or prime) positive integer + 1.

The "0th prime" is defined to be 1 (a unit, formerly considered to be prime).

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A000040, A006093, A008578, A008864.

Cf. A141130, A141132, A141133, A141136, A141138.

A028815:= func< n | n eq 0 select 2 else 1 + NthPrime(n) >;
[A028815(n): n in [0..70]]; // G. C. Greubel, Aug 05 2024

Join[{2},Prime[Range[70]]+1] (* Harvey P. Dale, Oct 14 2019 *)

a(n) = if(n==0, 2, prime(n) + 1); \\ Altug Alkan, Mar 14 2018

def A028815(n): return 2 if n==0 else 1+nth_prime(n)
[A028815(n) for n in range(71)] # G. C. Greubel, Aug 05 2024

a(n) = prime(n) + 1 = A000040(n) + 1 = A008864(n) for n >= 1.
a(n) = A008578(n+1) + 1, n >= 0.
a(n) = 2*A006254(n-1), for n >= 2, with a(0) = 2, a(1) = 3. - G. C. Greubel, Aug 05 2024

A113935 a(n) = prime(n) + 3.

Original entry on oeis.org

5, 6, 8, 10, 14, 16, 20, 22, 26, 32, 34, 40, 44, 46, 50, 56, 62, 64, 70, 74, 76, 82, 86, 92, 100, 104, 106, 110, 112, 116, 130, 134, 140, 142, 152, 154, 160, 166, 170, 176, 182, 184, 194, 196, 200, 202, 214, 226, 230, 232, 236, 242, 244, 254, 260, 266, 272, 274

Offset: 1

Jorge Coveiro, Jan 30 2006

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A098090, A116366.

List([1..70], n-> Primes[n]+3); # G. C. Greubel, May 18 2019

[p+3: p in PrimesUpTo(500)]; // Vincenzo Librandi, Nov 27 2013

Prime[Range[70]]+3 (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)

PARI
```
for(x=1,100,print1(prime(x)+3,","))
```

[nth_prime(n)+3 for n in (1..70)] # G. C. Greubel, May 18 2019

a(n) = A116366(n-1,1) for n>1. - Reinhard Zumkeller, Feb 06 2006
a(n) = 2*A098090(n-1) for n > 1. - Reinhard Zumkeller, Sep 14 2006
a(n) = A000040(n) + 3 = A008864(n) + 2 = A052147(n) + 1 = A175221(n) - 1 = A175222(n) - 2 = A139049(n) - 3 = A175223(n) - 4 = A175224(n) - 5 = A140353(n) - 6 = A175225(n) - 7. - Jaroslav Krizek, Mar 06 2010

A239712 Primes of the form m = 2^i + 2^j - 1, where i > j >= 0.

Original entry on oeis.org

2, 5, 11, 17, 19, 23, 47, 67, 71, 79, 131, 191, 257, 263, 271, 383, 1031, 1039, 1087, 1151, 1279, 2063, 2111, 4099, 4111, 4127, 4159, 5119, 6143, 8447, 16447, 20479, 32771, 32783, 32831, 33023, 33791, 65537, 65539, 65543, 65551, 65599, 66047, 73727, 81919, 262147, 262151, 262271, 262399, 263167

Offset: 1

Hieronymus Fischer, Mar 28 2014 and Apr 22 2014

nonn

Numbers m such that b = 2 is the only base such that the base-b digital sum of m + 1 is equal to b.
Example: 5 + 1 = 110_2 which implies ds_2(5 + 1) = 2 = b, where ds_b = digital sum in base-b. However, ds_3(6) = 2 <> 3, ds_4(6) = 3 <> 4, ds_5(6) = 2 <> 5, ds_6(6) = 1 <> 6. For all other bases > 6 we have ds_b(6) = 6 <> b. It follows that b = 2 is the only such base.
The base-2 representation of a term 2^i + 2^j - 1 has a base-2 digital sum of 1 + j.
In base-2 representation the first terms are 10, 101, 1011, 10001, 10011, 10111, 101111, 1000011, 1000111, 1001111, 10000011, 10111111, 100000001, 100000111, 100001111, 101111111, 10000000111, 10000001111, 10000111111, 10001111111, ...
Numbers m = 2^i + 2^j - 1 with odd i and j are not terms. Example: 10239 = 2^13 + 2^11 - 1 is not a prime.

			a(1) = 2, since 2 = 2^1 + 2^0 - 1 is prime.
a(5) = 19, since 19 = 2^4 + 2^2 - 1 is prime.

Hieronymus Fischer, Table of n, a(n) for n = 1..250

Cf. A007953, A018900, A081091, A008864, A187813.

Cf. A239703, A239708, A239709, A239713 - A239720.

Select[Union[Total/@(2^#&/@Subsets[Range[0,20],{2}])-1],PrimeQ] (* Harvey P. Dale, Aug 08 2014 *)

A239712
"Answers the n-th term of A239712.
  Usage: n A239712
  Answer: a(n)"
  | a b i k m p q terms |
  terms := OrderedCollection new.
  b := 2.
  p := 1.
  k := 0.
  m := 0.
  [k < self] whileTrue:
         [m := m + 1.
         p := b * p.
         q := 1.
         i := 0.
         [i < m and: [k < self]] whileTrue:
                   [i := i + 1.
                   a := p + q - 1.
                   a isPrime
                        ifTrue:
                            [k := k + 1.
                            terms add: a].
                   q := b * q]].
  ^terms at: self
[by Hieronymus Fischer, Apr 22 2014]
-----------

floorPrimesWhichAreDistinctPowersOf: b withOffset: d
  "Answers an array which holds the primes < n that obey b^i + b^j + d, i>j>=0,
  where n is the receiver. b > 1 (here: b = 2, d = -1).
  Uses floorDistinctPowersOf: from A018900
  Usage:
  n floorPrimesWhichAreDistinctPowersOf: b withOffset: d
  Answer: #(2 5 11 17 19 23 ...) [terms < n]"
  ^((self - d floorDistinctPowersOf: b)
  collect: [:i | i + d]) select: [:i | i isPrime]
[by Hieronymus Fischer, Apr 22 2014]
------------

primesWhichAreDistinctPowersOf: b withOffset: d
  "Answers an array which holds the n primes of the form b^i + b^j + d, i>j>=0, where n is the receiver.
  Direct calculation by scanning b^i + b^j + d in increasing order and selecting terms which are prime.
  b > 1; this sequence: b = 2, d = 1.
  Usage:
  n primesWhichAreDistinctPowersOf: b withOffset: d
  Answer: #(2 5 11 17 19 23 ...) [a(1) ... a(n)]"
  | a k p q terms n |
  terms := OrderedCollection new.
  n := self.
  k := 0.
  p := b.
  [k < n] whileTrue:
         [q := 1.
         [q < p and: [k < n]] whileTrue:
                   [a := p + q + d.
                   a isPrime
                        ifTrue:
                            [k := k + 1.
                            terms add: a].
                   q := b * q].
         p := b * p].
  ^terms asArray
[by Hieronymus Fischer, Apr 22 2014]

a(n) = A239708(n) - 1.

a(n+1) = min(A018900(k) > a(n)| A018900(k) - 1 is prime, k >= 1) - 1.

Examples moved from Maple field to Examples field by Harvey P. Dale, Aug 08 2014

A055670 a(n) = prime(n) - (-1)^prime(n).

Original entry on oeis.org

1, 4, 6, 8, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 54, 60, 62, 68, 72, 74, 80, 84, 90, 98, 102, 104, 108, 110, 114, 128, 132, 138, 140, 150, 152, 158, 164, 168, 174, 180, 182, 192, 194, 198, 200, 212, 224, 228, 230, 234, 240, 242, 252, 258, 264, 270, 272, 278, 282, 284

Offset: 1

N. J. A. Sloane, Jun 09 2000

Number of right-inequivalent prime Hurwitz quaternions of norm p, where p = n-th rational prime (indexed by A000040).
Two primes are considered right-equivalent if they differ by right multiplication by one of the 24 units. - N. J. A. Sloane
Start of n-th run of consecutive nonprime numbers. Since 2 is the only even prime, for all other prime numbers the expression "- (-1)^(n-th prime)" works out to "+ 1." - Alonso del Arte, Oct 18 2011

			a(1) = 2 - (-1)^2 = 1, a(2) = 3 - (-1)^3 = 4.

L. E. Dickson, Algebras and Their Arithmetics, Dover, 1960, Section 91.
Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology, Dover, New York, 1978, page 134.

Shawn A. Broyles, Table of n, a(n) for n = 1..20000

Cf. A000040, A006093.
Cf. A055669-A055672.
a(n) = A083503(p) for n>1.

Join[{1},Prime[Range[2,70]]+1] (* Harvey P. Dale, Oct 29 2013 *)

a(n) = prime(n)+1 = A008864(n) for n >= 2. a(n) = A055669(n)/24.

More terms from David W. Wilson, May 02 2001
I would also like to get the sequences of inequivalent prime Hurwitz quaternions, where two primes are considered equivalent if they differ by left or right multiplication by one of the 24 units. This will give two more sequences, analogs of A055670 and A055672.
Edited by N. J. A. Sloane, Aug 16 2009

cp's OEIS Frontend

A008329 Number of divisors of p+1, p prime.

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A084921 a(n) = lcm(p-1, p+1) where p is the n-th prime.

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A377703 First differences of the sequence A345531(k) = least prime-power greater than the k-th prime.

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A373821 Run-lengths of run-lengths of first differences of odd primes.

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A377288 Numbers k such that there are exactly two prime-powers between prime(k)+1 and prime(k+1)-1.

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A023509 Greatest prime divisor of prime(n) + 1.

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A028815 a(n) = prime(n) + 1 (starting with 1).

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