A125266 -id:A125266

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A008864

a(n) = prime(n) + 1.

+10
179

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Sum of divisors of prime(n). - Labos Elemer, May 24 2001

For n > 1, there are a(n) more nonnegative Hurwitz quaternions than nonnegative Lipschitz quaternions, which are counted in A239396 and A239394, respectively. - T. D. Noe, Mar 31 2014

These are the numbers which are in A239708 or in A187813, but excluding the first 3 terms of A187813, i.e., a number m is a term if and only if m is a term > 2 of A187813, or m is the sum of two distinct powers of 2 such that m - 1 is prime. This means that a number m is a term if and only if m is a term > 2 such that there is no base b with a base-b digital sum of b, or b = 2 is the only base for which the base-b digital sum of m is b. a(6) is the only term such that a(n) = A187813(n); for n < 6, we have a(n) > A187813(n), and for n > 6, we have a(n) < A187813(n). - Hieronymus Fischer, Apr 10 2014

Does not contain any number of the format 1 + q + ... + q^e, q prime, e >= 2, i.e., no terms of A060800, A131991, A131992, A131993 etc. [Proof: that requires 1 + p = 1 + q + ... + q^e, or p = q*(1 + ... + q^(e-1)). This is not solvable because the left hand side is prime, the right hand side composite.] - R. J. Mathar, Mar 15 2018

1/a(n) is the asymptotic density of numbers whose prime(n)-adic valuation is odd. - Amiram Eldar, Jan 23 2021

REFERENCES

C. W. Trigg, Problem #1210, Series Formation, J. Rec. Math., 15 (1982), 221-222.

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

R. P. Boas and N. J. A. Sloane, Correspondence, 1974

N. J. A. Sloane and Brady Haran, Eureka Sequences, Numberphile video (2021).

FORMULA

a(n) = prime(n) + 1 = A000040(n) + 1.

a(n) = A000005(A034785(n)) = A000203(A000040(n)). - Labos Elemer, May 24 2001

a(n) = A084920(n) / A006093(n). - Reinhard Zumkeller, Aug 06 2007

A239703(a(n)) <= 1. - Hieronymus Fischer, Apr 10 2014

From Ilya Gutkovskiy, Jul 30 2016: (Start)

a(n) ~ n*log(n).

Product_{n>=1} (1 + 2/(a(n)*(a(n) - 2))) = 5/2. (End)

MAPLE

A008864:=n->ithprime(n)+1; seq(A008864(n), n=1..50); # Wesley Ivan Hurt, Apr 11 2014

MATHEMATICA

Prime[Range[70]]+1 (* Vladimir Joseph Stephan Orlovsky, Apr 27 2008 *)

PROG

(PARI) forprime(p=2, 1e3, print1(p+1", ")) \\ Charles R Greathouse IV, Jun 16 2011

(PARI) A008864(n) = (1+prime(n)); \\ Antti Karttunen, Mar 14 2021

(Haskell)

a008864 = (+ 1) . a000040

-- Reinhard Zumkeller, Sep 04 2012, Oct 08 2012

(Magma) [NthPrime(n)+1: n in [1..70]]; // Vincenzo Librandi, Jul 30 2016

(Sage) [nth_prime(n) +1 for n in (1..70)] # G. C. Greubel, May 20 2019

CROSSREFS

Cf. A000040, A060800, A131991, A131992, A131993, A141468.

Cf. A007953, A079696, A187813, A239703, A239708.

Column 1 of A341605, column 2 of A286623 and of A328464.

Partial sums of A125266.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, R. K. Guy

STATUS

approved

A007918

Least prime >= n (version 1 of the "next prime" function).

+10
107

2, 2, 2, 3, 5, 5, 7, 7, 11, 11, 11, 11, 13, 13, 17, 17, 17, 17, 19, 19, 23, 23, 23, 23, 29, 29, 29, 29, 29, 29, 31, 31, 37, 37, 37, 37, 37, 37, 41, 41, 41, 41, 43, 43, 47, 47, 47, 47, 53, 53, 53, 53, 53, 53, 59, 59, 59, 59, 59, 59, 61, 61, 67, 67, 67, 67, 67, 67, 71, 71, 71, 71, 73, 73

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

Version 2 of the "next prime" function is "smallest prime > n". This produces A151800.

Maple uses version 2.

According to the "k-tuple" conjecture, a(n) is the initial term of the lexicographically earliest increasing arithmetic progression of n primes; the corresponding common differences are given by A061558. - David W. Wilson, Sep 22 2007

It is easy to show that the initial term of an increasing arithmetic progression of n primes cannot be smaller than a(n). - N. J. A. Sloane, Oct 18 2007

Also, smallest prime bounded by n and 2n inclusively (in accordance with Bertrand's theorem). Smallest prime >n is a(n+1) and is equivalent to smallest prime between n and 2n exclusively. - Lekraj Beedassy, Jan 01 2007

Run lengths of successive equal terms are given by A125266. - Felix Fröhlich, May 29 2022

Conjecture: if n > 1, then a(n) < n^(n^(1/n)). - Thomas Ordowski, Feb 23 2023

LINKS

T. D. Noe, Table of n, a(n) for n = 0..10000

Jens Kruse Andersen, Records for primes in arithmetic progressions

K. Atanassov, On Some of Smarandache's Problems

K. Atanassov, On the 37th and 38th Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 2, 83-85.

Henry Bottomley, Prime number calculator

J. Castillo, Other Smarandache Type Functions: Inferior/Superior Smarandache f-part of x, Smarandache Notions Journal, Vol. 10, No. 1-2-3, 1999, 202-204.

Andrew Granville, Prime Number Patterns

Hans Gunter, Puzzle 145. The Inferior Smarandache Prime Part and Superior Smarandache Prime Part functions; Solutions by Jean Marie Charrier, Teresinha DaCosta, Rene Blanch, Richard Kelley and Jim Howell.

Jonathan Sondow and Eric Weisstein, Bertrand's Postulate, World of Mathematics.

Eric Weisstein's World of Mathematics, Next Prime, k-tuple conjecture

Index entries for sequences related to primes in arithmetic progressions

FORMULA

For n > 1: a(n) = A000040(A049084(A007917(n)) + 1 - A010051(n)). - Reinhard Zumkeller, Jul 26 2012

a(n) = A151800(n-1). - Seiichi Manyama, Apr 02 2018

MAPLE

A007918 := n-> nextprime(n-1); # M. F. Hasler, Apr 09 2008

MATHEMATICA

NextPrime[Range[-1, 72]] (* Jean-François Alcover, Apr 18 2011 *)

PROG

(PARI) A007918(n)=nextprime(n) \\ M. F. Hasler, Jun 24 2011

(PARI) for(x=0, 100, print1(nextprime(x)", ")) \\ Cino Hilliard, Jan 15 2007

(Haskell)

a007918 n = a007918_list !! n

a007918_list = 2 : 2 : 2 : concat (zipWith

(\p q -> (replicate (fromInteger(q - p)) q))

a000040_list $ tail a000040_list)

-- Reinhard Zumkeller, Jul 26 2012

(Magma) [2] cat [NextPrime(n-1): n in [1..80]]; // Vincenzo Librandi, Jan 14 2016

(Python)

from sympy import nextprime

def A007918(n): return nextprime(n-1) # Chai Wah Wu, Apr 22 2022

CROSSREFS

Cf. A000040, A007917, A008407, A020497, A061558, A125266, A151799, A151800, A171400.

KEYWORD

nonn,easy,nice

AUTHOR

R. Muller and Charles T. Le (charlestle(AT)yahoo.com)

STATUS

approved

A054541

Sum of first n terms equals n-th prime.

+10
15

2, 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, 10, 14, 4, 2, 4, 14, 6, 10, 2, 4, 6, 8, 6, 6, 4, 6, 8, 4, 8, 10, 2, 10, 2, 6, 4, 6, 8, 4, 2, 4, 12, 8, 4, 8, 4, 6

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Except for first term, same as A001223.

First differences of A182986. - Omar E. Pol, Oct 31 2013

A075526 is 1 together with A001223. This is 2 together with A001223. A125266 is 3 together with A001223. - Omar E. Pol, Nov 01 2013

Convolved with A024916 gives A086718. - Omar E. Pol, Dec 23 2021

LINKS

Table of n, a(n) for n=1..97.

MATHEMATICA

Join[{2}, Differences[Prime[Range[100]]]] (* Paolo Xausa, Oct 25 2023 *)

PROG

(PARI) a(n) = if (n==1, 2, prime(n) - prime(n-1)); \\ Michel Marcus, Oct 31 2013

CROSSREFS

Partial sums give A000040.

Cf. A001223, A075526, A086718, A024916, A125266, A182986.

KEYWORD

nonn

AUTHOR

G. L. Honaker, Jr., Apr 09 2000

EXTENSIONS

More terms from James A. Sellers, Apr 11 2000

STATUS

approved

A075526

a(n) = A008578(n+2) - A008578(n+1).

+10
14

1, 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, 10, 14, 4, 2, 4, 14, 6, 10, 2, 4, 6, 8, 6, 6, 4, 6, 8, 4, 8, 10, 2, 10, 2, 6, 4, 6, 8, 4, 2, 4, 12, 8, 4, 8, 4, 6

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

n appears this number of times in A000720. - Lekraj Beedassy, Jun 19 2006

a(0) = 1, for n >= 1: a(n) = differences between consecutive primes (A001223(n)) = A158611(n+2) - A158611(n+1). Partial sums give A006093 (shifted). - Jaroslav Krizek, Aug 04 2009

First differences of noncomposite numbers. - Juri-Stepan Gerasimov, Feb 17 2010

This is 1 together with A001223. A054541 is 2 together with A001223. A125266 is 3 together with A001223. - Omar E. Pol, Nov 01 2013

LINKS

Table of n, a(n) for n=0..96.

FORMULA

a(n) = A001223(n) for n>0.

MATHEMATICA

Prime[Range[100]] // Differences // Prepend[#, 1]& (* Jean-François Alcover, Dec 11 2021 *)

CROSSREFS

Cf. A000040, A001223, A075527, A158611.

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Sep 22 2002

EXTENSIONS

Correction for change of offset in A158611 and A008578 in Aug 2009 Jaroslav Krizek, Jan 27 2010

STATUS

approved

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