A014688 -id:A014688 - cp's OEIS frontend

A090178 a(1) = 2; for n > 1, a(n) = n + prime(n-1).

2, 4, 6, 9, 12, 17, 20, 25, 28, 33, 40, 43, 50, 55, 58, 63, 70, 77, 80, 87, 92, 95, 102, 107, 114, 123, 128, 131, 136, 139, 144, 159, 164, 171, 174, 185, 188, 195, 202, 207, 214, 221, 224, 235, 238, 243, 246, 259, 272, 277, 280, 285, 292, 295, 306, 313, 320, 327

Offset: 1

Pab Ter (pabrlos(AT)yahoo.com), May 29 2004

nonn

Sum of index n and the corresponding n-th term of noncomposite numbers (A008578).
Does n > 2 exist such that n*prime(n-1)/(n+prime(n-1)), i.e., A164931(n)/a(n) is an integer? - Jaroslav Krizek, Aug 31 2009
Complement of A171508(n). - Jaroslav Krizek, Dec 13 2009

			a(2) = 2 + prime(1) = 4; a(5) = 5 + prime(4) = 12; a(9) = 9 + prime(8) = 28.

M. Feenstra, P. J. Carter and C. P. Harding, The Ultimate I.Q. Book, Wardlock, see p. 99.

The published version (A048171) is said to be incorrect.

[n + NthPrime(n-1): n in [1..58] ]; // Klaus Brockhaus, Sep 09 2009

Join[{2}, Table[Prime[n - 1] + n, {n, 2, 60}]] (* Vincenzo Librandi, Dec 08 2015 *)

a(n) = n + A008578(n). - David Wasserman, May 20 2008

a(1) = 2; a(n) = 1 + A014688(n-1), for n >= 2. - Omar E. Pol, Nov 01 2013

Definition corrected, second comment and example edited by Klaus Brockhaus, Sep 09 2009

A106587 Sum of n-th prime squared and n-th perfect square.

Original entry on oeis.org

5, 13, 34, 65, 146, 205, 338, 425, 610, 941, 1082, 1513, 1850, 2045, 2434, 3065, 3770, 4045, 4850, 5441, 5770, 6725, 7418, 8497, 10034, 10877, 11338, 12233, 12722, 13669, 17090, 18185, 19858, 20477, 23426, 24097, 26018, 28013, 29410, 31529, 33722

Offset: 1

Alexandre Wajnberg, May 10 2005

			a(5)=146 because 121 (fifth prime^2) + 25 (fifth square) = 146.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000290, A001248, A014688, A075526, A076368.

[NthPrime(n)^2 + n^2: n in [1..50]]; // G. C. Greubel, Sep 07 2021

Mathematica
```
Table[Prime[n]^2 + n^2, {n, 50}]
```

a(n) = n^2 + prime(n)^2; \\ Michel Marcus, Sep 08 2021

[nth_prime(n)^2 + n^2 for n in (1..50)] # G. C. Greubel, Sep 07 2021

a(n) = n^2 + prime(n)^2.

Extended by Ray Chandler, May 13 2005

A115884 Numbers k such that the k-th prime plus k gives a palindrome.

Original entry on oeis.org

1, 2, 3, 4, 22, 45, 66, 71, 75, 88, 94, 97, 103, 105, 116, 140, 331, 432, 454, 565, 646, 703, 795, 1042, 1108, 1168, 1248, 1334, 1644, 1652, 1864, 1874, 1900, 2181, 2295, 2323, 2485, 2509, 2585, 2679, 2835, 2899, 2923, 3052, 3360, 3372, 3396, 3404

Offset: 1

Giovanni Resta, Feb 06 2006

			prime(103) + 103 = 666, a palindrome; so 103 is a term.

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A014688, A084122, A115885, A115888.

filter:= proc(n) local p,L;
   p:= ithprime(n)+n;
   L:= convert(p,base,10);
   ListTools:-Reverse(L) = L
end proc:
select(filter, [$1..10000]); # Robert Israel, Nov 04 2014

palQ[n_]:=Module[{idn=IntegerDigits[n]},idn==Reverse[idn]]; With[ {nn=3500}, Rest[Flatten[Position[Total/@Thread[{Prime[Range[nn]], Range[nn]}],?(palQ)]]]] (* _Harvey P. Dale, Oct 11 2011 *)
palQ[n_] := Reverse[x = IntegerDigits[n]] == x; Select[Range[3405], palQ[Prime[#] + #] &] (* Jayanta Basu, Jun 24 2013 *)

ispal(n) = my(e=digits(n));e == Vecrev(e) \\ A002113
for(k=1,10^6,b=k+prime(k);if(ispal(b),print1(k,", "))) \\ Alexandru Petrescu, Jun 15 2022

from sympy import nextprime
def ispal(n): s = str(n); return s == s[::-1]
def agen(): # generator of terms
    k, pk = 1, 2
    while True:
        if ispal(k+pk): yield k
        k, pk = k+1, nextprime(pk)
g = agen()
print([next(g) for n in range(1, 51)]) # Michael S. Branicky, Jun 15 2022

A332086 a(n) = pi(prime(n) + n) - n, where pi is the prime counting function.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 3, 3, 2, 3, 3, 4, 4, 4, 4, 3, 4, 4, 6, 5, 5, 4, 4, 4, 4, 6, 6, 6, 6, 7, 6, 7, 8, 7, 7, 6, 6, 8, 7, 8, 7, 8, 10, 9, 9, 10, 9, 9, 8, 9, 10, 9, 8, 8, 8, 7, 9, 10, 10, 9, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13

Offset: 1

Ya-Ping Lu, Aug 22 2020

nonn

This sequence is related to a theorem of Lu and Deng (see LINKS): “The prime gap of a prime number is less than or equal to the prime count of the prime number”, which is equivalent to “There exists at least one prime number between p and p+pi(p)+1”, or pi(p+pi(p)) - pi(p) > 1, where pi is prime counting function. The n-th term of the sequence, a(n), is the number of prime number between the n-th prime number p_n and p_n + pi(p_n) + 1. According to the theorem, a(n) >= 1.

			a(1) = pi(p_1 + 1) - 1 = pi(2 + 1) - 1 = 2 - 1 = 1;
a(2) = pi(p_2 + 2) - 2 = pi(3 + 2) - 2 = 3 - 2 = 1;
a(6) = pi(p_6 + 6) - 6 = pi(13 + 6) - 6 = 8 - 6 = 2;
a(80) = pi(p_80 + 80) - 80 = pi(409 + 80) - 80 = 93 - 80 = 13.

Robert Israel, Table of n, a(n) for n = 1..10000
Ya-Ping Lu and Shu-Fang Deng, An upper bound for the prime gap, arXiv:2007.15282 [math.GM], 2020.

Cf. A000720 (pi), A014688 (prime(n)+n).

f:= n -> numtheory:-pi(ithprime(n)+n)-n:
map(f, [$1..100]); # Robert Israel, Sep 08 2020

a[n_] := PrimePi[Prime[n] + n] - n; Array[a, 100] (* Amiram Eldar, Aug 23 2020 *)

a(n) = primepi(prime(n) + n) - n; \\ Michel Marcus, Aug 23 2020

from sympy import prime, primepi
for n in range(1, 1001):
    a = primepi(prime(n) + n) - n
    print(a)

a(n) = pi(prime(n) + n) - n.

a(n) = A000720(A014688(n)) - n. - Michel Marcus, Aug 23 2020

Name edited by Michel Marcus, Sep 02 2020

A093572 Greatest prime factor of Product(k+prime(k): 1<=k<=n).

Original entry on oeis.org

3, 5, 5, 11, 11, 19, 19, 19, 19, 19, 19, 19, 19, 19, 31, 31, 31, 79, 79, 79, 79, 101, 101, 113, 113, 127, 127, 127, 127, 127, 127, 163, 163, 173, 173, 173, 173, 173, 173, 173, 173, 223, 223, 223, 223, 223, 223, 271, 271, 271, 271, 271, 271, 271, 271, 271, 271

Offset: 1

Reinhard Zumkeller, Apr 01 2004

nonn

a(n) = A006530(A093570(n)) = A006530(A093571(n));

a(n) = Max(A076556(k): 1<=k<=n).

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

Cf. A014688, A092602.

PrimeFactors[n_Integer] := Flatten[ Table[ # [[1]], {1}] & /@ FactorInteger[n]]; Table[ PrimeFactors[ Product[k + Prime[k], {k, n}]][[ -1]], {n, 60}] (* Robert G. Wilson v, Apr 07 2004 *)
FactorInteger[#][[-1,1]]&/@FoldList[Times,Table[n+Prime[n],{n,60}]] (* Harvey P. Dale, May 27 2016 *)

More terms from Robert G. Wilson v, Apr 07 2004

A097935 Number of primes that are not less than prime(n)-n and not greater than prime(n)+n.

Original entry on oeis.org

2, 3, 4, 4, 3, 5, 5, 5, 5, 5, 5, 6, 7, 7, 7, 8, 8, 9, 8, 9, 9, 10, 10, 12, 10, 10, 10, 11, 11, 12, 13, 13, 12, 13, 12, 12, 14, 16, 15, 15, 14, 15, 17, 17, 17, 17, 17, 18, 18, 19, 19, 19, 20, 18, 18, 20, 19, 19, 20, 21, 21, 21, 20, 21, 21, 23, 22, 23, 21, 22, 22, 22, 23, 24, 25

Offset: 1

Reinhard Zumkeller, Sep 05 2004

nonn

			a(10) = #{p prime: A000040(10)-10 <= p <= A000040(10)+10} = #{p prime: 19 <= p <= 39} = #{19,23,29,31,37} = 5.

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

Cf. A000720, A014688, A014689.

a[n_] := PrimePi[Prime[n] + n] - PrimePi[Prime[n] - n - 1];
Array[a, 100] (* Jean-François Alcover, Jun 11 2019 *)

a(n) = my(p=prime(n)); primepi(p+n) - primepi(p-n-1); \\ Michel Marcus, Jun 11 2019

[len([k for k in (nth_prime(n)-n..nth_prime(n)+n) if is_prime(k)]) for n in (1..75)]  # Peter Luschny, Sep 03 2013

a(n) = A000720(A014688(n)) - A000720(A014689(n)-1).

A098390 Prime(n)+Log2(prime(n)), where Log2=A000523.

Original entry on oeis.org

3, 4, 7, 9, 14, 16, 21, 23, 27, 33, 35, 42, 46, 48, 52, 58, 64, 66, 73, 77, 79, 85, 89, 95, 103, 107, 109, 113, 115, 119, 133, 138, 144, 146, 156, 158, 164, 170, 174, 180, 186, 188, 198, 200, 204, 206, 218, 230, 234, 236, 240, 246, 248, 258, 265, 271, 277, 279

Offset: 1

Reinhard Zumkeller, Sep 06 2004

nonn

a(n) = A000040(n) + A098388(n).

			a(10) = A000040(10) + A098388(10) = 29 + 4 = 33.

Cf. A098389, A014688, A098387, A098393.

#+Floor[Log[2,#]]&/@Prime[Range[60]] (* Harvey P. Dale, Dec 30 2011 *)

A025003 a(1) = 2; a(n+1) = a(n)-th nonprime, where nonprimes begin at 1.

Original entry on oeis.org

2, 4, 8, 14, 22, 33, 48, 66, 90, 120, 156, 202, 256, 322, 400, 494, 604, 734, 888, 1067, 1272, 1512, 1790, 2107, 2472, 2890, 3364, 3903, 4515, 5207, 5990, 6875, 7868, 8984, 10238, 11637, 13207, 14959, 16909, 19075, 21483, 24173, 27149, 30436, 34080, 38103

Offset: 1

David W. Wilson

nonn

Index of first occurrence of n in A090532.

Let b(n) (n >= 0) be the smallest integer k >= 1 that takes n steps to reach 1 iterating the map f: k -> k - pi(k). The sequence {b(n), n >= 0} begins 1, 2, 4, 8, 14, 22, 33, 48, 66, 90, 120, 156, ... and agrees with the present sequence except for b(0). - Ya-Ping Lu, Sep 07 2020

			From _Ya-Ping Lu_, Sep 07 2020: (Start)
a(1) = 2 because f(2) = 2 - pi(2) = 1 and m(2) = 1;
For the integer 3, since f(3) = 1. m(3) = 1, which is not bigger than m(1) or m(2). So, 3 is not a term in the sequence;
a(2) = 4 because f^2(4) = f(2) = 1 and m(4) = 2;
a(3) = 8 because f^3(8) = f^2(4) = 1 and m(8) = 3. (End)

Cf. A000040, A000720, A014688, A014689, A062298, A090532, A332086, A337334.

N:= 50: # to get a(0)..a(N)
V:= Array(0..N):
V[0]:= 1: V[1]:= 2:
m:= 2: p:= 3: g:= 1: n:= 1:
do
  if g+p-m-1 >= V[n] then
    m:= V[n]+m-g;
    n:= n+1;
    V[n]:= m;
    if n = N then break fi;
    g:= V[n-1];
  else
    g:= g+p-m;
    m:= p+1;
    p:= nextprime(m);
  fi;
od;
convert(V, list); # Robert Israel, Sep 08 2020

from sympy import prime, primepi
n_last = 0
pi_last = 0
ct_max = -1
for n in range(1, 100001):
    ct = 0
    pi = pi_last + primepi(n) - primepi(n_last)
    n_c = n
    pi_c = pi
    while n_c > 1:
        nc -= pi_c
        ct += 1
        pi_c -= primepi(n_c + pi_c) - primepi(n_c)
    if ct > ct_max:
        print(n)
        ct_max = ct
    n_last = n
    pi_last = pi # Ya-Ping Lu, Sep 07 2020

a(n) = min(k: f^n(k) = 1), where f = A062298 and n-fold iteration of f is denoted by f^n. - Ya-Ping Lu, Sep 07 2020

A064427 a(n) = n + (number of primes < n).

Original entry on oeis.org

1, 2, 4, 6, 7, 9, 10, 12, 13, 14, 15, 17, 18, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 40, 41, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 55, 56, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 77, 78, 80, 81, 82

Offset: 1

Santi Spadaro, Sep 30 2001

nonn

From Jaroslav Krizek, Dec 10 2009: (Start)
Complement of A014688.
Numbers that are not the sum of k and the k-th prime for any k >= 1. (End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Carlos Rivera, Puzzle 821. Prime numbers and complementary sequences, The Prime Puzzles & Problems Connection.
Eric Weisstein's World of Mathematics, Prime Counting Function.
Wikipedia, Prime-counting function.

Cf. A000720, A014688, A095117.

a064427 1 = 1
a064427 n = a000720 (n - 1) + toInteger n
-- Reinhard Zumkeller, Apr 17 2012

[1] cat [#PrimesUpTo(n-1)+n: n in [2..100]]; // Vincenzo Librandi, Feb 13 2016

a[n_] := PrimePi[a[n-1]]+n; a[1]=1
Table[PrimePi[n-1]+n,{n,60}] (* Harvey P. Dale, Apr 03 2015 *)

a(n) = if (n==1, 1, primepi(n-1)+n); \\ Michel Marcus, Feb 13 2016

For n > 1: a(n) = n + A000720(n-1).

Definition improved by Reinhard Zumkeller, Apr 16 2012

Edited by Jon E. Schoenfield, Nov 24 2023

A093570 a(n) = Product_{k=1..n}(k + prime(k)).

Original entry on oeis.org

3, 15, 120, 1320, 21120, 401280, 9630720, 260029440, 8320942080, 324516741120, 13629703127040, 667855453224960, 36064194474147840, 2055659085026426880, 127450863271638466560, 8794109565743054192640, 668352326996472118640640

Offset: 1

Reinhard Zumkeller, Apr 01 2004

nonn

a(n) < A000142(n) + A002110(n) for n>1;

a(n) = Product_{k=1..n}A014688(k).

Harvey P. Dale, Table of n, a(n) for n = 1..342

Cf. A093572, A093571.

Table[ Product[k + Prime[k], {k, n}], {n, 17}] (* Robert G. Wilson v, Apr 07 2004 *)
FoldList[Times,Table[m+Prime[m],{m,20}]] (* Harvey P. Dale, Jun 04 2023 *)

More terms from Robert G. Wilson v, Apr 07 2004

cp's OEIS Frontend

A090178 a(1) = 2; for n > 1, a(n) = n + prime(n-1).

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A106587 Sum of n-th prime squared and n-th perfect square.

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A115884 Numbers k such that the k-th prime plus k gives a palindrome.

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A332086 a(n) = pi(prime(n) + n) - n, where pi is the prime counting function.

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Maple

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A093572 Greatest prime factor of Product(k+prime(k): 1<=k<=n).

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Mathematica

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A097935 Number of primes that are not less than prime(n)-n and not greater than prime(n)+n.

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Mathematica

PARI

Sage

Formula

A098390 Prime(n)+Log2(prime(n)), where Log2=A000523.

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