A014683 -id:A014683 - cp's OEIS frontend

A140706 A054525 * A014683; a(n) = Sum_{d|n} mu(d)*A014683(n/d).

1, 2, 3, 1, 5, 0, 7, 4, 5, 2, 11, 5, 13, 4, 6, 8, 17, 7, 19, 9, 10, 8, 23, 8, 19, 10, 18, 13, 29, 11, 31, 16, 18, 14, 22, 12, 37, 16, 22, 16, 41, 15, 43, 21, 25, 20, 47, 16, 41, 21, 30, 25, 53, 18, 38, 24, 34, 26, 59, 15, 61, 28, 37, 32, 46, 23, 67, 33, 42, 27, 71, 24, 73, 34, 41

Offset: 1

Gary W. Adamson, May 24 2008

nonn

a(n) = n iff n is prime.

			a(4) = 1 = (0, -1, 0, 1) dot (1, 3, 4, 4), where (0, -1, 0, 1) = row 4 of triangle A054525.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A008683, A014683, A054525.

read("transforms") : A014683 := proc(n) if isprime(n) then 1+n; else n; fi; end: a014683 := [seq(A014683(n),n=1..150)] ; a140706 := MOBIUS(a014683) ; for i from 1 to nops(a140706) do printf("%d,",op(i,a140706)) ; od: # R. J. Mathar, Jan 19 2009

Table[Sum[MoebiusMu[d] (# + Boole@ PrimeQ@ #) &[n/d], {d, Divisors@ n}], {n, 75}] (* Michael De Vlieger, Jul 29 2017 *)

A014683(n) = (n+isprime(n));
A140706(n) = sumdiv(n,d,moebius(d)*A014683(n/d)); \\ Antti Karttunen, Jul 28 2017

from sympy import isprime, mobius, divisors
def a014683(n): return n + isprime(n)
def a140706(n): return sum(mobius(d)*a014683(n//d) for d in divisors(n))
print([a140706(n) for n in range(1,51)]) # Indranil Ghosh, Jul 29 2017

Möbius transform of A014683: (1, 3, 4, 4, 6, 6, 8, 8, 9, 10, ...); where A014683(n) = n if n is not prime; but (n+1) if n is prime.

a(n) = Sum_{d|n} A008683(d)*A014683(n/d), where A008683 is Moebius mu function. - Antti Karttunen, Jul 28 2017

More terms from R. J. Mathar, Jan 19 2009

Second part added to the name by Antti Karttunen, Jul 28 2017

A000086 Number of solutions to x^2 - x + 1 == 0 (mod n).

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0

Offset: 1

N. J. A. Sloane

Number of elliptic points of order 3 for Gamma_0(n).
Equivalently, number of fixed points of Gamma_0(n) of type rho.
Values are 0 or a power of 2.
Shadow transform of central polygonal numbers A002061. - Michel Marcus, Jun 06 2013
Empirical: a(n) == A001615(n) (mod 3) for all natural numbers n. - John M. Campbell, Apr 01 2018
From Jianing Song, Jul 03 2018: (Start)
The comment above is true. Since both a(n) and A001615(n) are multiplicative we just have to verify that for prime powers. Note that A001615(p^e) = (p+1)*p^(e-1). For p == 1 (mod 3), p+1 == 2 (mod 3) so (p+1)*p^(e-1) == 2 (mod 3); for p == 2 (mod 3), p+1 is a multiple of 3 so (p+1)*p^(e-1) == 0 (mod 3). For p = 3, if e = 1 then p+1 == 1 (mod 3); if e > 1 then (p+1)*p^(e-1) == 0 (mod 3).
Equivalently, number of solutions to x^2 + x + 1 == 0 (mod n). (End)

			G.f. = x + x^3 + 2*x^7 + 2*x^13 + 2*x^19 + 2*x^21 + 2*x^31 + 2*x^37 + 2*x^39 + ...

Bruno Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 101.
Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see p. 25, Eq. (3).

Christian G. Bower, Table of n, a(n) for n = 1..2000
Harriet Fell, Morris Newman, and Edward Ordman, Tables of genera of groups of linear fractional transformations, J. Res. Nat. Bur. Standards Sect. B 67B (1963), 61-68.
Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5(4) (1999), 138-150; see Definition 7 for the shadow transform.
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. A65 (2009), 156-163. [See Table 4.]
N. J. A. Sloane, Transforms.

Cf. A000089, A000091, A001616, A014683.
Cf. A027748, A079978, A038502, A007949, A165952.
Cf. A341422 (without zeros).

a000086 n = if n `mod` 9 == 0 then 0
  else product $ map ((* 2) . a079978 . (+ 2)) $ a027748_row $ a038502 n
-- Reinhard Zumkeller, Jun 23 2013

with(numtheory); A000086 := proc (n) local d, s; if modp(n,9) = 0 then RETURN(0) fi; s := 1; for d in divisors(n) do if isprime(d) then s := s*(1+eval(legendre(-3,d))) fi od; s end: # Gene Ward Smith, May 22 2006

Array[ Function[ n, If[ EvenQ[ n ] || Mod[ n, 9 ]==0, 0, Count[ Array[ Mod[ #^2-#+1, n ]&, n, 0 ], 0 ] ] ], 84 ]
a[ n_] := If[ n < 1, 0, Length[ Select[ (#^2 - # + 1)/n & /@ Range[n], IntegerQ]]]; (* Michael Somos, Aug 14 2015 *)
a[n_] := a[n] = Product[{p, e} = pe; Which[p==1 || p==3 && e==1, 1, p==3 && e>1, 0, Mod[p, 3]==1, 2, Mod[p, 3]==2, 0, True, a[p^e]], {pe, FactorInteger[n]}]; Array[a, 105] (* Jean-François Alcover, Oct 18 2018 *)

{a(n) = if( n<1, 0, sum( x=0, n-1, (x^2 - x + 1)%n==0))}; \\ Nov 15 2002

{a(n) = if( n<1, 0, direuler( p=2, n, if( p==3, 1 + X, if( p%3==2, 1, (1 + X) / (1 - X)))) [n])}; \\ Nov 15 2002

Multiplicative with a(p^e) = 1 if p = 3 and e = 1; 0 if p = 3 and e > 1; 2 if p == 1 (mod 3); 0 if p == 2 (mod 3). - David W. Wilson, Aug 01 2001
a(A226946(n)) = 0; a(A034017(n)) > 0. - Reinhard Zumkeller, Jun 23 2013
a(2*n) = a(3*n + 2) = a(9*n) = a(9*n + 6) = 0. - Michael Somos, Aug 14 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*sqrt(3)/(3*Pi) = 0.367552... (A165952). - Amiram Eldar, Oct 11 2022

A113646 a(n) is the smallest composite integer which is >= n.

Original entry on oeis.org

4, 4, 4, 4, 6, 6, 8, 8, 9, 10, 12, 12, 14, 14, 15, 16, 18, 18, 20, 20, 21, 22, 24, 24, 25, 26, 27, 28, 30, 30, 32, 32, 33, 34, 35, 36, 38, 38, 39, 40, 42, 42, 44, 44, 45, 46, 48, 48, 49, 50, 51, 52, 54, 54, 55, 56, 57, 58, 60, 60, 62, 62, 63, 64, 65, 66, 68, 68, 69, 70, 72, 72

Offset: 1

Leroy Quet, Jan 15 2006

nonn

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

Cf. A002808, A014683.

a113646 n = if n < 3 then 4 else a014683 n
-- Reinhard Zumkeller, Nov 01 2014

# This Maple program returns the smallest composite greater than n - N. J. A. Sloane, Sep 11 2019
iscomp := n-> if isprime(n) or (n=1) then false else true; fi;
f := proc(n) local a; global iscomp; a:=n+1; while not iscomp(a) do a:=a+1; od; a; end;

Table[k = n; While[! CompositeQ@ k, k++]; k, {n, 72}] (* Michael De Vlieger, Sep 06 2017 *)

from sympy import isprime
def a(n):
  an = max(4, n)
  while isprime(an): an += 1
  return an
print([a(n) for n in range(1, 73)]) # Michael S. Branicky, Apr 04 2021

from sympy import isprime
def A113646(n): return n+isprime(n) if n>2 else 4 # Chai Wah Wu, Oct 03 2024

a(1) = a(2) = 4. For n >= 3, a(n) = A014683(n).

A135681 a(n)=n if n=1 or if n=prime. Otherwise, n=4 if n is even and n=1 if n is odd.

Original entry on oeis.org

1, 2, 3, 4, 5, 4, 7, 4, 1, 4, 11, 4, 13, 4, 1, 4, 17, 4, 19, 4, 1, 4, 23, 4, 1, 4, 1, 4, 29, 4, 31, 4, 1, 4, 1, 4, 37, 4, 1, 4, 41, 4, 43, 4, 1, 4, 47, 4, 1, 4, 1, 4, 53, 4, 1, 4, 1, 4, 59, 4, 61, 4, 1, 4, 1, 4, 67, 4, 1, 4, 71, 4, 73

Offset: 1

Mohammad K. Azarian, Dec 01 2007

nonn

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

Cf. A135679; A135580; A014681; A014682; A014683; A014684; A014685; A014686; A014687; A014688; A014689; A014690.

a[n_] := If[PrimeQ[n] || n == 1, n, If[EvenQ[n], 4, 1] ]; Table[a[n], {n,1,25}] (* G. C. Greubel, Oct 26 2016 *)

A113636 In the sequence of positive integers add 1 to each nonprime number.

Original entry on oeis.org

2, 2, 3, 5, 5, 7, 7, 9, 10, 11, 11, 13, 13, 15, 16, 17, 17, 19, 19, 21, 22, 23, 23, 25, 26, 27, 28, 29, 29, 31, 31, 33, 34, 35, 36, 37, 37, 39, 40, 41, 41, 43, 43, 45, 46, 47, 47, 49, 50, 51, 52, 53, 53, 55, 56, 57, 58, 59, 59, 61, 61, 63, 64, 65, 66, 67, 67, 69, 70, 71, 71, 73

Offset: 1

Cino Hilliard, Jan 15 2006

This is the complement of sequence A014683.

Möbius transform of A380449(n). - Wesley Ivan Hurt, Jun 21 2025

Cf. A005171, A014683, A014684, A113523, A113638, A179278, A380449.

Array[# + Boole[! PrimeQ@ #] &, 72] (* Michael De Vlieger, Nov 05 2020 *)

a(n) = if (!isprime(n), n+1, n); \\ Michel Marcus, Nov 06 2020

a(n) = A014684(n) + 1. - Bill McEachen, Nov 01 2020
From Wesley Ivan Hurt, Jun 21 2025: (Start)
a(n) = n + c(n), where c = A005171.
a(n) = Sum_{d|n} A380449(d) * mu(n/d). (End)

Offset 1 from Michel Marcus, Nov 06 2020

A135682 a(n)=n if n=1 or if n=prime. Otherwise, n=4 if n is even and n=7 if n is odd.

Original entry on oeis.org

1, 2, 3, 4, 5, 4, 7, 4, 7, 4, 11, 4, 13, 4, 7, 4, 17, 4, 19, 4, 7, 4, 23, 4, 7, 4, 7, 4, 29, 4, 31, 4, 7, 4, 7, 4, 37, 4, 7, 4, 41, 4, 43, 4, 7, 4, 47, 4, 7, 4, 7, 4, 53, 4, 7, 4, 7, 4, 59, 4, 61, 4, 7, 4, 7, 4, 67, 4, 7, 4, 71, 4, 73

Offset: 1

Mohammad K. Azarian, Dec 01 2007

nonn

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

Cf. A000027; A000040; A135679; A135680; A014681; A014682; A014683; A014684; A014685; A014686; A014687; A014688; A014689; A014690; A135681.

a[n_] := If[PrimeQ[n] || n == 1, n, If[EvenQ[n], 4, 7] ]; Table[a[n], {n,1,25}] (* G. C. Greubel, Oct 26 2016 *)

A135684 a(n)=11 if n is a prime number. Otherwise, a(n)=n.

Original entry on oeis.org

1, 11, 11, 4, 11, 6, 11, 8, 9, 10, 11, 12, 11, 14, 15, 16, 11, 18, 11, 20, 21, 22, 11, 24, 25, 26, 27, 28, 11, 30, 11, 32, 33, 34, 35, 36, 11, 38, 39, 40, 11, 42, 11, 44, 45, 46, 11, 48, 49, 50, 51, 52, 11, 54, 55, 56, 57, 58, 11

Offset: 1

Mohammad K. Azarian, Dec 01 2007

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

Cf. A000027; A000040; A135679; A135680; A014681; A014682; A014683; A014684; A014685; A014686; A014687; A014688; A014689; A014690; A135681; A135682; A135683.

[IsPrime(n) select 11 else n: n in [1..70]]; // Vincenzo Librandi, Feb 22 2013

Table[If[PrimeQ[n], 11, n], {n, 70}] (* Vincenzo Librandi, Feb 22 2013 *)

A374426 a(n) = n*(n + 1)/2 + pi(n), where pi(n) = A000720(n) is the prime counting function.

Original entry on oeis.org

1, 4, 8, 12, 18, 24, 32, 40, 49, 59, 71, 83, 97, 111, 126, 142, 160, 178, 198, 218, 239, 261, 285, 309, 334, 360, 387, 415, 445, 475, 507, 539, 572, 606, 641, 677, 715, 753, 792, 832, 874, 916, 960, 1004, 1049, 1095, 1143, 1191, 1240, 1290, 1341, 1393, 1447

Offset: 1

James C. McMahon, Jul 08 2024

nonn

James C. McMahon, Table of n, a(n) for n = 1..10000

Cf. A000217, A000720, A010051, A256885.

Partial sums of A014683.

Mathematica
```
Table[n(n+1)/2+PrimePi[n],{n,53}]
```

a(n) = n*(n+1)/2 + primepi(n); \\ Michel Marcus, Jul 31 2024

a(n) = A000217(n) + A000720(n).

a(1) = 1; for n > 1: a(n) = a(n-1) + n + A010051(n).

A380457 Sum of divisors of n plus the number of distinct prime divisors of n: a(n) = sigma(n) + omega(n).

Original entry on oeis.org

1, 4, 5, 8, 7, 14, 9, 16, 14, 20, 13, 30, 15, 26, 26, 32, 19, 41, 21, 44, 34, 38, 25, 62, 32, 44, 41, 58, 31, 75, 33, 64, 50, 56, 50, 93, 39, 62, 58, 92, 43, 99, 45, 86, 80, 74, 49, 126, 58, 95, 74, 100, 55, 122, 74, 122, 82, 92, 61, 171, 63, 98, 106, 128, 86, 147, 69, 128, 98, 147, 73, 197, 75, 116, 126, 142, 98, 171, 81, 188, 122, 128, 85, 227, 110, 134, 122

Offset: 1

Wesley Ivan Hurt, Jun 22 2025

Inverse Möbius transform of A014683(n).

For each divisor d of n, add d+1 if d is prime, else add d.

			a(12) = 1 + (2+1) + (3+1) + 4 + 6 + 12 = 30.

Cf. A000203 (sigma), A001221 (omega), A014683.

Table[DivisorSigma[1, n] + PrimeNu[n], {n, 100}]

a(n) = Sum_{d|n} A014683(d).

a(p^k) = (p^(k+1)+p-2)/(p-1) for p prime, k>=1. - Wesley Ivan Hurt, Jul 02 2025

A079778 a(n) = smallest prime number such that a(n) - n is composite.

Original entry on oeis.org

5, 11, 7, 13, 11, 31, 11, 17, 13, 19, 17, 37, 17, 23, 19, 31, 23, 43, 23, 29, 29, 31, 29, 59, 29, 41, 31, 37, 37, 79, 37, 41, 37, 43, 41, 61, 41, 47, 43, 61, 47, 67, 47, 53, 53, 61, 53, 73, 53, 59

Offset: 1

Amarnath Murthy, Feb 03 2003

nonn

Cf. A014683.

Corrected and extended by Mark Hudson (mrmarkhudson(AT)hotmail.com), Feb 04 2003

cp's OEIS Frontend

A140706 A054525 * A014683; a(n) = Sum_{d|n} mu(d)*A014683(n/d).

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A000086 Number of solutions to x^2 - x + 1 == 0 (mod n).

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A113646 a(n) is the smallest composite integer which is >= n.

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A135681 a(n)=n if n=1 or if n=prime. Otherwise, n=4 if n is even and n=1 if n is odd.

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Mathematica

A113636 In the sequence of positive integers add 1 to each nonprime number.

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A135682 a(n)=n if n=1 or if n=prime. Otherwise, n=4 if n is even and n=7 if n is odd.

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A135684 a(n)=11 if n is a prime number. Otherwise, a(n)=n.

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