A005171 -id:A005171 - cp's OEIS frontend

A341466 Number of partitions of n into 7 distinct nonprime parts.

1, 0, 1, 1, 2, 2, 4, 3, 6, 6, 9, 9, 14, 15, 21, 23, 30, 33, 43, 47, 61, 67, 81, 91, 112, 123, 150, 165, 194, 217, 255, 281, 330, 363, 417, 461, 529, 582, 665, 730, 823, 905, 1018, 1115, 1253, 1368, 1519, 1662, 1844, 2010, 2227, 2419, 2659, 2894, 3175, 3442

Offset: 50

Ilya Gutkovskiy, Feb 12 2021

nonn

Cf. A005171, A018252, A096258, A219201, A302479, A341454, A341461, A341462, A341464, A341465, A341467.

b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t=0, 1, 0), `if`(i<1 or t<1, 0, b(n, i-1, t)+
      `if`(isprime(i), 0, b(n-i, min(n-i, i-1), t-1))))
    end:
a:= n-> b(n$2, 7):
seq(a(n), n=50..105);  # Alois P. Heinz, Feb 12 2021

b[n_, i_, t_] := b[n, i, t] = If[n == 0,
     If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] +
     If[PrimeQ[i], 0, b[n - i, Min[n - i, i - 1], t - 1], 0]]];
a[n_] := b[n, n, 7];
Table[a[n], {n, 50, 105}] (* Jean-François Alcover, Feb 23 2022, after Alois P. Heinz *)

A341467 Number of partitions of n into 8 distinct nonprime parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 5, 6, 7, 10, 12, 16, 19, 24, 28, 36, 41, 52, 60, 73, 85, 102, 116, 142, 161, 192, 217, 256, 287, 339, 382, 442, 496, 574, 639, 737, 821, 937, 1041, 1184, 1309, 1483, 1640, 1845, 2037, 2283, 2508, 2807, 3081, 3430, 3761, 4170, 4553, 5045

Offset: 64

Ilya Gutkovskiy, Feb 12 2021

nonn

Cf. A005171, A018252, A096258, A219202, A302479, A341455, A341461, A341462, A341464, A341465, A341466.

b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t=0, 1, 0), `if`(i<1 or t<1, 0, b(n, i-1, t)+
      `if`(isprime(i), 0, b(n-i, min(n-i, i-1), t-1))))
    end:
a:= n-> b(n$2, 8):
seq(a(n), n=64..118);  # Alois P. Heinz, Feb 12 2021

b[n_, i_, t_] := b[n, i, t] = If[n == 0,
     If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] +
     If[PrimeQ[i], 0, b[n - i, Min[n - i, i - 1], t - 1], 0]]];
a[n_] := b[n, n, 8];
Table[a[n], {n, 64, 118}] (* Jean-François Alcover, Feb 22 2022, after Alois P. Heinz *)

A341480 Number of ways to write n as an ordered sum of 3 nonprime numbers.

Original entry on oeis.org

1, 0, 0, 3, 0, 3, 3, 3, 9, 4, 9, 12, 12, 15, 21, 19, 27, 30, 30, 39, 42, 46, 54, 60, 61, 75, 72, 91, 90, 108, 99, 129, 123, 142, 147, 168, 156, 201, 180, 217, 213, 246, 235, 279, 255, 304, 297, 336, 327, 375, 342, 412, 390, 447, 423, 492, 453, 529, 507, 573, 538, 630, 579

Offset: 3

Ilya Gutkovskiy, Feb 13 2021

nonn

Robert Israel, Table of n, a(n) for n = 3..10000

Cf. A005171, A018252, A052284, A076608, A098238, A341408, A341461, A341481, A341482, A341483, A341484, A341485, A341486.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
      `if`(isprime(j), 0, b(n-j, t-1)), j=1..n)))
    end:
a:= n-> b(n, 3):
seq(a(n), n=3..65);  # Alois P. Heinz, Feb 13 2021

nmax = 65; CoefficientList[Series[Sum[Boole[!PrimeQ[k]] x^k, {k, 1, nmax}]^3, {x, 0, nmax}], x] // Drop[#, 3] &

G.f. g(x)^3 where g(x) is the G.f. of A005171.

A341481 Number of ways to write n as an ordered sum of 4 nonprime numbers.

Original entry on oeis.org

1, 0, 0, 4, 0, 4, 6, 4, 16, 8, 18, 28, 25, 40, 50, 56, 76, 92, 98, 136, 147, 176, 212, 240, 272, 328, 352, 420, 471, 524, 592, 668, 747, 808, 938, 996, 1127, 1232, 1354, 1456, 1658, 1720, 1966, 2052, 2279, 2408, 2700, 2772, 3144, 3232, 3568, 3740, 4117, 4228, 4722

Offset: 4

Ilya Gutkovskiy, Feb 13 2021

nonn

Robert Israel, Table of n, a(n) for n = 4..10000

Cf. A005171, A018252, A052284, A076608, A340960, A341451, A341462, A341480, A341482, A341483, A341484, A341485, A341486.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
      `if`(isprime(j), 0, b(n-j, t-1)), j=1..n)))
    end:
a:= n-> b(n, 4):
seq(a(n), n=4..58);  # Alois P. Heinz, Feb 13 2021

nmax = 58; CoefficientList[Series[Sum[Boole[!PrimeQ[k]] x^k, {k, 1, nmax}]^4, {x, 0, nmax}], x] // Drop[#, 4] &

G.f. g(x)^4, where g(x) is the G.f. of A005171.

A005451 a(n) = 1 if n is a prime number, otherwise a(n) = n.

Original entry on oeis.org

1, 1, 1, 4, 1, 6, 1, 8, 9, 10, 1, 12, 1, 14, 15, 16, 1, 18, 1, 20, 21, 22, 1, 24, 25, 26, 27, 28, 1, 30, 1, 32, 33, 34, 35, 36, 1, 38, 39, 40, 1, 42, 1, 44, 45, 46, 1, 48, 49, 50, 51, 52, 1, 54, 55, 56, 57, 58, 1, 60

Offset: 1

N. J. A. Sloane

Denominator of (1 + Gamma(n))/n.

Möbius transform of A380441(n). - Wesley Ivan Hurt, Jun 21 2025

Paulo Ribenboim, The little book of big primes, Springer 1991, p. 106.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Achilleas Sinefakopoulos, Problem 10578 (Submitted solution.)
H. S. Wilf, Problem 10578, Amer. Math. Monthly, 104 (1997), 270.

Cf. A005171, A005450 (numerators).

Cf. A088140, A089026, A135684, A181569.

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

[Denominator((1 + Factorial(n-1))/n): n in [1..70]]; // G. C. Greubel, Nov 22 2022

seq(denom((1 + (n-1)!)/n), n=1..80); # G. C. Greubel, Nov 22 2022

Table[If[PrimeQ[n], 1, n], {n, 70}] (* Vincenzo Librandi, Feb 22 2013 *)
a[n_] := ((n-1)! + 1)/n - Floor[(n-1)!/n] // Denominator; Table[a[n] , {n, 70}] (* Jean-François Alcover, Jul 17 2013, after Minac's formula *)
Table[Denominator[(1 + Gamma[n])/n], {n,2,70}] (* G. C. Greubel, Nov 22 2022 *)

def A005451(n):
    if n == 4: return n
    f = factorial(n-1)
    return 1/((f + 1)/n - f//n)
[A005451(n) for n in (1..71)]   # Peter Luschny, Oct 16 2013

[denominator((1+gamma(n))/n) for n in range(1,71)] # G. C. Greubel, Nov 22 2022

Define b(n) = ( (n-1)*(n^2-3*n+1)*b(n-1) - (n-2)^3*b(n-2) )/(n*(n-3)); b(2) = b(3) = 1; a(n) = denominator(b(n)).
a(n) = A088140(n), n >= 3. - R. J. Mathar, Oct 28 2008
a(n) = gcd(n, (n!*n!!)/n^2). - Lechoslaw Ratajczak, Mar 09 2019
From Wesley Ivan Hurt, Jun 21 2025: (Start)
a(n) = n^c(n), where c = A005171.
a(n) = Sum_{d|n} A380441(d) * mu(n/d). (End)

Name edited and a(1)=1 prepended by G. C. Greubel, Nov 22 2022. Name further edited by N. J. A. Sloane, Nov 22 2022

A087624 a(n)=0 if n is prime, A001221(n) otherwise.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 1, 1, 2, 0, 2, 0, 2, 2, 1, 0, 2, 0, 2, 2, 2, 0, 2, 1, 2, 1, 2, 0, 3, 0, 1, 2, 2, 2, 2, 0, 2, 2, 2, 0, 3, 0, 2, 2, 2, 0, 2, 1, 2, 2, 2, 0, 2, 2, 2, 2, 2, 0, 3, 0, 2, 2, 1, 2, 3, 0, 2, 2, 3, 0, 2, 0, 2, 2, 2, 2, 3, 0, 2, 1, 2, 0, 3, 2, 2, 2, 2, 0, 3, 2, 2, 2, 2, 2, 2, 0, 2, 2, 2, 0, 3, 0, 2, 3

Offset: 1

Michele Dondi (bik.mido(AT)tiscalinet.it), Sep 14 2003

Number of prime divisors of n, but excluding n itself if n is prime.
Number of non-associated primes in the ring Z_n.
Also for n > 1 the number of times n is crossed off in the sieve of Eratosthenes (A000040). - Reinhard Zumkeller, Oct 17 2008
Number of primes that are proper divisors of n. - Omar E. Pol, Dec 27 2008

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
Index entries for sequences generated by sieves [From _Reinhard Zumkeller_, Oct 17 2008]

Cf. A001221, A010051, A087625.

A144489 gives partial sums.

a087624 n = if a010051 n == 1 then 0 else a001221 n
-- Reinhard Zumkeller, Apr 05 2013

with(numtheory); f:=proc(n) if isprime(n) then nops(factorset(n))-1 else nops(factorset(n)) fi; end;

Array[If[PrimeQ[#],0,PrimeNu[#]]&,110] (* Harvey P. Dale, Mar 27 2013 *)

a(n) = if (isprime(n), 0, omega(n)); \\ Michel Marcus, Nov 06 2022

a(n) = A001221(n) * A005171(n). - Jason Kimberley, Nov 19 2014
G.f.: Sum_{k>=1} x^(2*prime(k)) / (1 - x^prime(k)). - Ilya Gutkovskiy, Apr 13 2021
a(n) = omega(n) - c(n), where c = A010051. - Wesley Ivan Hurt, Jun 23 2024

Edited by N. J. A. Sloane, Dec 11 2008

A341482 Number of ways to write n as an ordered sum of 5 nonprime numbers.

Original entry on oeis.org

1, 0, 0, 5, 0, 5, 10, 5, 25, 15, 30, 55, 45, 85, 105, 126, 180, 220, 260, 360, 415, 510, 650, 745, 915, 1101, 1270, 1525, 1800, 2045, 2440, 2780, 3225, 3660, 4250, 4771, 5465, 6185, 6930, 7840, 8816, 9790, 11015, 12240, 13505, 15146, 16595, 18385, 20240, 22325, 24255

Offset: 5

Ilya Gutkovskiy, Feb 13 2021

nonn

Cf. A005171, A018252, A052284, A076608, A340961, A341452, A341464, A341480, A341481, A341483, A341484, A341485, A341486.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
      `if`(isprime(j), 0, b(n-j, t-1)), j=1..n)))
    end:
a:= n-> b(n, 5):
seq(a(n), n=5..55);  # Alois P. Heinz, Feb 13 2021

nmax = 55; CoefficientList[Series[Sum[Boole[!PrimeQ[k]] x^k, {k, 1, nmax}]^5, {x, 0, nmax}], x] // Drop[#, 5] &

A341483 Number of ways to write n as an ordered sum of 6 nonprime numbers.

Original entry on oeis.org

1, 0, 0, 6, 0, 6, 15, 6, 36, 26, 45, 96, 75, 156, 201, 242, 375, 456, 586, 816, 987, 1256, 1656, 1962, 2512, 3102, 3717, 4616, 5577, 6612, 8067, 9516, 11283, 13372, 15678, 18378, 21412, 24966, 28719, 33388, 38244, 43872, 50248, 57288, 64914, 74074, 83328, 94248

Offset: 6

Ilya Gutkovskiy, Feb 13 2021

nonn

Cf. A005171, A018252, A052284, A076608, A340962, A341453, A341465, A341480, A341481, A341482, A341484, A341485, A341486.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
      `if`(isprime(j), 0, b(n-j, t-1)), j=1..n)))
    end:
a:= n-> b(n, 6):
seq(a(n), n=6..53);  # Alois P. Heinz, Feb 13 2021

nmax = 53; CoefficientList[Series[Sum[Boole[!PrimeQ[k]] x^k, {k, 1, nmax}]^6, {x, 0, nmax}], x] // Drop[#, 6] &

A141100 Number of unordered pairs of odd composite numbers that sum to 2n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 1, 2, 0, 1, 3, 1, 1, 3, 2, 1, 4, 1, 2, 5, 1, 3, 5, 1, 4, 5, 3, 3, 6, 3, 3, 7, 3, 3, 9, 3, 4, 7, 4, 6, 9, 5, 5, 8, 6, 6, 10, 5, 5, 12, 4, 6, 12, 5, 9, 11, 7, 7, 11, 9, 9, 13, 8, 8, 16, 7, 11, 14, 8, 11, 14, 9, 9, 17, 13, 10, 16, 11, 11, 19, 11, 12, 18, 10

Offset: 1

T. D. Noe, Jun 02 2008, Jun 05 2008

nonn

See A141099 for pairs of odd nonprime numbers. We have a(n) > 0 except for the 14 values of 2n given in A118081.

			a(18)=2 because 36 = 9+27 = 15+21.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A005171, A118081, A141095, A141097, A141099.

Table[cnt=0; Do[If[ !PrimeQ[i] && !PrimeQ[2n-i], cnt++ ], {i,3,n,2}]; cnt, {n,100}]

a(n) = 1 - floor(n/2) + Sum_{i=3..n} c(i) * c(2n-i), n>1, where c = A005171. - Wesley Ivan Hurt, Dec 27 2013

A275647 Decimal expansion of Pi^2/6 - Sum_{k>=1} 1/prime(k)^2.

Original entry on oeis.org

1, 1, 9, 2, 6, 8, 6, 6, 4, 6, 8, 0, 7, 1, 6, 0, 9, 3, 7, 9, 6, 5, 8, 7, 1, 8, 0, 1, 8, 1, 3, 7, 7, 7, 2, 5, 5, 0, 4, 5, 7, 1, 8, 5, 5, 7, 9, 6, 6, 9, 0, 6, 0, 1, 5, 9, 9, 9, 1, 3, 9, 2, 9, 9, 0, 1, 8, 8, 4, 2, 4, 4, 3, 0, 3, 9, 2, 8, 9, 9, 9, 9, 3, 8, 4, 6, 7, 1, 3, 5, 6, 2, 6, 4, 8, 5, 1, 8, 4, 7, 0, 8, 1, 1, 9, 8, 0, 9, 1, 2, 9, 5, 4, 1, 6, 5, 5, 9, 6, 6

Offset: 1

Ilya Gutkovskiy, Aug 04 2016

Decimal expansion of sum of squares of reciprocals of nonprime numbers.
Decimal expansion of the nonprime zeta function at 2.
Continued fraction [1; 5, 5, 3, 1, 2, 2, 6, 2, 2, 4, 1, 1, 93, 2, 1, 1, 5, 3, 5, 3, 2, 1, 2, 6, 1, 4, 5, 1, 34, 1, ...]
More generally, the nonprime zeta function at s equals Sum_{k>=1} (1/k^s - 1/prime(k)^s) = Product_{k>=1} 1/(1 - prime(k)^(-s)) - Sum_{k>=1} 1/prime(k)^s.
Floor(1/(zeta(s)-prime zeta(s)-1)) gives second term in continued fraction for nonprime zeta(s): 5, 36, 187, 852, 3663, 15280, 62692, 254760, 1029279, 4143617, ...
Dirichlet g.f. of A005171: nonprime zeta(s).

			1/1^2 + 1/4^2 + 1/6^2 + 1/8^2 + 1/9^2 + 1/10^2 + ... = 1.192686646807160937965871801813777255045718557966906015999...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Ilya Gutkovskiy, Nonprime zeta function.
Eric Weisstein's World of Mathematics, Riemann Zeta Function 2.
Eric Weisstein's World of Mathematics, Prime Zeta Function.

Cf. A002808, A008683, A013661, A018252, A062312, A085548, A111003, A222171.

RealDigits[Pi^2/6 - PrimeZetaP[2], 10, 120][[1]]
RealDigits[Zeta[2] - PrimeZetaP[2], 10, 120][[1]]

eps()=2.>>bitprecision(1.)
primezeta(s)=my(lm=s*log(2)); lm=lambertw(lm/eps())\lm; sum(k=1,lm, moebius(k)*log(abs(zeta(k*s)))/k)
zeta(2) - primezeta(2) \\ Charles R Greathouse IV, Aug 05 2016

Pi^2/6 - sumeulerrat(1/p, 2) \\ Amiram Eldar, Mar 19 2021

Equals zeta(2) - prime zeta(2) = A013661 - A085548.
Equals Sum_{k>=1} (1 - k*mu(k)*log(zeta(2*k)))/k^2, where mu(k) is the Moebius function (A008683).
Equals Sum_{k>=1} 1/A062312(k).
Equals Sum_{k>=1} 1/A018252(k)^2.
Equals 1 + Sum_{k>=1} 1/A002808(k)^2.
Equals A222171 + A111003 - A085548.

cp's OEIS Frontend

A341466 Number of partitions of n into 7 distinct nonprime parts.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

A341467 Number of partitions of n into 8 distinct nonprime parts.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

A341480 Number of ways to write n as an ordered sum of 3 nonprime numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A341481 Number of ways to write n as an ordered sum of 4 nonprime numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A005451 a(n) = 1 if n is a prime number, otherwise a(n) = n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Magma

Maple

Mathematica

Sage

SageMath

Formula

Extensions

A087624 a(n)=0 if n is prime, A001221(n) otherwise.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

Extensions

A341482 Number of ways to write n as an ordered sum of 5 nonprime numbers.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

A341483 Number of ways to write n as an ordered sum of 6 nonprime numbers.

Views

Author

Keywords