A052284 -id:A052284 - cp's OEIS frontend

A157424 Triangle read by rows, A157423 * (A052284 * 0^(n-k)).

1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 2, 1, 0, 1, 0, 0, 3, 0, 1, 0, 1, 0, 0, 5, 1, 0, 1, 0, 1, 0, 0, 7, 1, 1, 0, 1, 0, 3, 0, 0, 11, 1, 1, 1, 0, 2, 0, 5, 0, 0, 0, 17, 0, 1, 1, 1, 0, 3, 0, 7, 0, 0, 27, 1, 0, 1, 1, 2, 0, 5, 0, 11, 0, 0, 40

Offset: 1

Gary W. Adamson & Mats Granvik, Feb 28 2009

Row sums = A052284 starting at n=1: (1, 1, 1, 2, 3, 5, 7, 11, 17,...). As a property of eigentriangles, sum of n-th row terms = rightmost term of next row.

			First few rows of the triangle =
1;
0, 1;
0, 0, 1;
1, 0, 0, 1;
0, 1, 0, 0, 2;
1, 0, 1, 0, 0, 3;
0, 1, 0, 1, 0, 0, 5;
1, 0, 1, 0, 2, 0, 0, 7;
1, 1, 0, 1, 0, 3, 0, 0, 11;
1, 1, 1, 0, 2, 0, 5,0, 0, 17;
0, 1, 1, 1, 0, 3, 0, 7, 0, 0, 27;
1, 0, 1, 1, 2, 0, 5, 0, 11, 0, 0, 40;
0, 1, 0, 1, 2, 3, 0, 7, 0, 17, 0, 0, 61;
1, 0, 1, 0, 2, 3, 5, 0, 11, 0, 27, 0, 0, 92;
...
Example: row 5 = (0, 1, 0, 0, 2) = termwise products of (0, 1, 0, 0, 1) and
(1, 1, 1, 1, 2); where (0, 1, 0, 0, 2) = row 5 of triangle A157423 and
(1, 1, 1, 1, 2) = the first 5 terms of A052284.

Cf. A157423, A005171, A052284

Triangle read by rows, A157423 * (A052284 * 0^(n-k)). A157423 = an infinite lower triangular matrix with A005171 in every column. (A052284 * 0^(n-k)) = an infinite lower triangular matrix with A052284: (1, 1, 1, 1, 2, 3, 5, 7, 11, 17, 27,...) as the main diagonal and the rest zeros.

A005171 Characteristic function of nonprimes: 0 if n is prime, else 1.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1

Offset: 1

Russ Cox

Number of orbits of length n in map whose periodic points are A023890. - Thomas Ward
Characteristic function of nonprimes A018252. - Jonathan Vos Post, Dec 30 2007
Triangle A157423 = A005171 in every column. A052284 = INVERT transform of A005171, and the eigensequence of triangle A157423. - Gary W. Adamson, Feb 28 2009

Douglas Hofstadter, Fluid Concepts and Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought.

Antti Karttunen, Table of n, a(n) for n = 1..65537
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
Yash Puri and Thomas Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Yash Puri and Thomas Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.
Index entries for characteristic functions

Cf. A010051, A018252, A023890.

Cf. A157423, A157424, A052284, A050374, A033273, A008683.

a005171 = (1 -) . a010051  -- Reinhard Zumkeller, Mar 30 2014

A005171 := proc(n)
    if isprime(n) then
        0 ;
    else
        1 ;
    end if;
end proc: # R. J. Mathar, May 26 2017

a[n_] := If[PrimeQ@ n, 0, 1]; Array[a, 105] (* Robert G. Wilson v, Jun 20 2011 *)
nn = 105; t[n_, k_] :=  t[n, k] = If[n == k, 1, If[k == 1, 1 - Product[t[n, k + i], {i, 1, n - 1}], If[Mod[n, k] == 0, t[n/k, 1], 1], 1]]; Table[t[n, 1], {n, 1, nn}] (* Mats Granvik, Sep 21 2013 *)

a(n)=if(n<1, 0, !isprime(n)) /* Michael Somos, Jun 08 2005 */

from sympy import isprime
def a(n): return int(not isprime(n))
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Oct 28 2021

a(n) = (1/n)* Sum_{ d divides n } mu(d)*A023890(n/d). E.g., a(6) = 1 since the 6th term of A023890 is 7 and the first term is 1. [edited by Michel Marcus, Dec 14 2023]
a(n) = 1 - A010051(n). - Jonathan Vos Post, Dec 30 2007
a(n) equals the first column in a table T defined by the recurrence: If n = k then T(n,k) = 1 else if k = 1 then T(n,k) = 1 - Product_{k divides n} of T(n,k), else if k divides n then T(n,k) = T(n/k,1). This is true since T(n,k) = 0 when k divides n and n/k is prime which results in Product_{k divides n} = 0 for the composite numbers and where k ranges from 2 to n. Therefore there is a remaining 1 in the expression 1-Product_{k divides n}, in the first column. Provided below is a Mathematica program as an illustration. - Mats Granvik, Sep 21 2013
a(n) = A057427(A239968(n)). - Reinhard Zumkeller, Mar 30 2014
a(n) = Sum_{d|n} A033273(d)*A008683(n/d). - Ridouane Oudra, Jul 03 2025

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.

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] &

A341484 Number of ways to write n as an ordered sum of 7 nonprime numbers.

Original entry on oeis.org

1, 0, 0, 7, 0, 7, 21, 7, 49, 42, 63, 154, 119, 259, 357, 420, 707, 861, 1169, 1666, 2072, 2752, 3703, 4557, 5999, 7637, 9422, 12089, 14931, 18354, 22904, 27825, 33866, 41328, 49539, 59753, 71386, 85071, 100800, 119455, 140448, 164794, 193179, 224826, 261464, 303422

Offset: 7

Ilya Gutkovskiy, Feb 13 2021

nonn

Cf. A005171, A018252, A052284, A076608, A340963, A341454, A341466, A341480, A341481, A341482, A341483, 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, 7):
seq(a(n), n=7..52);  # Alois P. Heinz, Feb 13 2021

nmax = 52; CoefficientList[Series[Sum[Boole[!PrimeQ[k]] x^k, {k, 1, nmax}]^7, {x, 0, nmax}], x] // Drop[#, 7] &

A341485 Number of ways to write n as an ordered sum of 8 nonprime numbers.

Original entry on oeis.org

1, 0, 0, 8, 0, 8, 28, 8, 64, 64, 84, 232, 182, 400, 596, 680, 1232, 1520, 2128, 3144, 3970, 5504, 7532, 9584, 12945, 16920, 21464, 28288, 35778, 45264, 57856, 72024, 90036, 112456, 138140, 170600, 208874, 254192, 309088, 373584, 449731, 539408, 645584, 767776

Offset: 8

Ilya Gutkovskiy, Feb 13 2021

nonn

Cf. A005171, A018252, A052284, A076608, A340964, A341455, A341467, A341480, A341481, A341482, A341483, A341484, 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, 8):
seq(a(n), n=8..51);  # Alois P. Heinz, Feb 13 2021

nmax = 51; CoefficientList[Series[Sum[Boole[!PrimeQ[k]] x^k, {k, 1, nmax}]^8, {x, 0, nmax}], x] // Drop[#, 8] &

A341486 Number of ways to write n as an ordered sum of 9 nonprime numbers.

Original entry on oeis.org

1, 0, 0, 9, 0, 9, 36, 9, 81, 93, 108, 333, 270, 585, 945, 1047, 2016, 2547, 3612, 5571, 7101, 10227, 14256, 18621, 25830, 34497, 44955, 60610, 78480, 101754, 133092, 169380, 217008, 276852, 347967, 439272, 549786, 683244, 849528, 1047678, 1288017, 1577934

Offset: 9

Ilya Gutkovskiy, Feb 13 2021

nonn

Cf. A005171, A018252, A052284, A076608, A340965, A341457, A341468, A341480, A341481, A341482, A341483, A341484, A341485.

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, 9):
seq(a(n), n=9..50);  # Alois P. Heinz, Feb 13 2021

nmax = 50; CoefficientList[Series[Sum[Boole[!PrimeQ[k]] x^k, {k, 1, nmax}]^9, {x, 0, nmax}], x] // Drop[#, 9] &

A224344 Number T(n,k) of compositions of n using exactly k primes (counted with multiplicity); triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 2, 5, 1, 3, 8, 5, 5, 13, 13, 1, 7, 23, 27, 7, 11, 39, 52, 25, 1, 17, 65, 99, 66, 9, 27, 106, 186, 151, 41, 1, 40, 177, 340, 323, 133, 11, 61, 293, 608, 666, 358, 61, 1, 92, 482, 1076, 1330, 867, 236, 13, 142, 781, 1894, 2581, 1971, 737, 85, 1

Offset: 0

Alois P. Heinz, May 23 2013

			A(5,1) = 8: [2,1,1,1], [1,2,1,1], [1,1,2,1], [1,1,1,2], [3,1,1], [1,3,1], [1,1,3], [5].
Triangle T(n,k) begins:
   1;
   1;
   1,   1;
   1,   3;
   2,   5,   1;
   3,   8,   5;
   5,  13,  13,   1;
   7,  23,  27,   7;
  11,  39,  52,  25,   1;
  17,  65,  99,  66,   9;
  27, 106, 186, 151,  41,  1;
  40, 177, 340, 323, 133, 11;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Column k=0 gives: A052284.
Row sums are: A011782.
Row lengths are: A008619.
T(floor(n/2)) = A093178(n).
T(2n,n-1) = A001844(n-1) for n>0.
Cf. A000040, A000079, A004526, A102291, A121303, A222656.

T:= proc(n) option remember; local j; if n=0 then 1
      else []; for j to n do zip((x, y)->x+y, %,
      [`if`(isprime(j), 0, NULL), T(n-j)], 0) od; %[] fi
    end:
seq(T(n), n=0..16);

zip[f_, x_List, y_List, z_] :=  With[{m = Max[Length[x], Length[y]]},  Thread[f[PadRight[x, m, z], PadRight[y, m, z]]]]; T[n_] := T[n] =  Module[{j, pc}, If[n == 0, {1}, pc = {}; For[j = 1, j <= n, j++, pc = zip[Plus, pc, Join[If[PrimeQ[j], {0}, {}], T[n-j]], 0]]; pc]]; Table[T[n], {n, 0, 16}] // Flatten (* Jean-François Alcover, Jan 29 2014, after Alois P. Heinz *)

Sum_{k=1..floor(n/2)} k * T(n,k) = A102291(n).

cp's OEIS Frontend

A157424 Triangle read by rows, A157423 * (A052284 * 0^(n-k)).

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A005171 Characteristic function of nonprimes: 0 if n is prime, else 1.

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

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

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

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

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A341484 Number of ways to write n as an ordered sum of 7 nonprime numbers.

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A341485 Number of ways to write n as an ordered sum of 8 nonprime numbers.

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A341486 Number of ways to write n as an ordered sum of 9 nonprime numbers.

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