A005171 -id:A005171 - cp's OEIS frontend

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

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] &

A139119 Primes whose binary representation shows the distribution of prime numbers using "0" for primes and "1" for nonprime numbers.

Original entry on oeis.org

2, 37, 149, 599, 153437, 39279991, 628479869, 11056334789265976156021, 3263254052013454238294691704608897001027543, 7524551543123483484068003542235060639999919940760883731360687

Offset: 1

Omar E. Pol, Apr 11 2008

Primes in A118255.
Primes whose binary representation is also the concatenation of the initial terms of A005171, the characteristic function of nonprimes. - Omar E. Pol, Oct 07 2013
a(11) is a 120-digit number 377859...798653. - Robert Price, Apr 03 2019

Cf. A000040, A118255, A118256, A118257, A139101, A139102, A139103, A139104.

Select[Table[FromDigits[Boole /@ Not /@ PrimeQ /@ Range@k, 2], {k, 1, 100}], PrimeQ] (* Federico Provvedi, Oct 07 2013 *)

f(n) = fromdigits(vector(n, k, !isprime(k)), 2); \\ A118255
lista(nn) = for (n=1, nn, if (isprime(p=f(n)), print1(p, ", "))); \\ Michel Marcus, Apr 04 2019

a(8)-a(10) from Donovan Johnson, Oct 07 2013

A191558 a(n) = 0 if n prime, otherwise n.

Original entry on oeis.org

1, 0, 0, 4, 0, 6, 0, 8, 9, 10, 0, 12, 0, 14, 15, 16, 0, 18, 0, 20, 21, 22, 0, 24, 25, 26, 27, 28, 0, 30, 0, 32, 33, 34, 35, 36, 0, 38, 39, 40, 0, 42, 0, 44, 45, 46, 0, 48, 49, 50, 51, 52, 0, 54, 55, 56, 57, 58, 0, 60, 0, 62, 63, 64, 65, 66, 0, 68, 69, 70, 0, 72, 0, 74, 75, 76, 77, 78, 0, 80, 81, 82, 0, 84, 85, 86, 87, 88, 0, 90, 91, 92, 93, 94

Offset: 1

Vincenzo Librandi, Jun 07 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Cf. A002808.

[IsPrime(n) select 0 else n: n in [1..100]]; // Vincenzo Librandi, May 17 2014

A191558 := proc(n) if isprime(n) then 0; else n; end if; end proc:
seq(A191558(n),n=1..80) ; # R. J. Mathar, Jun 11 2011

Table[If[PrimeQ[n], 0, n], {n, 150}] (* Vincenzo Librandi, May 17 2014 *)
(*recurrence*)
Clear[t];
nn = 94;
t[1, 1] = 1;
t[n_, k_] :=
  t[n, k] =
   If[n == k, n*(1 - Product[t[n, k - i], {i, 1, k - 1}]),
    If[n > k, t[n - k, k], 1]];
Table[t[n, n], {n, 1, nn}](* Mats Granvik, Jul 05 2014 *)
Table[n (1 - PrimePi[n] + PrimePi[n - 1]), {n, 50}] (* Wesley Ivan Hurt, Jul 06 2014 *)

a(n) = n * A005171(n) = n - A061397(n). - R. J. Mathar, Jun 11 2011

A347788 Number of compositions (ordered partitions) of n into at most 2 nonprime parts.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 2, 2, 3, 5, 2, 4, 4, 5, 5, 8, 4, 8, 6, 8, 7, 11, 6, 12, 9, 13, 9, 14, 10, 16, 12, 14, 13, 19, 13, 22, 14, 17, 17, 22, 16, 24, 18, 22, 19, 25, 18, 28, 21, 28, 21, 28, 22, 32, 25, 30, 25, 33, 26, 38, 28, 31, 29, 38, 29, 42, 30, 34, 33, 42

Offset: 0

Ilya Gutkovskiy, Sep 13 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A005171, A018252, A052284, A076608, A347796, A347797, A347798, A347799, A358638.

Table[Length@Flatten[Permutations/@IntegerPartitions[n,2,Select[Range@n,!PrimeQ@#&]],1],{n,0,70}] (* Giorgos Kalogeropoulos, Sep 13 2021 *)

A347788(n) = if(n<2,1,!isprime(n)+sum(k=1,n-1,!(isprime(k)+isprime(n-k)))); \\ Antti Karttunen, Nov 25 2022

A051352 a(0) = 0; for n>0, a(n) = a(n-1) + n if n not prime else a(n-1) - n.

Original entry on oeis.org

0, 1, -1, -4, 0, -5, 1, -6, 2, 11, 21, 10, 22, 9, 23, 38, 54, 37, 55, 36, 56, 77, 99, 76, 100, 125, 151, 178, 206, 177, 207, 176, 208, 241, 275, 310, 346, 309, 347, 386, 426, 385, 427, 384, 428, 473, 519, 472, 520, 569, 619, 670, 722, 669, 723, 778

Offset: 0

Armand Turpel armandt(AT)unforgettable.com

Sequence is not monotonic.

Difference between sum of nonprime numbers and prime numbers <= n. - Zak Seidov, Sep 27 2003

William A. Tedeschi, Table of n, a(n) for n = 0..10000

Cf. A005171, A010051.

Cf. A000217, A034387.

a051352 n = a051352_list !! n
a051352_list = 0 : zipWith (+)
   (a051352_list) (zipWith (*) [1..] $ map ((1 -) . (* 2)) a010051_list)
-- Reinhard Zumkeller, Jan 02 2015

A034387 := proc(n)
    option remember;
    if n <= 1 then
        0;
    else
        procname(n-1)+ `if`(isprime(n), n, 0)
    end if;
end proc:
A051352 := proc(n)
    n*(n+1)/2 - 2*A034387(n) ;
end proc:
seq(A051352(n),n=0..40) ; # R. J. Mathar, Jun 26 2024

a[0]=0;a[n_]:=a[n]=If[PrimeQ[n],a[n-1]-n,a[n-1]+n]; Table[a[i], {i,0,60}] (* Harvey P. Dale, Apr 07 2011 *)
nxt[{n_,a_}]:={n+1,If[PrimeQ[n+1],a-n-1,a+n+1]}; NestList[nxt,{0,0},60][[All,2]] (* Harvey P. Dale, Sep 07 2022 *)

a(n) = my(v=primes([1, n])); n*(n+1)/2 -2*vecsum(v); \\ Michel Marcus, Jun 24 2024

a(n) = a(n-1) + n * (1 - 2*A010051(n)) = a(n-1) + n * (2*A005171(n) - 1) = a(n-1) + n * (A005171(n) - A010051(n)). - Reinhard Zumkeller, Nov 25 2009

a(n) = A000217(n) - 2*A034387(n). - Michel Marcus, Jun 24 2024

A118257 Numbers k such that A118255(k) is prime.

Original entry on oeis.org

2, 6, 8, 10, 18, 26, 30, 74, 142, 203, 398, 651, 792, 1314, 3487, 5978, 6240, 7814, 8054, 8673, 21436, 23947, 52985, 91784, 157537, 164901

Offset: 1

Pierre CAMI, Apr 19 2006

A118255(1314) is prime with 396 digits

A118255(23947) is a probable prime with 7209 digits. - Giovanni Resta, Apr 26 2006

			A118255(2) = 2 prime, A118255(6) = 149 prime, A118255(8) = 599 prime.

Cf. A005171, A118255, A118256.

A118255[n_] := Module[{},
   If[n == 1, A118255[1] = 1,
    If[PrimeQ[n], A118255[n] = 2 A118255[n - 1],
     A118255[n] = 2 A118255[n - 1] + 1]]];
Select[Range[5000], PrimeQ[A118255[#]] &] (* Robert Price, Apr 03 2019 *)

a(15)-a(22) from Giovanni Resta, Apr 26 2006

a(23)-a(26) from Michael S. Branicky, Dec 11 2024

A358638 Number of partitions of n into at most 2 distinct nonprime parts.

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 1, 1, 1, 2, 3, 1, 2, 2, 3, 3, 4, 2, 4, 3, 4, 4, 6, 3, 6, 5, 7, 5, 7, 5, 8, 6, 7, 7, 10, 7, 11, 7, 9, 9, 11, 8, 12, 9, 11, 10, 13, 9, 14, 11, 14, 11, 14, 11, 16, 13, 15, 13, 17, 13, 19, 14, 16, 15, 19, 15, 21, 15, 17, 17, 21, 16, 22, 17, 21, 18, 22, 18, 25, 18, 22

Offset: 0

Ilya Gutkovskiy, Nov 24 2022

nonn

Cf. A005171, A018252, A096258, A302479, A347788, A358639, A358640.

A358638(n) = if(n<2,1,!isprime(n)+sum(k=1,(n-1)\2,!(isprime(k)+isprime(n-k)))); \\ Antti Karttunen, Nov 25 2022

For n > 0, a(n) = A005171(n) + A302479(n).

A050374 Number of ordered factorizations of n into composite factors.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 1, 0, 3, 1, 1, 1, 1, 0, 1, 0, 3, 1, 1, 1, 4, 0, 1, 1, 3, 0, 1, 0, 1, 1, 1, 0, 5, 1, 1, 1, 1, 0, 3, 1, 3, 1, 1, 0, 5, 0, 1, 1, 5, 1, 1, 0, 1, 1, 1, 0, 7, 0, 1, 1, 1, 1, 1, 0, 5, 2, 1, 0, 5, 1, 1, 1, 3, 0, 5, 1, 1, 1, 1, 1, 10, 0, 1, 1, 4, 0, 1

Offset: 1

Christian G. Bower, Nov 15 1999

nonn

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

The Dirichlet inverse is given by A005171, all but the first term in A005171 turned negative. - R. J. Mathar, Jul 15 2010

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n

Cf. A002033, A002808, A005171, A050370-A050375, A000045, A002110, A032032.

read(transforms):
[1, seq(-A005171(n), n=2..100)] ;
a050374 := DIRICHLETi(%) ; # R. J. Mathar, May 26 2017

A050374(n) = if(1==n,n,sumdiv(n,d,if(dA050374(d),0))); \\ Antti Karttunen, Oct 20 2017

Dirichlet g.f.: 1/(1-B(s)) where B(s) is D.g.f. of characteristic function of composite numbers.
a(n) = A050375(A101296(n)). - R. J. Mathar, May 26 2017
For n >= 1, a(p^n) = A000045(n-1), for any prime p.
For n >= 0, a(A002110(n)) = A032032(n).

cp's OEIS Frontend

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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Maple

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

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Maple

Mathematica

A139119 Primes whose binary representation shows the distribution of prime numbers using "0" for primes and "1" for nonprime numbers.

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A191558 a(n) = 0 if n prime, otherwise n.

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Magma

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A347788 Number of compositions (ordered partitions) of n into at most 2 nonprime parts.

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A051352 a(0) = 0; for n>0, a(n) = a(n-1) + n if n not prime else a(n-1) - n.

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A118257 Numbers k such that A118255(k) is prime.

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A358638 Number of partitions of n into at most 2 distinct nonprime parts.

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