A163300 -id:A163300 - cp's OEIS frontend

A163301 a(n) = Sum_{x=n-th even nonprime..n-th odd nonprime} -x*(-1)^x.

1, 3, 5, 7, 8, 8, 10, 10, 11, 13, 14, 14, 15, 15, 17, 17, 18, 20, 20, 21, 22, 22, 23, 23, 23, 24, 26, 28, 29, 29, 29, 29, 29, 29, 30, 31, 31, 33, 33, 33, 33, 35, 35, 36, 36, 37, 38, 38, 39, 39, 41, 41, 41, 41, 43, 45, 45, 45, 45, 45, 46, 46, 46, 46, 46, 47, 49, 50, 50, 52, 52

Offset: 1

Juri-Stepan Gerasimov, Jul 26 2009

nonn

Here n-th even nonprime = A163300(n), n-th odd nonprime = A014076(n) and A163300 U A014076 = A141468.

Seems to be essentially the same as A008508. - R. J. Mathar, May 30 2025

			a(1) = -0*(-1)^0 - 1*(-1)^1 = 0 + 1 = 1;
a(2) = -4*(-1)^4 - 5*(-1)^5 - 6*(-1)6 - 7*(-1)^7 - 8*(-1)^8 - 9*(-1)^9 = -4 + 5 - 6 + 7 - 8 + 9 = 3.

Cf. A014076, A141468, A163300.

A163300 := proc(n) if n <= 2 then op(n,[0,4]) ; else for a from procname(n-1)+2 by 2 do if not isprime(a) then return a; end if; end do; end if; end proc:
A014076 := proc(n) if n = 1 then 1; else for a from procname(n-1)+2 by 2 do if not isprime(a) then return a ; end if; end do: end if; end proc:
A001057 := proc(n) (1-(-1)^n*(2*n+1))/4; end proc:
A163301 := proc(n) A001057( A014076(n)) - A001057(A163300(n)-1) ; end proc: seq(A163301(n),n=1..120) ; # R. J. Mathar, May 21 2010

a(n) = Sum_{x=A163300(n)..A014076(n)}-x*(-1)^x.

a(n) = A001057( A014076(n)) - A001057(A163300(n)-1). - R. J. Mathar, May 21 2010

Corrected from a(39) onwards by R. J. Mathar, May 21 2010

A163750 a(n) = (n-th even nonprime mod n-th prime).

Original entry on oeis.org

0, 1, 1, 1, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 132

Offset: 1

Juri-Stepan Gerasimov, Aug 03 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000040, A163300.

A163300 := proc(n) if n = 1 then 0; else for a from procname(n-1)+2 by 2 do if not isprime(a) then return(a) ; end if; end do: end if; end proc: A163750 := proc(n) A163300(n) mod ithprime(n) ; end proc: seq(A163750(n),n=1..120) ; # R. J. Mathar, Oct 10 2009

A087156[n_] := Mod[n, DivisorSigma[1, n]]; Table[Mod[2*A087156[n], Prime[n]], {n, 1, 50}] (* G. C. Greubel, Aug 02 2017 *)

a(n) = (A163300(n) mod A000040(n)).

A163751 a(n) = n-th even nonprime minus n-th nonprime.

Original entry on oeis.org

0, 3, 2, 2, 2, 3, 4, 4, 4, 5, 6, 6, 6, 7, 8, 8, 9, 10, 11, 12, 12, 12, 13, 14, 15, 16, 16, 17, 18, 18, 18, 19, 20, 20, 21, 22, 23, 24, 24, 25, 26, 27, 28, 28, 28, 29, 30, 31, 32, 32, 33, 34, 34, 34, 35, 36, 37, 38, 38, 39, 40, 40, 41, 42, 43, 44, 44, 45, 46, 47, 48, 49, 50, 50, 51

Offset: 1

Juri-Stepan Gerasimov, Aug 03 2009

nonn

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

Cf. A141468, A163300.

nonPrime[n_Integer] := FixedPoint[n + PrimePi@# &, n + PrimePi@n];
A087156[n_] := Mod[n, DivisorSigma[1, n]]; A141468 := Array[nonPrime, 60, 0]; Table[2*A087156[n] - A141468[[n]], {n, 1, 50}] (* G. C. Greubel, Aug 02 2017 *)

a(n) = (A163300(n) - A141468(n)).

Entries checked by R. J. Mathar, May 21 2010

A165955 n-th odd nonprime plus n-th even nonprime.

Original entry on oeis.org

1, 13, 21, 29, 35, 39, 47, 51, 57, 65, 71, 75, 81, 85, 93, 97, 103, 111, 115, 121, 127, 131, 137, 141, 145, 151, 159, 167, 173, 177, 181, 185, 189, 193, 199, 205, 209, 217, 221, 225, 229, 237, 241, 247, 251, 257, 263, 267, 273, 277, 285, 289, 293, 297, 305, 313, 317, 321, 325

Offset: 1

Juri-Stepan Gerasimov, Oct 01 2009

nonn

			a(1) = 1+0 = 1. a(2) = 9+4 = 13. a(3) = 15+6 = 21.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A014076, A141468, A163300.

A014076 := Select[Range@500, ! PrimeQ@# && OddQ@# &]; A163300 := Drop[Range[0, 500, 2], {2}]; Table[(A163300[[n]] + A014076[[n]]), {n, 1, 50}] (* G. C. Greubel, Sep 17 2017 *)
Module[{nn=300,np,enp,onp,len},np=Select[Range[0,nn],!PrimeQ[#]&];enp= Select[ np,EvenQ];onp=Select[np,OddQ];len=Min[Length[enp], Length[ onp]]; Total/@Thread[{Take[enp,len],Take[onp,len]}]] (* Harvey P. Dale, Nov 28 2018 *)

from sympy import primepi
def A165955(n):
    if n == 1: return 1
    m, k = n-1, primepi(n) + n - 1 + (n>>1)
    while m != k:
        m, k = k, primepi(k) + n - 1 + (k>>1)
    return m+(n<<1) # Chai Wah Wu, Aug 01 2024

a(n) = A014076(n) + A163300(n).

Entries checked by R. J. Mathar, Oct 10 2009

A165971 The n-th odd nonprime minus the n-th even nonprime.

Original entry on oeis.org

1, 5, 9, 13, 15, 15, 19, 19, 21, 25, 27, 27, 29, 29, 33, 33, 35, 39, 39, 41, 43, 43, 45, 45, 45, 47, 51, 55, 57, 57, 57, 57, 57, 57, 59, 61, 61, 65, 65, 65, 65, 69, 69, 71, 71, 73, 75, 75, 77, 77, 81, 81, 81, 81, 85, 89, 89, 89, 89, 89, 91, 91, 91, 91, 91, 93, 97, 99, 99, 103

Offset: 1

Juri-Stepan Gerasimov, Oct 02 2009

			a(1) = 1-0 = 1. a(2) = 9-4 = 5. a(3) = 15-6 = 9.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A014076, A163300, A141468, A160522.

A014076 := Select[Range@500, ! PrimeQ@# && OddQ@# &]; A163300 := Drop[Range[0, 500, 2], {2}]; Table[(-A163300[[n]] + A014076[[n]]), {n, 1, 50}] (* G. C. Greubel, Sep 17 2017 *)

from sympy import primepi
def A165971(n):
    if n == 1: return 1
    m, k = n-1, primepi(n) + n - 1 + (n>>1)
    while m != k:
        m, k = k, primepi(k) + n - 1 + (k>>1)
    return m-(n<<1) # Chai Wah Wu, Aug 01 2024

a(n) = A014076(n) - A163300(n).

Equals: {1} U A160522.

77 duplicated by R. J. Mathar, Oct 10 2009

A171576 a(n) = abs(n-th noncomposite number minus n-th even nonprime number).

Original entry on oeis.org

1, 2, 3, 3, 3, 1, 1, 1, 1, 3, 7, 7, 11, 13, 13, 15, 19, 23, 23, 27, 29, 29, 33, 35, 39, 45, 47, 47, 49, 49, 51, 63, 65, 69, 69, 77, 77, 81, 85, 87, 91, 95, 95, 103, 103, 105, 105, 115, 125, 127, 127, 129, 133, 133, 141, 145, 149, 153, 153, 157, 159, 159, 167, 179, 181

Offset: 1

Juri-Stepan Gerasimov, Dec 12 2009

nonn

			a(1) = abs(1-0) = 1, a(2) = abs(2-4) = 2, a(3) = abs(3-6) = 3.

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

Cf. A008578 (noncomposite), A163300 (even nonprime).

Module[{nn=400,nc,enp,len},nc=Select[Range[nn],!CompositeQ[#]&];enp= Join[{0},Range[4,nn,2]];len=Min[Length[nc],Length[enp]];Abs[#[[1]]- #[[2]]]&/@Thread[{Take[nc,len],Take[enp,len]}]] (* Harvey P. Dale, Mar 01 2020 *)

a(n) = abs(A008578(n) - A163300(n)).

Corrected (a 67 replaced by another 69) by R. J. Mathar, May 22 2010

A174009 Numbers n such that A174008(k)=n-th prime.

Original entry on oeis.org

1, 4, 5, 11, 13, 16, 18, 19, 20, 25, 33, 37, 38, 39, 40, 48, 52, 59, 60, 69, 72, 73, 76, 79, 84, 85, 86, 87, 96, 98, 104, 110, 117, 122, 135, 136, 140, 142, 145, 151, 153, 160, 162, 173, 179, 183, 186, 191, 192, 199, 200, 206, 214, 218, 221, 226, 232, 234, 239, 242

Offset: 1

Juri-Stepan Gerasimov, Mar 05 2010

nonn

n-th prime in sequence A174008.

			a(1)=1 because A174008(1)=2=1st prime;
a(2)=4 because A174008(2)=7=4th prime;
a(3)=5 because A174008(3)=11=5th prime;
a(4)=11 because A174008(7)=31=11th prime.

Cf. A000040, A001477, A078916, A163300, A174008.

From R. J. Mathar, Apr 28 2010: (Start)
A163300 := proc(n) option remember ; if n = 1 then 0; else for a from procname(n-1)+2 by 2 do if not isprime(a) then return a ; end if; end do; end if; end proc:
A174008 := proc(n) ithprime(n)+A163300(n) ; end proc:
A174009 := proc(k) p := A174008(k) ; if isprime(p) then printf("%d,", numtheory[pi](p) ) ; end if; return ; end proc:
seq(A174009(k),k=1..400 ) ; (End)

More terms from R. J. Mathar, Apr 28 2010

A171575 n-th noncomposite number plus n-th even nonprime number.

Original entry on oeis.org

1, 6, 9, 13, 17, 23, 27, 33, 37, 43, 51, 55, 63, 69, 73, 79, 87, 95, 99, 107, 113, 117, 125, 131, 139, 149, 155, 159, 165, 169, 175, 191, 197, 205, 209, 221, 225, 233, 241, 247, 255, 263, 267, 279, 283, 289, 293, 307, 321, 327, 331, 337, 345, 349, 361, 369, 377

Offset: 1

Juri-Stepan Gerasimov, Dec 12 2009

nonn

			a(1) = 1 + 0 = 1, a(2) = 2 + 4 = 6, a(3) = 3 + 6 = 9.

Cf. A008578(the noncomposite numbers), A163300(the even nonprime numbers).

Module[{nc=Select[Range[300],!CompositeQ[#]&],len},len=Length[nc];Join[ {1},Rest[Total/@Thread[{nc,Range[2,2len,2]}]]]] (* Harvey P. Dale, Mar 08 2018 *)

a(n) = A008578(n) + A163300(n).

Entries checked by R. J. Mathar, May 23 2010

A174184 Prime(n)+even nonprime(n) is prime.

Original entry on oeis.org

1, 2, 3, 7, 9, 11, 12, 13, 14, 18, 24, 27, 28, 29, 30, 36, 38, 43, 44, 53, 54, 55, 57, 60, 63, 64, 65, 66, 72, 74, 80, 84, 90, 93, 102, 103, 108, 110, 111, 117, 118, 125, 126, 135, 138, 141, 143, 148, 150, 155, 156, 162, 165, 171, 174, 180, 183, 186, 188, 190, 198

Offset: 1

Juri-Stepan Gerasimov, Mar 11 2010

nonn

Unit together with prime(n)+s*n is prime, s=2.

			1 is in the sequence because A000040(1) + A163300(1) = 2 (1st prime) + 0 (1st even nonprime) is prime;
2 is in the sequence because A000040(2) + A163300(2) = 3 (2nd prime) + 4 (2nd even nonprime) is prime.

Cf. A000040, A076297, A163300.

From R. J. Mathar, Apr 20 2010: (Start)
A163300 := proc(n) option remember ; if n = 1 then 0; else for a from procname(n-1)+2 by 2 do if not isprime(a) then return a ; end if; end do; end if; end proc:
for n from 1 to 200 do if isprime( ithprime(n) + A163300(n)) then printf("%d,",n) ; end if; end do: (End)

a(n+1)=A076297(n).

Entries checked by R. J. Mathar, Apr 20 2010

A174185 Numbers k such that the k-th prime minus the k-th even nonprime is prime.

Original entry on oeis.org

1, 7, 8, 9, 12, 15, 19, 20, 21, 23, 24, 25, 30, 32, 33, 34, 35, 36, 37, 39, 41, 42, 45, 46, 48, 51, 56, 63, 67, 71, 75, 78, 81, 82, 85, 86, 88, 89, 90, 96, 102, 107, 112, 115, 116, 117, 120, 121, 123, 126, 128, 132, 135, 137, 146, 150, 152, 153, 156, 158, 159, 163, 164

Offset: 1

Juri-Stepan Gerasimov, Mar 11 2010

nonn

Numbers k such that prime(k) - (even nonprime)(k) is prime.

			1 is a term because A000040(1) - A163300(1) = 2 (prime);
7 is a term because A000040(7) - A163300(7) = 3 (prime).

Cf. A000040, A163300.

A163300 := proc(n) option remember ; if n = 1 then 0; else for a from procname(n-1)+2 by 2 do if not isprime(a) then return a ; end if; end do; end if; end proc:
for n from 1 to 300 do if isprime( ithprime(n) - A163300(n)) then printf("%d,",n) ; end if; end do: # R. J. Mathar, Apr 20 2010

Corrected (11 replaced with 12) by R. J. Mathar, Apr 20 2010

cp's OEIS Frontend

A163301 a(n) = Sum_{x=n-th even nonprime..n-th odd nonprime} -x*(-1)^x.

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A163750 a(n) = (n-th even nonprime mod n-th prime).

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A163751 a(n) = n-th even nonprime minus n-th nonprime.

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A165955 n-th odd nonprime plus n-th even nonprime.

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A165971 The n-th odd nonprime minus the n-th even nonprime.

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A171576 a(n) = abs(n-th noncomposite number minus n-th even nonprime number).

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A174009 Numbers n such that A174008(k)=n-th prime.

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A171575 n-th noncomposite number plus n-th even nonprime number.

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