A014076 -id:A014076 - cp's OEIS frontend

A163504 a(n) = abs(n-th prime minus n-th odd nonprime).

1, 6, 10, 14, 14, 14, 16, 16, 16, 16, 18, 14, 14, 14, 16, 12, 10, 14, 10, 10, 12, 8, 8, 4, 2, 2, 2, 4, 6, 4, 8, 10, 14, 14, 20, 18, 22, 22, 24, 28, 32, 28, 36, 34, 36, 34, 42, 52, 52, 52, 50, 54, 54, 62, 62, 62, 66, 66, 70, 72, 70, 78, 90, 92, 92, 92, 100, 102, 110, 106, 108, 112

Offset: 1

Juri-Stepan Gerasimov, Jul 29 2009

nonn

			a(1)=abs(2-1)=1; a(2)=abs(3-9)=6.

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

Cf. A000040, A014076, A129146.

A014076 := proc(n) option remember; local a; 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: A163504 := proc(n) abs(ithprime(n)-A014076(n)) ; end: seq(A163504(n),n=1..80) ; # R. J. Mathar, Oct 10 2009

A014076 := Select[Range[1, 10000, 2], PrimeOmega[#] != 1 &]; Table[Abs[Prime[n] - A014076[[n]]], {n,1,50}] (* G. C. Greubel, Jul 27 2017 *)
Module[{nn=250,onp},onp=Select[Range[1,nn,2],!PrimeQ[#]&];Abs[#[[1]]-#[[2]]]&/@Thread[{Prime[Range[Length[onp]]],onp}]] (* Harvey P. Dale, Dec 07 2024 *)

a(n) = abs(A000040(n)-A014076(n)).

An 8 inserted by R. J. Mathar, Oct 10 2009

A163505 a(n) = (n-th odd nonprime) mod (n-th odd number).

Original entry on oeis.org

0, 0, 0, 0, 7, 5, 7, 5, 5, 7, 7, 5, 5, 3, 5, 3, 3, 5, 3, 3, 3, 1, 1, 46, 46, 48, 52, 1, 1, 58, 58, 58, 58, 58, 60, 62, 62, 66, 66, 66, 66, 70, 70, 72, 72, 74, 76, 76, 78, 78, 82, 82, 82, 82, 86, 90, 90, 90, 90, 90, 92, 92, 92, 92, 92, 94, 98, 100, 100, 104, 104, 104, 104, 106, 106

Offset: 1

Juri-Stepan Gerasimov, Jul 29 2009

nonn

			a(1) =  1 mod 1 = 0;
a(2) =  9 mod 3 = 0;
a(3) = 15 mod 5 = 0;
a(4) = 21 mod 7 = 0;
a(5) = 25 mod 9 = 7.

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

Cf. A005408, A014076.

A014076 := proc(n) option remember; local a; 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:
A163505 := proc(n) A014076(n) mod (2*n-1); end: seq(A163505(n),n=1..80) ; # R. J. Mathar, Oct 10 2009

A014076 := Select[Range[1, 10000, 2], PrimeOmega[#] != 1 &]; Table[Mod[A014076[[n]], 2*n - 1], {n,1,50}] (* G. C. Greubel, Jul 27 2017 *)

a(n) = A014076(n) mod A005408(n-1). [corrected by R. J. Mathar, Oct 10 2009]

Missing term between a(53) and a(54) inserted by G. C. Greubel, Jul 27 2017

A163506 a(n) = n-th odd nonprime * n-th odd number.

Original entry on oeis.org

1, 27, 75, 147, 225, 297, 429, 525, 663, 855, 1029, 1173, 1375, 1539, 1827, 2015, 2277, 2625, 2849, 3159, 3485, 3741, 4095, 4371, 4655, 5049, 5565, 6105, 6555, 6903, 7259, 7623, 7995, 8375, 8901, 9443, 9855, 10575, 11011, 11455, 11907, 12699, 13175, 13833

Offset: 1

Juri-Stepan Gerasimov, Jul 29 2009

nonn

			a(1) =  1*1 =   1;
a(2) =  9*3 =  27;
a(3) = 15*5 =  75;
a(4) = 21*7 = 147;
a(5) = 25*9 = 225.

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

Cf. A005408, A014076.

A014076 := proc(n) option remember; local a; 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:
A163506 := proc(n) A014076(n) *(2*n-1); end: seq(A163506(n),n=1..80) ; # R. J. Mathar, Oct 10 2009

A014076 := Select[Range[1, 10000, 2], PrimeOmega[#] != 1 &]; Table[A014076[[n]]*(2*n - 1), {n, 1, 50}] (* G. C. Greubel, Jul 27 2017 *)

a(n) = A014076(n)*A005408(n-1). - corrected by R. J. Mathar, Oct 10 2009

7279 replaced with 7259 by R. J. Mathar, Oct 10 2009

A163636 The sum of all odd numbers from 2n-1 up to the n-th odd nonprime.

Original entry on oeis.org

1, 24, 60, 112, 153, 171, 253, 275, 336, 448, 525, 555, 640, 672, 828, 864, 969, 1155, 1197, 1320, 1449, 1495, 1632, 1680, 1728, 1875, 2133, 2407, 2580, 2640, 2700, 2760, 2820, 2880, 3069, 3264, 3328, 3672, 3740, 3808, 3876, 4248, 4320, 4551, 4625, 4864

Offset: 1

Juri-Stepan Gerasimov, Aug 02 2009

			a(1)=1. a(2)=3+5+7+9=24. a(3)=5+7+9+11+13+15=60.

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

Cf. A005408, A014076.

A014076 := proc(n) option remember; local a; if n = 1 then 1; else for a from procname(n-1)+2 by 2 do if not isprime(a) then RETURN(a) ; fi; od: fi; end:
A163636 := proc(n) local onpr; onpr := A014076(n) ; (onpr+2*n-1)*(onpr-2*n+3)/4; end: seq(A163636(n),n=1..80) ; # R. J. Mathar, Aug 08 2009

A014076 := Select[Range[1, 10299, 2], PrimeOmega[#] != 1 &]; Table[(A014076[[n]] + 2*n - 1)*(A014076[[n]] - 2*n + 3)/4, {n, 1, 50}] (* G. C. Greubel, Jul 31 2017 *)
Module[{nn=201,onp},onp=Select[Range[1,nn,2],!PrimeQ[#]&];Table[Total[ Range[ 2n-1,onp[[n]],2]],{n,Length[onp]}]] (* Harvey P. Dale, Jul 03 2020 *)

from sympy import primepi
def A163636(n):
    if n == 1: return 1
    m, k, n2 = n-1, primepi(n) + n - 1 + (n>>1), (n<<1)-1
    while m != k:
        m, k = k, primepi(k) + n - 1 + (k>>1)
    return (lambda x: (x+n2)*(x-n2+2)>>2)(m) # Chai Wah Wu, Jul 31 2024

a(n) = A005408(n-1)+A005408(n)+...+A014076(n);

a(n) = ( A014076(n)+2*n-1 ) *( A014076(n)-2*n+3 )/4.

Edited and a(21) corrected by R. J. Mathar, Aug 08 2009

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

A166160 a(n) = (n-th odd prime + n-th odd nonprime)/2 - 1.

Original entry on oeis.org

1, 6, 10, 15, 18, 21, 25, 28, 33, 37, 42, 45, 48, 51, 57, 61, 64, 70, 73, 76, 81, 84, 89, 94, 97, 100, 105, 109, 113, 121, 124, 128, 130, 136, 139, 144, 148, 153, 157, 161, 163, 171, 173, 177, 179, 187, 195, 198, 201, 204, 210, 212, 218, 222, 228, 234, 236, 240, 243

Offset: 1

Juri-Stepan Gerasimov, Oct 09 2009

nonn

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

Cf. A001477, A005097, A014076, A047845, A065091.

A065091 := proc(n) ithprime(n+1) ; 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:
A166160 := proc(n) (A065091(n)+A014076(n))/2-1 ; end proc: seq(A166160(n),n=1..120) ; # R. J. Mathar, May 21 2010

A014076 := Select[Range@250, ! PrimeQ@# && OddQ@# &]; A166160 := Prime[Range[2, 150]]; Table[(A166160[[n]] + A014076[[n]])/2 - 1, {n, 1, 50}] (* G. C. Greubel, Sep 17 2017 *)
Module[{nn=250,op,onp},op=Prime[Range[2,nn]];onp=Select[Range[1,nn,2],!PrimeQ[#]&];Total[#]/2-1&/@Thread[{Take[op,Length[ onp]],onp}]] (* Harvey P. Dale, Mar 02 2025 *)

a(n) = A005097(n) + A047845(n), where A005097 U A047845 = A001477.

a(n) = (A065091(n) + A014076(n))/2 - 1.

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

A166236 Absolute value of (n-th odd prime minus n-th odd nonprime)/2.

Original entry on oeis.org

1, 2, 4, 5, 6, 5, 7, 6, 5, 7, 6, 5, 6, 5, 5, 3, 4, 4, 3, 4, 3, 2, 1, 2, 3, 2, 1, 1, 1, 5, 6, 8, 8, 12, 11, 12, 14, 13, 15, 17, 17, 19, 19, 19, 19, 23, 27, 28, 27, 28, 28, 28, 32, 34, 34, 34, 34, 36, 37, 37, 40, 46, 47, 47, 48, 53, 53, 56, 56, 55, 57, 60, 62, 63, 64, 65, 68, 68, 71, 73

Offset: 1

Juri-Stepan Gerasimov, Oct 09 2009

nonn

			a(1)=abs(3-1)/2=1; a(2)=abs(5-9)/2=abs(-2)=2; a(3)=abs(7-15)/2=abs(-4)=4.

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

Cf. A014076, A065091.

A014076 := Select[Range@500, ! PrimeQ@# && OddQ@# &]; A065091 := Prime[Range[2, 500]]; Table[Abs[A065091[[n]] - A014076[[n]]]/2, {n, 1, 50}] (* G. C. Greubel, Sep 17 2017 *)

a(n) = abs(A065091(n) - A014076(n))/2.

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

A173136 Odd nonprimes n such that exactly one of 2n-1 and 2n+1 is prime.

Original entry on oeis.org

1, 27, 33, 35, 39, 45, 49, 55, 57, 63, 65, 81, 87, 91, 95, 105, 111, 115, 117, 119, 121, 125, 129, 147, 153, 155, 159, 165, 169, 175, 177, 183, 187, 189, 195, 201, 205, 209, 215, 217, 219, 221, 225, 243, 245, 249, 255, 273, 279, 289, 297, 299, 301, 303, 315

Offset: 1

Juri-Stepan Gerasimov, Feb 10 2010

			a(1)=1 because 2*1-1=1 is nonprime and 2*1+1=3 is prime.

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

Cf. A014076, A172458.

Select[Range[1,341,2],!PrimeQ[#]&&Total[Boole[PrimeQ[2 #+{1,-1}]]]==1&] (* Harvey P. Dale, Sep 01 2017 *)

Corrected at 3 or more places by R. J. Mathar, Mar 09 2010

A279051 Sum of odd nonprime divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 16, 1, 1, 10, 1, 1, 22, 1, 1, 1, 26, 1, 37, 1, 1, 16, 1, 1, 34, 1, 36, 10, 1, 1, 40, 1, 1, 22, 1, 1, 70, 1, 1, 1, 50, 26, 52, 1, 1, 37, 56, 1, 58, 1, 1, 16, 1, 1, 94, 1, 66, 34, 1, 1, 70, 36, 1, 10, 1, 1, 116, 1, 78, 40, 1, 1, 118, 1, 1, 22, 86, 1, 88, 1, 1, 70, 92, 1, 94, 1, 96, 1, 1

Offset: 1

Ilya Gutkovskiy, Jan 17 2017

nonn

			a(9) = 10 because 9 has 3 divisors {1, 3, 9} among which 2 are odd nonprime {1, 9} therefore 1 + 9 = 10.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Ilya Gutkovskiy, Extended graphical example
Eric Weisstein's World of Mathematics, Odd Divisor Function
Index entries for sequences related to sums of divisors

Cf. A000593, A014076, A023890, A005069, A033272, A093641.

with(numtheory):
a:= n-> add(`if`(d::even or d::prime, 0, d), d=divisors(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 18 2017

Table[DivisorSum[n, #1 &, Mod[#1, 2] == 1 && ! PrimeQ[#1] &], {n, 97}]
nmax = 97; Rest[CoefficientList[Series[Sum[k x^k/(1 + x^k), {k, 1, nmax}] - Sum[Prime[k] x^Prime[k]/(1 - x^Prime[k]), {k, 2, nmax}], {x, 0, nmax}], x]]

a(n) = sumdiv(n, d, !isprime(d)*(d%2)*d); \\ Michel Marcus, Sep 18 2017

G.f.: A(x) = B(x) - C(x), where B(x) = Sum_{k>=1} k*x^k/(1 + x^k), C(x) = Sum_{k>=2} prime(k)*x^prime(k)/(1 - x^prime(k)).
a(n) = Sum_{d|n, d odd nonprime} d.
a(A093641(n)) = 1.

cp's OEIS Frontend

A163504 a(n) = abs(n-th prime minus n-th odd nonprime).

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A163505 a(n) = (n-th odd nonprime) mod (n-th odd number).

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A163506 a(n) = n-th odd nonprime * n-th odd number.

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A163636 The sum of all odd numbers from 2n-1 up to the n-th odd 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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A166160 a(n) = (n-th odd prime + n-th odd nonprime)/2 - 1.

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