A038529 -id:A038529 - cp's OEIS frontend

A014237 a(n) = (n-th prime) - (n-th nonprime).

1, -1, -1, -1, 2, 3, 5, 5, 8, 13, 13, 17, 20, 21, 23, 28, 33, 34, 39, 41, 41, 46, 49, 54, 61, 63, 64, 67, 67, 69, 82, 85, 89, 90, 99, 100, 105, 109, 112, 117, 122, 123, 131, 131, 134, 135, 146, 157, 159, 160, 163, 167, 167, 176, 181, 186, 191, 191, 196

Offset: 1

Clark Kimberling

sign

a(n) = A000040(n) - A018252(n). - Reinhard Zumkeller, Apr 30 2014

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A038529, A026233, A127118.

a014237 n = a000040 n - a018252 n  -- Reinhard Zumkeller, Apr 30 2014

nonPrime[n_Integer] := FixedPoint[n + PrimePi@# &, n+PrimePi@n];
Table[Prime[n] - nonPrime[n], {n, 1, 70}] (* G. C. Greubel, Jun 22 2019 *)

from sympy import prime, composite
def A014237(n):
    return 1 if n == 1 else prime(n)-composite(n-1) # Chai Wah Wu, Dec 27 2018

A067563 Product of n-th prime number and n-th composite number.

Original entry on oeis.org

8, 18, 40, 63, 110, 156, 238, 285, 368, 522, 620, 777, 902, 1032, 1175, 1378, 1593, 1708, 2010, 2272, 2409, 2686, 2905, 3204, 3686, 3939, 4120, 4494, 4796, 5085, 5842, 6288, 6713, 6950, 7599, 7852, 8478, 8965, 9352, 9861, 10382, 10860, 11842, 12159

Offset: 1

Rick L. Shepherd, Jan 29 2002

			E.g. a(4)=63 because the fourth prime is 7 and the fourth composite is 9.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A002808, A064799.

Cf. A127118, A038529.

a067563 n = a000040 n * a002808 n  -- Reinhard Zumkeller, Apr 30 2014

Module[{nn=50,prs,comps,len},prs=Prime[Range[nn]];comps=Complement[ Range[ 4,3nn],prs];len=Min[nn,Length[comps]];Times@@@Thread[ {Take[ prs,len], Take[comps,len]}]](* Harvey P. Dale, Sep 02 2015 *)

from sympy import prime, composite
def A067563(n):
    return prime(n)*composite(n) # Chai Wah Wu, Dec 27 2018

a(n) = prime(n) * composite(n).

A066277 Primes p(m) such that a prime number q exists so that p(m)-q = c(m), the m-th composite number.

Original entry on oeis.org

2, 3, 5, 7, 17, 23, 29, 31, 41, 43, 67, 89, 97, 131, 139, 157, 281, 311, 313, 331, 353, 379, 401, 431, 449, 499, 569, 571, 607, 631, 683, 733, 743, 751, 787, 829, 881, 883, 947, 967, 983, 1033, 1091, 1117, 1123, 1151, 1301, 1303, 1327, 1373, 1543, 1559

Offset: 1

Labos Elemer, Dec 10 2001

nonn

Number of terms < 10^k: 4, 13, 41, 177, 1119, 6963, 48647, 359109, 2766164, ..., . - Robert G. Wilson v, Dec 11 2017

			p(25) = A000040(25) = 97; 97 - 61 = A002808(25) = c(25) = 38 and 61 is prime.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000

Cf. A000040, A002808, A038529, A060253.

Composite[n_Integer] := FixedPoint[n + PrimePi@# +1 &, n + PrimePi@n +1]; fQ[n_] := PrimeQ[Prime@n - Composite@n]; Prime@ Select[ Range@250, fQ] (* Robert G. Wilson v, Dec 11 2017 *)

a(n) = prime(A060253(n)) or A000040(A060253(n)). - Michel Marcus, Dec 11 2017

A060253 Numbers n such that difference between n-th prime and n-th composite number is prime.

Original entry on oeis.org

1, 2, 3, 4, 7, 9, 10, 11, 13, 14, 19, 24, 25, 32, 34, 37, 60, 64, 65, 67, 71, 75, 79, 83, 87, 95, 104, 105, 111, 115, 124, 130, 132, 133, 138, 145, 152, 153, 161, 163, 166, 174, 182, 187, 188, 190, 212, 213, 217, 220, 243, 246, 251, 255, 257, 264, 275, 279, 281

Offset: 1

Robert G. Wilson v, Mar 22 2001

			n=10: p(10)=29, c(10)=18, c(10)-p(10)=11, so 10=a(7) is here.

Numbers n such that A038529(n) is prime. Cf. A000040, A002808.

f[ n_Integer ] := Block[ {k = n + PrimePi[ n ] + 1}, While[ k - PrimePi[ k ] - 1 != n, k++ ]; k ]; Select[ Range[ 500 ], PrimeQ[ Prime[ # ] - f[ # ] ] & ]
Module[{nn=2000,pr,cm,len,th},pr=Prime[Range[PrimePi[nn]]];cm=Select[ Range[ nn],CompositeQ];len=Min[Length[pr],Length[cm]];th=Thread[{Take[ pr,len],Take[ cm,len]}];Position[th,?(PrimeQ[Abs[#[[1]]-#[[2]]]]&)]]// Quiet//Flatten (* _Harvey P. Dale, Jun 29 2020 *)

Values of n such that A000040(n)-A002808(n)=p(n)-c(n) is a prime number.

A065870 n-th prime - n-th semiprime.

Original entry on oeis.org

-2, -3, -4, -3, -3, -2, -4, -3, -2, 3, -2, 3, 6, 5, 8, 7, 10, 10, 12, 14, 15, 17, 18, 20, 23, 24, 21, 22, 23, 26, 36, 38, 43, 44, 43, 40, 42, 45, 48, 52, 57, 58, 62, 60, 63, 58, 69, 80, 82, 83, 78, 81, 82, 90, 91, 94, 92, 93, 94, 96, 96, 99, 106, 109, 110, 112, 125, 128, 134, 135, 138, 142, 149, 154, 158, 157, 154, 160, 154, 160

Offset: 1

Robert A. Stump (bee_ess107(AT)yahoo.com), Dec 07 2001

			Prime_1 - semiprime_1 = 2 - 4 = -2, prime_2 - semiprime_2 = 3 - 6 = -3, prime_3 - semiprime_3 = 5 - 9 = -4, etc.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000040, A001358, A038529 (n-th prime - n-th composite).

  nn=250;Module[{pr=Prime[Range[nn]],sp=Select[Range[nn],PrimeOmega[ #]==2&],len}, len=Min[nn,Length[sp]];#[[1]]-#[[2]]&/@ Thread[ {Take[ pr,len],Take[sp,len]}]](* Harvey P. Dale, Aug 14 2013 *)

m=0; for (n=4, 250, if (bigomega(n) == 2, m = m + 1; print1(prime(m) - n,",")))

n=m=0; for (k=1, 10^9, if (bigomega(k) != 2, next); m = m + 1; write("b065870.txt", n++, " ", prime(m) - k); if (n==1000, return) ) \\ Harry J. Smith, Nov 02 2009

a(n) = A000040(n) - A001358(n). - Zak Seidov, Apr 29 2015

More terms from Rick L. Shepherd, Mar 09 2002

A071261 Smallest k such that |p(k)-c(k)| is an n-digit number where p(k) is the k-th prime and c(k) is the k-th composite number.

Original entry on oeis.org

1, 10, 37, 206, 1401, 10626, 85316, 712597, 6117420, 53593619, 476889480, 4295913223, 39084524707, 358521889961, 3311439169717, 30765562102926, 287282165065268, 2694418859703791, 25368958219747617

Offset: 1

Amarnath Murthy, May 30 2002

Cf. A038529.

Corrected and extended by Gabriel Cunningham (gcasey(AT)mit.edu), Apr 08 2003
More terms from Sean A. Irvine, Nov 30 2009
a(12)-a(15) from Donovan Johnson, Dec 30 2010
a(16)-a(19) from Chai Wah Wu, Apr 17 2018

A072476 Difference between the sum of first n prime numbers and the sum of first n composite numbers.

Original entry on oeis.org

-2, -5, -8, -10, -9, -8, -5, -1, 6, 17, 28, 44, 63, 82, 104, 131, 163, 196, 233, 272, 312, 357, 405, 458, 517, 579, 642, 707, 772, 840, 921, 1004, 1092, 1181, 1279, 1378, 1481, 1589, 1700, 1816, 1937, 2058, 2187, 2317, 2450, 2584, 2729, 2884, 3042, 3201, 3362, 3527, 3693, 3868, 4048

Offset: 1

Amarnath Murthy, Jun 20 2002

sign

Cf. A071411.

Partial sums of A038529.

Composite[n_Integer] := FixedPoint[n + PrimePi[ # ] + 1 &, n + PrimePi[n] + 1]; Table[ Sum[ Prime[i] - Composite[i], {i, 1, n}], {n, 1, 55}]
Module[{nn=60,pr,cm},pr=Prime[Range[nn]];cm=Take[Select[Range[2nn], CompositeQ], nn]; Accumulate[ pr-cm]](* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 08 2017 *)

Edited by Robert G. Wilson v, Jun 21 2002

cp's OEIS Frontend

A014237 a(n) = (n-th prime) - (n-th nonprime).

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A067563 Product of n-th prime number and n-th composite number.

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A066277 Primes p(m) such that a prime number q exists so that p(m)-q = c(m), the m-th composite number.

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A060253 Numbers n such that difference between n-th prime and n-th composite number is prime.

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A065870 n-th prime - n-th semiprime.

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A071261 Smallest k such that |p(k)-c(k)| is an n-digit number where p(k) is the k-th prime and c(k) is the k-th composite number.

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A072476 Difference between the sum of first n prime numbers and the sum of first n composite numbers.

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