A141468 -id:A141468 - cp's OEIS frontend

A161672 Half the product of all numbers from A141468(n) up to prime(n).

0, 3, 10, 21, 3960, 77220, 490089600, 1523733120, 2075793350400, 50710801232931840000, 3144069676441774080000, 19348101737674447004467200000, 137500509682406403378413568000000

Offset: 1

Juri-Stepan Gerasimov, Jun 16 2009

nonn

			a(1)=0 is the product 0*1*2/2. a(2) is the product 1*2*3/2 = 3. a(3)=10 is the product 4*5/2.

Michael De Vlieger, Table of n, a(n) for n = 1..99

Cf. A000040, A141468.

f[n_] := FixedPoint[n + PrimePi@ # &, n + PrimePi@ n]; Array[1/2*Times @@ Range[f[# - 1], Prime[#]] &, 13] (* Michael De Vlieger, Mar 22 2022, after Robert G. Wilson v at A141468 *)

a(n) = prime(n)!/ (2*(A141468(n)-1)!).

a(0) replaced by 0 - R. J. Mathar, May 21 2010

A161845 a(n) = A002808(n)^A141468(n).

Original entry on oeis.org

1, 6, 4096, 531441, 100000000, 5159780352, 289254654976, 129746337890625, 72057594037927936, 6746640616477458432, 655360000000000000000, 630880792396715529789561, 705429498686404044207947776, 96479729228174488169059713024

Offset: 1

Juri-Stepan Gerasimov, Jun 20 2009

nonn

The n-th composite raised to the power m, where m = n-th nonprime.

			a(1) = 4^0 = 1;
a(2) = 6^1 = 6;
a(3) = 8^4 = 4096;
a(4) = 9^6 = 531441;
a(5) = 10^8 = 100000000;
a(6) = 12^9 = 5159780352.

Cf. A002808, A141468.

A002808 := proc(n) option remember; local a; if n = 1 then 4; else for a from procname(n-1)+1 do if not isprime(a) then RETURN(a) : fi; od: fi; end:
A141468 := proc(n) if n >= 3 then A002808(n-2) ; else n-1 ; fi; end:
A161845 := proc(n) A002808(n)^A141468(n) ; end:
seq(A161845(n),n=1..20) ; # R. J. Mathar, Jun 23 2009

a(4) corrected and extended by R. J. Mathar, Jun 23 2009

Edited by Jon E. Schoenfield, Feb 23 2019

A163296 Absolute value of the Sum_{x=0..A141468(n)} x*(-1)^x.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 5, 6, 7, 8, 8, 9, 10, 11, 11, 12, 13, 13, 14, 14, 15, 16, 17, 17, 18, 18, 19, 20, 20, 21, 22, 23, 23, 24, 25, 25, 26, 26, 27, 28, 28, 29, 29, 30, 31, 32, 32, 33, 33, 34, 35, 35, 36, 37, 38, 38, 39, 39, 40, 41, 41, 42, 43, 43, 44, 44, 45, 46, 46, 47, 47, 48, 48, 49, 50, 50, 51, 52, 53, 53

Offset: 1

Juri-Stepan Gerasimov, Jul 24 2009

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

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

Cf. A141468.

A130472 := proc(n) (-1)^n*floor((n+1)/2) ; end: A141468 := proc(n) option remember; local a; if n = 1 then 0 ; else for a from procname(n-1)+1 do if not isprime(a) then RETURN(a) ; fi; od: fi; end: A163296 := proc(n) abs(A130472( A141468(n))) ; end: seq(A163296(n),n=1..100) ; # R. J. Mathar, Jul 26 2009

nonPrime[n_Integer] := FixedPoint[n + PrimePi@# &, n + PrimePi@n]; Table[Abs[Sum[k*(-1)^k, {k, 0, nonPrime[n]}]], {n,0,50}] (* G. C. Greubel, Dec 18 2016 *)

a(n) = |A130472(A141468(n))|. - R. J. Mathar, Jul 26 2009

Corrected by R. J. Mathar, Jul 26 2009

A167609 Primes which are not the sum of two consecutive nonprimes A141468.

Original entry on oeis.org

2, 3, 7, 11, 13, 23, 37, 47, 59, 61, 73, 83, 107, 157, 167, 179, 193, 227, 263, 277, 313, 347, 359, 383, 397, 421, 457, 467, 479, 503, 541, 563, 587, 613, 661, 673, 719, 733, 757, 839, 863, 877, 887, 983, 997, 1019, 1093, 1153, 1187, 1201, 1213, 1237, 1283

Offset: 1

Juri-Stepan Gerasimov, Nov 07 2009

nonn

2, and primes p such that floor(p/2) or ceiling(p/2) is prime, but not both. - Robert Israel, Jan 23 2024

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A005383, A005385, A060254, A079149, A141468.

filter:= proc(n) if n mod 4 = 1
  then isprime(n) and isprime((n+1)/2)
  else isprime(n) and isprime((n-1)/2)
fi end proc:
2, 3, op(select(filter, [seq(i,i=7 .. 10000, 2)])); # Robert Israel, Jan 23 2024

Entries checked by R. J. Mathar, May 30 2010

A171691 Number of unordered partitions {k1, k2} of n such that k1 and k2 are nonnegative nonprimes A141468.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 2, 2, 3, 1, 3, 2, 3, 3, 5, 2, 5, 3, 5, 4, 6, 3, 7, 5, 7, 5, 8, 5, 9, 6, 8, 7, 10, 7, 12, 7, 9, 9, 12, 8, 13, 9, 12, 10, 13, 9, 15, 11, 15, 11, 15, 11, 17, 13, 16, 13, 17, 13, 20, 14, 16, 15, 20, 15, 22, 15, 18, 17, 22, 16, 23, 17, 21, 18, 23, 18, 26, 18, 23

Offset: 1

Juri-Stepan Gerasimov, Dec 15 2009

nonn

			a(1) = 1 because 1 = 0 + 1.
a(2) = 1 because 2 = 1 + 1.
a(3) = 0.
a(4) = 1 because 4 = 0 + 4.
a(5) = 1 because 5 = 1 + 4.
a(6) = 1 because 6 = 0 + 6.
a(7) = 1 because 7 = 1 + 6.
a(8) = 2 because 8 = 0 + 8 = 4 + 4.

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A062610, A141468, A224708.

a(n)={sum(i=0, n\2, (i<2 || !isprime(i)) && !isprime(n-i))} \\ Andrew Howroyd, Jan 05 2020

Name clarified and terms a(55) and beyond from Andrew Howroyd, Jan 05 2020

A177416 Triangle read by rows: T(n,k) = 2^A141468(n) mod prime(k).

Original entry on oeis.org

1, 0, 2, 0, 1, 1, 0, 1, 4, 1, 0, 1, 1, 4, 3, 0, 2, 2, 1, 6, 5, 0, 1, 4, 2, 1, 10, 4, 0, 1, 1, 1, 4, 1, 16, 11, 0, 1, 4, 4, 5, 4, 13, 6, 8, 0, 2, 3, 1, 10, 8, 9, 12, 16, 27, 0, 1, 1, 2, 9, 3, 1, 5, 9, 25, 2, 0, 1, 4, 1, 3, 12, 4, 1, 13, 13, 8, 36, 0, 1, 1, 4, 1, 9, 16, 4, 6, 23, 1, 33, 1, 0, 2, 2, 1, 2, 5, 15, 8, 12, 17, 2, 29, 2, 42

Offset: 1

Juri-Stepan Gerasimov, Dec 10 2010

			The triangle begins at row n = 1 with columns 1 <= k <= n:
  1;
  0, 2;
  0, 1, 1;
  0, 1, 4, 1;
  0, 1, 1, 4, 3;
  0, 2, 2, 1, 6, 5;

Cf. A000079, A141468, A174620.

MapIndexed[PowerMod[2, #, Prime[Range[First[#2]]]] &, Join[{0, 1}, Select[Range[25], CompositeQ]]] (* Paolo Xausa, Jul 01 2024 *)

A 23 replaced with 33 by R. J. Mathar, Dec 13 2010

A105840 Primes of the form r(r(n)+1)+1, where r(n) = A141468(n) = n-th nonprime.

Original entry on oeis.org

2, 11, 17, 23, 31, 37, 41, 47, 59, 67, 71, 73, 79, 83, 89, 97, 101, 103, 109, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 191, 197, 199, 211, 223, 227, 233, 239, 241, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 331, 347, 349, 353, 359, 367

Offset: 1

Juri-Stepan Gerasimov, Aug 25 2008

nonn

			If n=1, then r(r(1)+1)+1=r(0+1)+1=r(1)+1=0+1=1 (nonprime).
If n=2, then r(r(2)+1)+1=r(1+1)+1=r(2)+1=1+1=2=a(1).
If n=3, then r(r(3)+1)+1=r(4+1)+1=r(5)+1=8+1=9 (nonprime).
If n=4, then r(r(4)+1)+1=r(6+1)+1=r(7)+1=10+1=11=a(2).
If n=5, then r(r(5)+1)+1=r(8+1)+1=r(9)+1=14+1=15 (nonprime).
If n=6, then r(r(6)+1)+1=r(9+1)+1=r(10)+1=15+1=16 (nonprime).
If n=7, then r(r(7)+1)+1=r(10+1)+1=r(11)+1=16+1=17=a(3).
If n=8, then r(r(8)+1)+1=r(12+1)+1=r(13)+1=20+1+21 (nonprime).
If n=9, then r(r(9)+1)+1=r(14+1)+1=r(15)+1=22+1=23=a(4), etc.

Cf. A000040, A141468.

53 removed, more terms added by R. J. Mathar, Sep 05 2008

A105882 Nonprimes of the form r(r(n)+1)+1, where A141468(n)=r(n)=n-th nonprime.

Original entry on oeis.org

1, 9, 15, 16, 21, 25, 26, 28, 33, 34, 36, 39, 40, 45, 49, 50, 51, 52, 55, 56, 57, 63, 64, 65, 69, 70, 76, 77, 78, 81, 86, 87, 88, 91, 93, 94, 95, 100, 105, 106, 111, 112, 115, 116, 117, 118, 119, 121, 122, 123, 124, 125, 126, 130, 133, 135, 141, 143, 145, 146, 147, 153

Offset: 1

Juri-Stepan Gerasimov, Aug 25 2008

nonn

			If n=1, then r(r(1)+1)+1=r(0+1)+1=r(1)+1=0+1=1=a(1).
If n=2, then r(r(2)+1)+1=r(1+1)+1=r(2)+1=1+1=2 (prime).
If n=3, then r(r(3)+1)+1=r(4+1)+1=r(5)+1=8+1=9=a(2).
If n=4, then r(r(4)+1)+1=r(6+1)+1=r(7)+1=10+1=11 (prime).
If n=5, then r(r(5)+1)+1=r(8+1)+1=r(9)+1=14+1=15=a(3).
If n=6, then r(r(6)+1)+1=r(9+1)+1=r(10)+1=15+1=16=a(4).
If n=7, then r(r(7)+1)+1=r(10+1)+1=r(11)+1=16+1=17 (prime).
If n=8, then r(r(8)+1)+1=r(12+1)+1=r(13)+1=20+1+21=a(5).
If n=9, then r(r(9)+1)+1=r(14+1)+1=r(15)+1=22+1=23 (prime).
If n=10, then r(r(10)+1)+1=r(15+1)+1=r(16)+1=24+1=25=a(6), etc.

Cf. A000040, A141468.

49 and 69 inserted by R. J. Mathar, Sep 05 2008

A106613 Nonprimes of the form r(r(r(n)+1)+1)+1, where A141468(n)=r(n)=n-th nonprime.

Original entry on oeis.org

1, 15, 25, 26, 34, 36, 40, 45, 49, 51, 52, 55, 56, 57, 63, 65, 69, 70, 76, 77, 78, 81, 86, 87, 88, 91, 93, 94, 95, 105, 106, 112, 116, 117, 118, 119, 121, 123, 124, 125, 133, 135, 143, 145, 146, 153, 154, 155, 159, 160, 161, 162, 165, 169, 170, 172, 175, 177, 183, 185

Offset: 1

Juri-Stepan Gerasimov, Aug 25 2008

nonn

			If n=1, then
r(r(r(1)+1)+1)+1=r(r(0+1)+1)+1=r(r(1)+1)+1=r(0+1)+1=r(1)+1=0+1=1=a(1).
If n=2, then
r(r(r(2)+1)+1)+1=r(r(1+1)+1)+1=r(r(2)+1)+1=r(1+1)+1=r(2)+1=1+1=2
(prime).
If n=3, then
r(r(r(3)+1)+1)+1=r(r(4+1)+1)+1=r(r(5)+1)+1=r(8+1)+1=r(9)+1=14+1=15=a(2).
If n=4, then
r(r(r(4)+1)+1)+1=r(r(6+1)+1)+1=r(r(7)+1)+1=r(10+1)+1=r(11)+1=16+1=17
(prime).
If n=5, then
r(r(r(5)+1)+1)+1=r(r(8+1)+1)+1=r(r(9)+1)+1=r(14+1)+1=r(15)+1=22+1=23
(prime).
If n=6, then
r(r(r(6)+1)+1)+1=r(r(9+1)+1)+1=r(r(10)+1)+1=r(15+1)+1=r(16)+1=24+1=25=a(3).
If n=7, then
r(r(r(7)+1)+1)+1=r(r(10+1)+1)+1=r(r(11)+1)+1=r(16+1)+1=r(17)+1=25+1=26=a(4).
If n=8, then
r(r(r(8)+1)+1)+1=r(r(12+1)+1)+1=r(r(13)+1)+1=r(20+1)+1=r(21)+1=30+1=31
(prime).
If n=9, then
r(r(r(9)+1)+1)+1=r(r(14+1)+1)+1=r(r(15)+1)+1=r(22+1)+1=r(23)+1=33+1=34=a(5).
If n=10, then
r(r(r(10)+1)+1)+1=r(r(15+1)+1)+1=r(r(16)+1)+1=r(24+1)+1=r(25)+1=35+1=36=a(6),
etc.

Cf. A000040, A141468.

28 removed, 93 added, 126 removed by R. J. Mathar, Sep 05 2008

A106622 Primes of the form r(r(r(n)+1)+1)+1, where A141468(n)=r(n)=n-th nonprime.

Original entry on oeis.org

2, 17, 23, 31, 37, 47, 67, 71, 97, 101, 103, 109, 127, 131, 137, 139, 149, 151, 157, 163, 179, 191, 197, 199, 211, 223, 227, 233, 239, 241, 257, 263, 269, 271, 277, 281, 283, 311, 313, 331, 347, 349, 353, 359, 367, 373, 379, 389, 397, 401, 419, 431, 443, 449

Offset: 1

Juri-Stepan Gerasimov, Aug 25 2008

nonn

			If n=1, then
r(r(r(1)+1)+1)+1=r(r(0+1)+1)+1=r(r(1)+1)+1=r(0+1)+1=r(1)+1=0+1=1
(nonprime).
If n=2, then
r(r(r(2)+1)+1)+1=r(r(1+1)+1)+1=r(r(2)+1)+1=r(1+1)+1=r(2)+1=1+1=2=a(1).
If n=3, then
r(r(r(3)+1)+1)+1=r(r(4+1)+1)+1=r(r(5)+1)+1=r(8+1)+1=r(9)+1=14+1=15
(nonprime).
If n=4, then
r(r(r(4)+1)+1)+1=r(r(6+1)+1)+1=r(r(7)+1)+1=r(10+1)+1=r(11)+1=16+1=17=a(2).
If n=5, then
r(r(r(5)+1)+1)+1=r(r(8+1)+1)+1=r(r(9)+1)+1=r(14+1)+1=r(15)+1=22+1=23=a(3).
If n=6, then
r(r(r(6)+1)+1)+1=r(r(9+1)+1)+1=r(r(10)+1)+1=r(15+1)+1=r(16)+1=24+1=25
(nonprime).
If n=7, then
r(r(r(7)+1)+1)+1=r(r(10+1)+1)+1=r(r(11)+1)+1=r(16+1)+1=r(17)+1=25+1=26
(nonprime).
If n=8, then
r(r(r(8)+1)+1)+1=r(r(12+1)+1)+1=r(r(13)+1)+1=r(20+1)+1=r(21)+1=30+1=31=a(4).
If n=9, then
r(r(r(9)+1)+1)+1=r(r(14+1)+1)+1=r(r(15)+1)+1=r(22+1)+1=r(23)+1=33+1=34
(nonprime).
If n=10, then
r(r(r(10)+1)+1)+1=r(r(15+1)+1)+1=r(r(16)+1)+1=r(24+1)+1=r(25)+1=35+1=36
(nonprime)
If n=11, then
r(r(r(11)+1)+1)+1=r(r(16+1)+1)+1=r(r(17)+1)+1=r(25+1)+1=r(26)+1=36+1=37=a(5),
etc.

Cf. A000040, A141468.

67 inserted, 73 removed, 227 and 233 inserted and extended by R. J. Mathar, Sep 05 2008

cp's OEIS Frontend

A161672 Half the product of all numbers from A141468(n) up to prime(n).

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A161845 a(n) = A002808(n)^A141468(n).

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A163296 Absolute value of the Sum_{x=0..A141468(n)} x*(-1)^x.

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A167609 Primes which are not the sum of two consecutive nonprimes A141468.

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A171691 Number of unordered partitions {k1, k2} of n such that k1 and k2 are nonnegative nonprimes A141468.

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PARI

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A177416 Triangle read by rows: T(n,k) = 2^A141468(n) mod prime(k).

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A105840 Primes of the form r(r(n)+1)+1, where r(n) = A141468(n) = n-th nonprime.

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A105882 Nonprimes of the form r(r(n)+1)+1, where A141468(n)=r(n)=n-th nonprime.

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A106613 Nonprimes of the form r(r(r(n)+1)+1)+1, where A141468(n)=r(n)=n-th nonprime.

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