A141468 -id:A141468 - cp's OEIS frontend

A161671 a(n) = prime(n) - A141468(n).

2, 2, 1, 1, 3, 4, 7, 7, 9, 14, 15, 19, 21, 22, 25, 29, 34, 35, 40, 43, 43, 47, 50, 55, 62, 65, 65, 68, 69, 71, 83, 86, 91, 91, 100, 101, 106, 111, 113, 118, 123, 124, 133, 133, 135, 136, 147, 158, 161, 161, 164, 169, 169, 177, 182, 187, 192, 193, 197, 200, 201, 209

Offset: 1

Juri-Stepan Gerasimov, Jun 16 2009, Dec 03 2009

nonn

			2(=2-0), 2(=3-1), 1(=5-4), 1(=7-6), 3(=11-8), 4(=13-9), 7(=17-10), 7(=19-12), 9(=23-14), 14(=29-15), etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Scatterplot of a(n), n = 1..120, showing and labeling primes p (in A214627) in red and blue, the red primes are duplicated and are listed in A220220. We plot in green duplicated composite terms.

Cf. A000040, A002808, A141468, A141559, A168563, A214627, A220220.

A161671 := proc(n)
    ithprime(n)-A141468(n) ;
end proc: # R. J. Mathar, Aug 08 2012

f[n_] := FixedPoint[n + PrimePi@ # &, n + PrimePi@ n]; Array[Prime[#] - f[# - 1] &, 62] (* Michael De Vlieger, Mar 22 2022, after Robert G. Wilson v at A141468 *)

a(n) = A000040(n) - A141468(n).
a(n+2) = A168563(n).
a(n) = A000040(n) - A018252(n-1), if n >= 2. - Omar E. Pol, Oct 21 2011
a(n) ~ n log n. - Charles R Greathouse IV, Dec 21 2011

Edited by N. J. A. Sloane, Jun 30 2009
207 replaced with 209 by R. J. Mathar, Oct 04 2009
Edited by Omar E. Pol, Oct 21 2011

A161753 Squares of nonprime numbers A141468.

Original entry on oeis.org

0, 1, 16, 36, 64, 81, 100, 144, 196, 225, 256, 324, 400, 441, 484, 576, 625, 676, 729, 784, 900, 1024, 1089, 1156, 1225, 1296, 1444, 1521, 1600, 1764, 1936, 2025, 2116, 2304, 2401, 2500, 2601, 2704, 2916, 3025, 3136, 3249, 3364, 3600, 3844, 3969, 4096

Offset: 1

Juri-Stepan Gerasimov, Jun 18 2009

Essentially the same as A062312: a(1)=0, a(n)=A062312(n-1) for n>=2. - R. J. Mathar, Sep 11 2012

			0=0^2, 1=1^2, 16=4^2, 36=6^2, etc.

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

Cf. A001248, A141468.

With[{nn=100},Complement[Range[0,nn],Prime[Range[PrimePi[nn]]]]^2] (* Harvey P. Dale, Jul 04 2013 *)

Corrected and edited by Omar E. Pol, Jun 29 2009

A174620 a(n) = 2^A141468(n) mod prime(n), where A141468(n) is the n-th nonnegative nonprime.

Original entry on oeis.org

1, 2, 1, 1, 3, 5, 4, 11, 8, 27, 2, 36, 1, 42, 24, 13, 11, 19, 45, 5, 8, 50, 71, 2, 86, 78, 15, 74, 16, 112, 4, 68, 14, 106, 66, 32, 79, 26, 25, 18, 76, 6, 78, 150, 163, 63, 69, 98, 189, 17, 184, 40, 1, 125, 249, 187, 229, 69, 169, 81, 264, 172, 18, 209, 114, 277, 1, 128, 46, 21

Offset: 1

Juri-Stepan Gerasimov, Nov 29 2010

nonn

			a(14) = 12 because (2^A141468(14) mod prime(14)) = (2097152 mod 43) = 12.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Main diagonal of A177416.

Cf. A000040, A000079, A141468.

MapIndexed[PowerMod[2, #, Prime[First[#2]]] &, Join[{0, 1}, Select[Range[100], CompositeQ]]] (* Paolo Xausa, Jul 01 2024 *)

c=-1; forprime(p=1,999,while(isprime(c++),);print1(lift( Mod(2,p)^c )", ")) \\ M. F. Hasler, Nov 29 2010

A166039 Sums of three consecutive nonprimes A141468.

Original entry on oeis.org

5, 11, 18, 23, 27, 31, 36, 41, 45, 49, 54, 59, 63, 67, 71, 75, 78, 81, 85, 90, 95, 99, 102, 105, 109, 113, 117, 121, 126, 131, 135, 139, 143, 147, 150, 153, 157, 161, 165, 168, 171, 175, 180, 185, 189, 192, 195, 199, 203, 207, 211, 216, 221, 225, 228, 231, 235

Offset: 1

Juri-Stepan Gerasimov, Oct 05 2009

nonn

			a(1) = 0 + 1 + 4 =  5;
a(2) = 1 + 4 + 6 = 11;
a(3) = 4 + 6 + 8 = 18.

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

Cf. A034961, A141468.

A002808 := proc(n) option remember; 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 <= 2 then n-1 ; else A002808(n-2) ; fi; end: A166039 := proc(n) add(A141468(j),j=n..n+2) ; end: seq(A166039(n),n=1..120) ; # R. J. Mathar, Oct 10 2009

With[{nn=100},Total/@Partition[Complement[Range[0,nn],Prime[ Range[ PrimePi[ nn]]]],3,1]] (* Harvey P. Dale, Aug 05 2015 *)

A167915 Primes which are the sums of two consecutive nonprimes (A141468).

Original entry on oeis.org

5, 17, 19, 29, 31, 41, 43, 53, 67, 71, 79, 89, 97, 101, 103, 109, 113, 127, 131, 137, 139, 149, 151, 163, 173, 181, 191, 197, 199, 211, 223, 229, 233, 239, 241, 251, 257, 269, 271, 281, 283, 293, 307, 311, 317, 331, 337, 349, 353, 367, 373, 379, 389, 401, 409

Offset: 1

Juri-Stepan Gerasimov, Nov 15 2009

nonn

Five together with primes that are the sum of two consecutive composite numbers.

			a(1)=1+4=5, a(2)=8+9=17.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000040, A060254, A141468, A176902.

[2*n+1: n in [1..300] | (not IsPrime(n) eq not IsPrime(n+1)) and IsPrime(2*n+1)]; // G. C. Greubel, Nov 10 2023

2*Select[Range[300], !PrimeQ[#] == !PrimeQ[#+1] && PrimeQ[2*#+1] &] + 1 (* G. C. Greubel, Jul 01 2016; Nov 10 2023 *)

[2*n+1 for n in (1..300) if  (not is_prime(n)) - (not is_prime(n+1)) == 0 and is_prime(2*n+1)] # G. C. Greubel, Nov 10 2023

a(n+1) = A060254(n) = A176902(n+1). - Juri-Stepan Gerasimov, Apr 28 2010

Typo corrected and terms checked by D. S. McNeil, Nov 17 2010

A141559 Primes of form (p(n)-r(n)), where A141468(n)=r(n)=n-th nonprime and p(n)=n-th prime.

Original entry on oeis.org

2, 2, 3, 7, 7, 19, 29, 43, 43, 47, 71, 83, 101, 113, 193, 197, 229, 241, 271, 283, 293, 311, 311, 347, 383, 439, 457, 463, 491, 491, 499, 523, 587, 619, 643, 683, 733, 797, 827, 827, 857, 863, 919, 991, 1021, 1031, 1091, 1151, 1187, 1289, 1367, 1367, 1549, 1567

Offset: 1

Juri-Stepan Gerasimov, Aug 14 2008

nonn

			If n=1, then p(1)-r(1)=2-0=2=a(1).
If n=2, then p(2)-r(2)=3-1=2=a(2).
If n=3, then p(3)-r(3)=5-4=1 (nonprime).
If n=4, then p(4)-r(4)=7-6=1 (nonprime).
If n=5, then p(5)-r(5)=11-8=3=a(3).
If n=6, then p(6)-r(6)=13-9=4 (composite).
If n=7, then p(7)-r(7)=17-10=7=a(4).
If n=8, then p(8)-r(8)=19-12=7=a(5).
If n=9, then p(9)-r(9)=23-14=9 (composite).
If n=10, then=p(10)-r(10)=29-15=14 (composite).
If n=11, then p(11)-r(11)=31-16=15 (composite).
If n=12, then p(12)-r(12)=37--18=19=a(6).
If n=13, then p(13)-r(13)=41-20=21 (composite).
If n=14, then p(14)-r(14)=43-21=22 (composite).
If n=15, then p(15)-r(15)=47-22=25 (composite).
If n=16, then p(16)-r(16)=53-24=29=a(7), etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Mohammad Javaheri, Nikolai A. Krylov, Permutations with a distinct divisor property, arXiv:1904.04227 [math.GR], 2019.

Cf. A000040, A141468.

Block[{nn = 2000, p, r}, p = Prime@ Range@ PrimePi@ nn; r = Complement[Range[0, nn], p]; Select[Array[p[[#]] - r[[#]] &, Min[Length /@ {p, r}]], PrimeQ]] (* Michael De Vlieger, May 21 2019 *)

Edited and extended by Ray Chandler, Aug 19 2008

A166037 Numbers that are the sum of 2 successive nonprimes A141468.

Original entry on oeis.org

1, 5, 10, 14, 17, 19, 22, 26, 29, 31, 34, 38, 41, 43, 46, 49, 51, 53, 55, 58, 62, 65, 67, 69, 71, 74, 77, 79, 82, 86, 89, 91, 94, 97, 99, 101, 103, 106, 109, 111, 113, 115, 118, 122, 125, 127, 129, 131, 134, 137, 139, 142, 146, 149, 151, 153, 155, 158, 161, 163, 166

Offset: 1

Juri-Stepan Gerasimov, Oct 05 2009

nonn

a(n) = (n-1)-th nonprimes + (n-1)-th composites for n >= 2. a(n) = A018252(n-1) + A002808(n-1) for n >= 2. - Jaroslav Krizek, Dec 13 2009

			a(1) = 0 + 1 =  1;
a(2) = 1 + 4 =  5;
a(3) = 4 + 6 = 11.

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Cf. A001043, A141468.

Cf. A167915 (primes that are the sums of two consecutive composites).

A002808 := proc(n) option remember; 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 <= 2 then n-1 ; else A002808(n-2) ; fi; end: A166037 := proc(n) A141468(n)+A141468(n+1) ; end: seq(A166037(n),n=1..120) ; # R. J. Mathar, Oct 10 2009

With[{nn=100},Join[{1},Total/@Partition[Complement[Range[nn],Prime[ Range[ PrimePi[ nn]]]],2,1]]] (* Harvey P. Dale, Aug 03 2014 *)

A179899 Integers of the form A179896(n)/A141468(n+1).

Original entry on oeis.org

0, 12, 21, 30, 36, 39, 48, 51, 57, 66, 72, 75, 81, 84, 93, 96, 102, 111, 114, 120, 126, 129, 135, 138, 141, 147, 156, 165, 171, 174, 177, 180, 183, 186, 192, 198, 201, 210, 213, 216, 219, 228, 231, 237, 240, 246, 252, 255, 261, 264, 273, 276, 279, 282, 291

Offset: 1

Odimar Fabeny, Jul 31 2010

			0 = 0/1, 12 = 108/9, 21 = 315/15, 30 = 630/21, 36 = 900/25 and so on.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A000040, A179545, A179896.

ithnonprime := proc(n)local k: option remember: if(n=1)then return 1: else k := procname(n-1)+1: while true do if(not isprime(k))then return k fi: k:=k+1: od: fi: end: A179899ind := proc(n) option remember: local k: if(n=1)then return 1:fi: for k from procname(n-1)+1 do if(ithnonprime(k) mod 2 <> 0)then return k: fi: od: end: A179899 := proc(n) return 3*(ithnonprime(A179899ind(n))-1)/2: end: seq(A179899(n),n=1..55); # Nathaniel Johnston, May 05 2011

More terms from Odimar Fabeny, Aug 12 2010

Definition rephrased by R. J. Mathar, Sep 01 2010

A102885 Index of n in the primes A000040 or nonprimes A141468.

Original entry on oeis.org

1, 2, 1, 2, 3, 3, 4, 4, 5, 6, 7, 5, 8, 6, 9, 10, 11, 7, 12, 8, 13, 14, 15, 9, 16, 17, 18, 19, 20, 10, 21, 11, 22, 23, 24, 25, 26, 12, 27, 28, 29, 13, 30, 14, 31, 32, 33, 15, 34, 35, 36, 37, 38, 16, 39, 40, 41, 42, 43, 17, 44, 18, 45, 46, 47, 48, 49, 19, 50, 51, 52, 20, 53, 21, 54

Offset: 0

Juri-Stepan Gerasimov, Aug 17 2008

The nonnegative numbers n occur exactly once in either A000040 or A141468. The sequence lists the corresponding index. It is a permutation of A008619.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A141468, A000040, A026233.

Module[{nn=80,pr,np},pr=Prime[Range[PrimePi[nn]]];np=Complement[ Range[ 0,nn],pr];Table[If[PrimeQ[n],Position[pr,n],Position[np,n]],{n,0,nn}]]//Flatten (* Harvey P. Dale, Sep 10 2022 *)

A141468(a(n))=n or A000040(a(n))=n.

Edited by R. J. Mathar, Aug 19 2008

A144551 a(n) = nonprime(n)*nonprime(n+1)/2, where nonprime(n) = A141468(n).

Original entry on oeis.org

0, 2, 12, 24, 36, 45, 60, 84, 105, 120, 144, 180, 210, 231, 264, 300, 325, 351, 378, 420, 480, 528, 561, 595, 630, 684, 741, 780, 840, 924, 990, 1035, 1104, 1176, 1225, 1275, 1326, 1404, 1485, 1540, 1596, 1653, 1740, 1860, 1953, 2016, 2080, 2145, 2244, 2346

Offset: 1

Juri-Stepan Gerasimov, Dec 31 2008

nonn

			a(1) = 0*1/2 = 0, a(2) = 1*4/2 = 2, a(3) = 4*6/2 = 12, etc.

Cf. A141468.

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: A144551 := proc(n) A141468(n)*A141468(n+1)/2 ; end: for n from 1 to 200 do printf("%d,",A144551(n)) ; od: # R. J. Mathar, Jan 03 2009

(Times@@#)/2&/@Partition[Select[Range[0,100],!PrimeQ[#]&],2,1] (* Harvey P. Dale, Feb 12 2020 *)

c(n) = {for(k=0, primepi(n), isprime(n++)&&k--); n};
t(n) = if(n<3,n-1,c(n-2));
vector(100, n, t(n)*t(n+1)/2) \\ Altug Alkan, Oct 17 2015

a(n) = my(A141468(n)=my(k=0); n--; while(-n+n+=-k+k=primepi(n), ); n); A141468(n)*A141468(n+1)/2; \\ Ruud H.G. van Tol, Jul 15 2024

1963 replaced by 1953 by R. J. Mathar, Jan 03 2009

cp's OEIS Frontend

A161671 a(n) = prime(n) - A141468(n).

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A161753 Squares of nonprime numbers A141468.

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A174620 a(n) = 2^A141468(n) mod prime(n), where A141468(n) is the n-th nonnegative nonprime.

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A166039 Sums of three consecutive nonprimes A141468.

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A167915 Primes which are the sums of two consecutive nonprimes (A141468).

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A141559 Primes of form (p(n)-r(n)), where A141468(n)=r(n)=n-th nonprime and p(n)=n-th prime.

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A166037 Numbers that are the sum of 2 successive nonprimes A141468.

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A179899 Integers of the form A179896(n)/A141468(n+1).