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approved

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proposed

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

If n=1, then:

If n=2, then:

If n=3, then:

If n=4, then:

If n=5, then:

If n=6, then:

If n=7, then:

If n=8, then:

If n=9, then:

If n=10, then:

If n=11, then:

A141468 := proc(n) option remember ; 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: rep := 4: for n from 1 to 400 do arep := n ; for i from 1 to rep do arep := A141468(arep)+1 ; od: if isprime(arep) then printf("%d, ", arep) ; fi; od: [From _# _R. J. Mathar_, Sep 05 2008]

approved

editing

_Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), _, Aug 25 2008

A141468 := proc(n) option remember ; 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: rep := 4: for n from 1 to 400 do arep := n ; for i from 1 to rep do arep := A141468(arep)+1 ; od: if isprime(arep) then printf("%d, ", arep) ; fi; od: [From _R. J. Mathar (mathar(AT)strw.leidenuniv.nl), _, Sep 05 2008]

97 removed and extended by _R. J. Mathar (mathar(AT)strw.leidenuniv.nl), _, Sep 05 2008

If n is composite, replace n by the concatenation of its proper divisors, otherwise a(n) = n.

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

1, 2, 3, 2, 5, 23, 7, 24, 3, 25, 11, 2346, 13, 27, 35, 248, 17, 2369, 19, 24510, 37, 67, 71, 101, 103, 109, 127, 137, 139, 151, 157, 179, 191, 197, 199, 211, 23, 2346812, 5, 213, 39, 24714, 29, 23561015, 31, 24816, 227, 233, 239, 241, 257, 263, 271, 277, 281, 283, 311, 217, 57, 234691218, 37, 219, 313, 24581020, 41, 23671421, 43, 241122, 35915, 223, 47, 23468121624, 7331, 347, 353, 359, 367, 373, 379, 389, 401, 419, 431, 443, 457, 461, 467, 499, 503, 509, 521, 523, 541, 547, 557, 563

1,21

If n=1, then

r(r(r(r(1)+1)+1)+1)+1=r(r(r(0+1)+1)+1)+1=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(r(2)+1)+1)+1)+1=r(r(r(1+1)+1)+1)+1=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(r(3)+1)+1)+1)+1=r(r(r(4+1)+1)+1)+1=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(2).

If n=4, then

r(r(r(r(4)+1)+1)+1)+1=r(r(r(6+1)+1)+1)+1=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=5, then

r(r(r(r(5)+1)+1)+1)+1=r(r(r(8+1)+1)+1)+1=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=6, then

r(r(r(r(6)+1)+1)+1)+1=r(r(r(9+1)+1)+1)+1=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=7, then

r(r(r(r(7)+1)+1)+1)+1=r(r(r(10+1)+1)+1)+1=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(3).

If n=8, then

r(r(r(r(8)+1)+1)+1)+1=r(r(r(12+1)+1)+1)+1=r(r(r(13)+1)+1)+1=r(r(20+1)+1)+1=r(r(21)+1)+1=r(30+1)+1=r(31)+1=44+1=45

(nonprime).

If n=9, then

r(r(r(r(9)+1)+1)+1)+1=r(r(r(14+1)+1)+1)+1=r(r(r(15)+1)+1)+1=r(r(22+1)+1)+1=r(r(23)+1)+1=r(33+1)+1=r(34)+1

48+1=49 (nonprime).

If n=10, then

r(r(r(r(10)+1)+1)+1)+1=r(r(r(15+1)+1)+1)+1=r(r(r(16)+1)+1)+1=r(r(24+1)+1)+1=r(r(25)+1)+1=r(35+1)+1=r(36)+1

50+1=51(nonprime)

If n=11, then

r(r(r(r(11)+1)+1)+1)+1=r(r(r(16+1)+1)+1)+1=r(r(r(17)+1)+1)+1=r(r(25+1)+1)+1=r(r(26)+1)+1=r(36+1)+1=r(37)+1=51+1=52(nonprime),

etc.

A106736 A141468 := proc(n) local dvs option remember ; if isprime(n) or n = 1 then n0; else dvs := [op(numtheory[divisors]for a from procname(n) minus {-1, n} )] ; dvs := sort+1 do if not isprime(dvsa) ; cat(opthen RETURN(dvs)a) ; fi ; od: fi; end: seq(A106736(rep := 4: for n), from 1 to 400 do arep := n ; for i from 1 to rep do arep := A141468(arep)+1..80 ; od: if isprime(arep) then printf("%d, ", arep) ; - fi; od: [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 23 2007Sep 05 2008]

Cf. A037278, A120712, A106708A000040, A141468.

nonn,base,new

nonn

njas, Jul 20 2007

Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Aug 25 2008

More terms from 97 removed and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 23 2007Sep 05 2008

A106736 := proc(n) local dvs ; if isprime(n) or n = 1 then n; else dvs := [op(numtheory[divisors](n) minus {1, n} )] ; dvs := sort(dvs) ; cat(op(dvs)) ; fi ; end: seq(A106736(n), n=1..80) ; - Richard R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 23 2007

nonn,base,new

More terms from Richard R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 23 2007

a(If n is composite, replace n) = semiprime equal to by the sum concatenation of two primes plus 1, its proper divisors, otherwise a(n) =0 n.

0, 4, 6, 9, 10, 14, 15, 21, 22, 1, 2, 3, 2, 5, 23, 7, 24, 3, 25, 26, 33, 0, 11, 2346, 13, 27, 35, 0, 248, 17, 2369, 19, 24510, 37, 211, 23, 2346812, 5, 213, 39, 0, 4924714, 29, 23561015, 31, 24816, 311, 217, 57, 234691218, 37, 219, 313, 24581020, 41, 23671421, 43, 241122, 35915, 223, 47, 23468121624, 7

0,1,2

Iff a(n)=0 then a(n) is even semiprime such that for the Golbach conjecture is equal to the sum of two primes.

a(1)=0 because 4 is not equal to the sum of two primes plus 1;

a(3)=6 because 6=2+3+1.

A106736 := proc(n) local dvs ; if isprime(n) or n = 1 then n; else dvs := [op(numtheory[divisors](n) minus {1, n} )] ; dvs := sort(dvs) ; cat(op(dvs)) ; fi ; end: seq(A106736(n), n=1..80) ; - Richard J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 23 2007

Cf. A037278, A120712, A106708.

easy,nonn,newbase

Giovanni Teofilatto (g.teofilatto(AT)tiscalinet.it), May 15 2005

njas, Jul 20 2007

More terms from Richard J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 23 2007

a(n) = semiprime equal to the sum of two primes plus 1, otherwise a(n)=0.

0, 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 0, 35, 0, 39, 0, 49

0,2

Iff a(n)=0 then a(n) is even semiprime such that for the Golbach conjecture is equal to the sum of two primes.

a(1)=0 because 4 is not equal to the sum of two primes plus 1;

a(3)=6 because 6=2+3+1.

easy,nonn

Giovanni Teofilatto (g.teofilatto(AT)tiscalinet.it), May 15 2005

approved