A163829 -id:A163829 - cp's OEIS frontend

A163830 The n-th composite minus the product of the indices of the primes in its prime factorization.

3, 4, 7, 5, 7, 10, 10, 9, 15, 14, 17, 13, 17, 22, 16, 20, 19, 24, 24, 31, 23, 27, 23, 32, 30, 27, 37, 34, 39, 33, 37, 46, 33, 41, 37, 46, 46, 40, 52, 41, 48, 54, 51, 47, 63, 47, 56, 61, 51, 58, 68, 62, 57, 68, 57, 66, 77, 65, 69, 76, 64, 72, 67, 83, 78, 67, 83, 71, 79, 71, 94

Offset: 1

Juri-Stepan Gerasimov, Aug 05 2009

nonn

The product of the indices of the primes (counted with multiplicity) is represented by A003963. An intermediate sequence m-A003963(m) = 0, 1, 1, 3, 2, 4, 3, 7, 5, 7, 6, ... at m=1, 2, 3, ... is defined and evaluated where m=A002808(n) is composite.

			At n=1, A002808(1) = 4 and A003963(4)=1, so a(1) = 4 - 1 = 3.
At n=2, A002808(2) = 6 and A003963(6)=2, so a(2) = 6 - 2 = 4.
At n=3, A002808(3) = 8 and A003963(8)=1, so a(3) = 8 - 1 = 7.

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

Cf. A002808, A163515, A003963.

A002808 := proc(n) local a; if n = 1 then 4; else for a from procname(n-1)+1 do if not isprime(a) then RETURN(a) ; end if; end do; end if; end proc:
A163829 := proc(n) local c; c := A002808(n) ; pfs := ifactors(c)[2] ; mul( numtheory[pi](op(1,p))^op(2,p), p=pfs) ; end:
A163830 := proc(n) A002808(n)-A163829(n) ; end: seq(A163830(n),n=1..100) ; # R. J. Mathar, Aug 08 2009

With[{nn=100},#-Times@@(PrimePi/@Flatten[Table[#[[1]],{#[[2]]}]&/@ FactorInteger[#]])&/@Complement[Range[2,nn],Prime[Range[ PrimePi[ nn]]]]](* Harvey P. Dale, Mar 29 2012 *)

a(n) = A002808(n) - A003963(A002808(n)).

Edited by R. J. Mathar, Jul 08 2009

A163831 a(n) is the n-th composite minus the sum of the indices of the primes in its prime factorization.

Original entry on oeis.org

2, 3, 5, 5, 6, 8, 9, 10, 12, 13, 15, 15, 16, 19, 19, 19, 21, 22, 24, 27, 26, 26, 28, 30, 29, 31, 34, 35, 37, 38, 36, 42, 41, 43, 42, 44, 47, 47, 49, 47, 47, 53, 50, 55, 58, 56, 58, 59, 58, 62, 65, 61, 67, 66, 68, 69, 73, 73, 68, 76, 75, 71, 75, 80, 82, 81, 81, 80, 78, 84, 89, 89, 90, 92

Offset: 1

Juri-Stepan Gerasimov, Aug 05 2009

nonn

For the n-th composite c(n) = A002808(n), determine its canonical factorization c(n) = Product_{j} prime(j)^e_j; then A163515(n) = Sum_{j} e_j*j; a(n) = c(n) - A163515(n).

			A002808(1) = 4 = prime(1)*prime(1), so a(1) = 4 - (1+1) = 2.
A002808(2) = 6 = prime(1)*prime(2), so a(2) = 6 - (1+2) = 3.
A002808(3) = 8 = prime(1)*prime(1)*prime(1), so a(3) = 8 - (1+1+1) = 5.

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

Cf. A002808, A163515, A163829.

A002808 := proc(n) local a; if n = 1 then 4; else for a from procname(n-1)+1 do if not isprime(a) then RETURN(a) ; end if; end do; end if; end proc:
A163515 := proc(n) local c; c := A002808(n) ; pfs := ifactors(c)[2] ; add( op(2,p)*numtheory[pi](op(1,p)), p=pfs) ; end:
A163831 := proc(n) A002808(n)-A163515(n) ; end: seq(A163831(n),n=1..100) ; # R. J. Mathar, Aug 05 2009

cmsi[n_]:=n-Total[(PrimePi/@(Flatten[Table[#[[1]],#[[2]]]&/@ FactorInteger[ n]]))]; cmsi/@Select[Range[100],CompositeQ] (* Harvey P. Dale, Dec 15 2021 *)

a(n) = A002808(n) - A163515(n).

Edited and corrected by R. J. Mathar, Aug 05 2009

cp's OEIS Frontend

A163830 The n-th composite minus the product of the indices of the primes in its prime factorization.

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