A069346 -id:A069346 - cp's OEIS frontend

A069350 Primes having no representation of the form n - A001222(n), complement of A069346.

3, 11, 29, 59, 61, 71, 79, 97, 101, 137, 149, 173, 191, 193, 197, 223, 229, 269, 271, 277, 331, 349, 383, 397, 419, 421, 457, 461, 523, 569, 613, 619, 659, 661, 673, 709, 733, 757, 773, 809, 853, 857, 859, 881, 907, 997

Offset: 1

Reinhard Zumkeller, Mar 18 2002

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A069351.

is(n)=for(k=1, log(n)\log(2)+1, if(bigomega(n+k)==k, return(0))); isprime(n) \\ Charles R Greathouse IV, Dec 10 2013

A069347 Number of ways to represent primes as A069346(n)=k-A001222(k).

Original entry on oeis.org

2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 4, 1, 3, 2, 2, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1

Offset: 1

Reinhard Zumkeller, Mar 18 2002

nonn

			a(12)=2 as: A069346(12)=47=A069345(49)=A069345(50)
47=7*7-2 and 47=5*5*2-3.

A069348 Primes of the form k + Omega(k), where Omega(k) is the number of prime factors of k, A001222(k).

Original entry on oeis.org

3, 11, 17, 23, 31, 37, 41, 47, 53, 59, 67, 71, 73, 79, 89, 97, 101, 113, 127, 131, 139, 151, 157, 163, 167, 173, 179, 181, 191, 193, 199, 211, 223, 229, 233, 239, 241, 251, 257, 269, 271, 277, 293, 307, 311, 313, 317, 331, 337

Offset: 1

Reinhard Zumkeller, Mar 18 2002

nonn

The representation is ambiguous, see A069349.

			a(10) = 59 = A064800(57) = 57 + 2 (as 57 = 3*19).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A069346, A069357.

Select[Table[n + PrimeOmega[n], {n, 1, 400}], PrimeQ] // Union (* Jean-François Alcover, Jun 08 2013 *)

is(n)=for(k=1,log(n+.5)\log(2),if(bigomega(n-k)==k,return(isprime(n)))); 0 \\ Charles R Greathouse IV, Dec 10 2013

Corrected by T. D. Noe, Nov 01 2006

A069356 Primes of the form 2^i*3^j - (i+j) with i, j >= 0.

Original entry on oeis.org

2, 5, 7, 43, 67, 103, 157, 281, 503, 641, 1451, 3061, 4597, 6553, 8179, 10357, 15541, 34981, 78721, 209939, 524269, 1062869, 2097131, 13436909, 25509151, 28311529, 63700969, 113246183, 153054989, 516560633, 573308903, 774840959, 805306339

Offset: 1

Reinhard Zumkeller, Mar 18 2002

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..300 [Corrected by Sean A. Irvine]

Cf. A069355, A005105, A069358, A069346.

Take[ Select[ Union[ Flatten[ Table[2^i*3^j - (i + j), {i, 0, 28}, {j, 0, 18}]]], PrimeQ[ # ] &], 33]

Edited and extended by Robert G. Wilson v, May 09 2003

A167133 Primes of the form (number of prime factors of k-th composite) plus (number of divisors of k-th composite).

Original entry on oeis.org

5, 7, 5, 5, 7, 11, 11, 13, 11, 5, 13, 11, 11, 17, 11, 13, 11, 11, 17, 11, 11, 5, 7, 11, 11, 11, 11, 5, 11, 11, 23, 11, 11, 11, 11, 13, 17, 11, 13, 11, 11, 11, 11, 11, 23, 11, 17, 11, 11, 11, 11, 11, 11, 5, 11, 23, 11, 11, 11, 7, 11, 11, 11, 5, 11, 11, 11, 11, 17, 23, 11, 11, 11, 11

Offset: 1

Juri-Stepan Gerasimov, Oct 28 2009

nonn

Contains every prime > 3 infinitely many times, as A000005(p^k)+A001222(p^k)=2*k+1 for prime p. - Robert Israel, Sep 30 2020

			a(1) = 2+3 = 5 (for 1st composite=4); a(2) = 3+4 = 7 (for 3rd composite=8).

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

Cf. A000005, A000040, A002808, A001222, A069346.

Cf. A035004, A062502.

f:= proc(n) local F,x;
  if isprime(n) then return NULL fi;
  F:= ifactors(n)[2];
  x:= add(t[2],t=F) + mul(1+t[2],t=F);
  if isprime(x) then x fi
end proc:
map(f, [$4..1000]); # Robert Israel, Sep 30 2020

Corrected and extended by R. J. Mathar, Oct 29 2009

A173032 Primes having no representation of the form n -+ Omega(n), where Omega is A001222.

Original entry on oeis.org

29, 61, 137, 149, 197, 349, 457, 523, 569, 619, 659, 1091, 1153, 1303, 1483, 1549, 1723, 1861, 1873, 2339, 2347, 2377, 2593, 2753, 2851, 3121, 3169, 3257, 3323, 3391, 3469, 3539, 4051, 4177, 4373, 4423, 4483, 4547, 4691, 4799, 4877, 4933, 4943, 5101, 5237

Offset: 1

Juri-Stepan Gerasimov, Nov 21 2010

nonn

Intersection of A069350 and A069351.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A069346, A069348, A001222.

is(n)=for(k=1, log(n)\log(2)+1, if(bigomega(n+k)==k||bigomega(n-k)==k, return(0))); isprime(n) \\ Charles R Greathouse IV, Dec 10 2013

cp's OEIS Frontend

A069350 Primes having no representation of the form n - A001222(n), complement of A069346.

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A069347 Number of ways to represent primes as A069346(n)=k-A001222(k).

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A069348 Primes of the form k + Omega(k), where Omega(k) is the number of prime factors of k, A001222(k).

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A069356 Primes of the form 2^i*3^j - (i+j) with i, j >= 0.

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Mathematica

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A167133 Primes of the form (number of prime factors of k-th composite) plus (number of divisors of k-th composite).

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A173032 Primes having no representation of the form n -+ Omega(n), where Omega is A001222.

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PARI