A081311 -id:A081311 - cp's OEIS frontend

A118955 Numbers of the form 2^k + prime.

3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 23, 24, 25, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 47, 48, 49, 51, 53, 54, 55, 57, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 77, 79, 80, 81, 83, 84, 85, 87, 89, 90, 91, 93

Offset: 1

Reinhard Zumkeller, May 07 2006

nonn

A109925(a(n)) > 0, complement of A118954;
The lower density is at least 0.09368 (Pintz) and upper density is at most 0.49095 (Habsieger & Roblot). The density, if it exists, is called Romanov's constant. Romani conjectures that it is around 0.434. - Charles R Greathouse IV, Mar 12 2008
Elsholtz & Schlage-Puchta improve the bound on lower density to 0.107648. Unpublished work by Jie Wu improves this to 0.110114, see Remark 1 in Elsholtz & Schlage-Puchta. - Charles R Greathouse IV, Aug 06 2021

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.1, p. 87.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000.
Christian Elsholtz and Jan-Christoph Schlage-Puchta, On Romanov's constant, Mathematische Zeitschrift, Vol. 288 (2018), pp. 713-724.
Laurent Habsieger and Xavier-Francois Roblot, On integers of the form p + 2^k, Acta Arithmetica 122:1 (2006), pp. 45-50.
J. Pintz, A note on Romanov's constant, Acta Mathematica Hungarica 112:1-2 (2006), pp. 1-14.
F. Romani, Computations concerning primes and powers of two, Calcolo 20 (1983), pp. 319-336.

Subsequence of A081311; A118957 is a subsequence.

Cf. A156695, A010051, A000079.

a118955 n = a118955_list !! (n-1)
a118955_list = filter f [1..] where
   f x = any (== 1) $ map (a010051 . (x -)) $ takeWhile (< x) a000079_list
-- Reinhard Zumkeller, Jan 03 2014

Select[Range[100], (For[r=False; k=1, #>k, k*=2, If[PrimeQ[#-k], r=True]]; r)& ] (* Jean-François Alcover, Dec 26 2013, after Charles R Greathouse IV *)

is(n)=my(k=1);while(n>k,if(isprime(n-k),return(1),k*=2));0 \\ Charles R Greathouse IV, Mar 12 2008

list(lim)=my(v=List(),t=1); while(tCharles R Greathouse IV, Aug 06 2021

from itertools import count, islice
from sympy import isprime
def A118955_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n: any(isprime(n-(1<A118955_list = list(islice(A118955_gen(),30)) # Chai Wah Wu, Nov 29 2023

A081308 Number of ways to write n as sum of a prime and an 3-smooth number.

Original entry on oeis.org

0, 0, 1, 2, 2, 3, 2, 3, 3, 2, 4, 2, 3, 4, 4, 2, 3, 2, 5, 4, 5, 2, 5, 1, 5, 3, 4, 1, 6, 2, 5, 4, 3, 3, 7, 0, 5, 4, 5, 3, 5, 1, 6, 3, 5, 3, 6, 1, 7, 4, 4, 1, 6, 1, 8, 4, 3, 1, 7, 1, 7, 3, 4, 2, 8, 1, 7, 3, 5, 3, 7, 1, 6, 4, 7, 2, 10, 0, 8, 3, 3, 2, 9, 2, 9, 3, 4, 3, 6, 1, 9, 3, 3, 2, 9, 0, 5, 5, 4, 3, 8, 1, 7, 3, 6

Offset: 1

Reinhard Zumkeller, Mar 17 2003

nonn

Asymptotically, for n coprime to 6, a(n) ~ C*n on the average, with C=3/(2*log(2)*log(3))~1.969796..., see the link. - M. F. Hasler, Oct 21 2011

a(A081310(n)) = 0; a(A081311(n)) > 0; a(A081312(n)) = 1; a(A081313(n)) > 1.

			a(12)=2: 12=11+1=3+3^2; a(13)=3: 13=11+2=7+2*3=5+2^3.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Mark Underwood, another goldbachian theme, Mar. 16, 2009 and follow-up on Oct. 21, 2011.
Mark Underwood and others, Another goldbachian theme, digest of 64 messages in primenumbers Yahoo group, Mar 16, 2009 - Oct 30, 2011. [Cached copy]

Cf. A081310, A081311, A081309, A000040, A003586.

Cf. A010051.

a081308 n = sum $ map (a010051' . (n -)) $ takeWhile (< n) a003586_list
-- Reinhard Zumkeller, Jul 04 2012

nmax = 1000;
S = Select[Range[nmax], Max[FactorInteger[#][[All, 1]]] <= 3&];
a[n_] := Count[TakeWhile[S, #Jean-François Alcover, Oct 13 2021 *)

A081308(n)=my(L2=log(2));sum(e3=0,log(n+.5)\log(3), sum(e2=0,log(n\3^e3)\L2, isprime(n-(3^e3)<M. F. Hasler, Oct 21 2011

A081310 Numbers having no representation as sum of a prime and an 3-smooth number.

Original entry on oeis.org

1, 2, 36, 78, 96, 120, 126, 144, 156, 162, 186, 204, 210, 216, 222, 276, 288, 300, 306, 324, 328, 330, 336, 342, 366, 372, 378, 396, 408, 414, 426, 438, 456, 474, 486, 498, 516, 528, 534, 540, 546, 552, 562, 576, 582, 606, 612, 624, 630, 636, 666, 672, 690

Offset: 1

Reinhard Zumkeller, Mar 17 2003

nonn

Complement of A081311.

			For all primes p<36 the greatest prime factor of 36-p is >3: 36-2=2*17, 36-3=3*11, 36-5=31, 36-7=29, 36-11=5*5, 36-13=23, 36-17=19, 36-19=17, 36-23=13, 36-29=7, 36-31=5, therefore 36 is a term.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A003586, A081308, A081311.

a081310 n = a081310_list !! (n-1)
a081310_list = filter ((== 0) . a081308) [1..]
-- Reinhard Zumkeller, Jul 04 2012

nmax = 1000;
S = Select[Range[nmax], Max[FactorInteger[#][[All, 1]]] <= 3 &];
A081308[n_] := Count[TakeWhile[S, # < n &], s_ /; PrimeQ[n - s]];
Select[Range[nmax], A081308[#] == 0 &] (* Jean-François Alcover, Oct 13 2021 *)

A081308(a(n)) = 0.

A081312 Numbers having a unique representation as sum of a prime and an 3-smooth number.

Original entry on oeis.org

3, 24, 28, 42, 48, 52, 54, 58, 60, 66, 72, 90, 102, 108, 114, 132, 138, 150, 168, 172, 174, 180, 192, 196, 198, 214, 228, 234, 240, 246, 252, 264, 268, 270, 282, 294, 298, 312, 318, 348, 354, 360, 384, 390, 402, 404, 420, 432, 444, 450, 462, 468, 478, 480, 492

Offset: 1

Reinhard Zumkeller, Mar 17 2003

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A003586, A081308, A081313.

Subsequence of A081311.

a081312 n = a081312_list !! (n-1)
a081312_list = filter ((== 1) . a081308) [1..]
-- Reinhard Zumkeller, Jul 04 2012

sp3sQ[n_]:=Length[Select[IntegerPartitions[n,{2}],(PrimeQ[#[[1]]]&&Max[ FactorInteger[#[[2]]][[All,1]]]<4)||(PrimeQ[#[[2]]]&&Max[ FactorInteger[ #[[1]]][[All,1]]]<4)&]]==1; Select[Range[500],sp3sQ]/.(5->Nothing) (* Harvey P. Dale, Feb 05 2019 *)
nmax = 1000;
S = Select[Range[nmax], Max[FactorInteger[#][[All, 1]]] <= 3 &];
A081308[n_] := Count[TakeWhile[S, # < n &], s_ /; PrimeQ[n - s]];
Select[Range[nmax], A081308[#] == 1 &] (* Jean-François Alcover, Oct 13 2021 *)

A081308(a(n)) = 1.

A081313 Numbers having more than one representation as sum of a prime and a 3-smooth number.

Original entry on oeis.org

4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 53, 55, 56, 57, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 84, 85, 86

Offset: 1

Reinhard Zumkeller, Mar 17 2003

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A003586, A081308, A081312.

Subsequence of A081311.

a081313 n = a081313_list !! (n-1)
a081313_list = filter ((> 1) . a081308) [1..]
-- Reinhard Zumkeller, Jul 04 2012

nmax = 1000;
S = Select[Range[nmax], Max[FactorInteger[#][[All, 1]]] <= 3 &];
A081308[n_] := Count[TakeWhile[S, # < n &], s_ /; PrimeQ[n - s]];
Select[Range[nmax], A081308[#] > 1 &] (* Jean-François Alcover, Oct 13 2021 *)

A081308(a(n)) > 1.

A081309 Smallest prime p such that n-p is a 3-smooth number, a(n)=0 if no such prime exists.

Original entry on oeis.org

0, 0, 2, 2, 2, 2, 3, 2, 3, 2, 2, 3, 5, 2, 3, 7, 5, 2, 3, 2, 3, 13, 5, 23, 7, 2, 3, 19, 2, 3, 7, 5, 17, 2, 3, 0, 5, 2, 3, 13, 5, 41, 7, 17, 13, 19, 11, 47, 13, 2, 3, 43, 5, 53, 7, 2, 3, 31, 5, 59, 7, 53, 31, 37, 11, 2, 3, 41, 5, 43, 7, 71, 19, 2, 3, 67, 5, 0, 7, 53, 17, 73, 2, 3, 13, 5, 23, 7, 17, 89

Offset: 1

Reinhard Zumkeller, Mar 17 2003

			a(25)=7: 25=7+2*3^2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A081310, A081311, A081308, A000040, A003586.

Cf. A065333.

a081309 n | null ps   = 0
          | otherwise = head ps
          where ps = [p | p <- takeWhile (< n) a000040_list,
                          a065333 (n - p) == 1]
-- Reinhard Zumkeller, Jul 04 2012

smooth3Q[n_] := n/2^IntegerExponent[n, 2]/3^IntegerExponent[n, 3] == 1;
a[n_] := Module[{p}, For[p = 2, p < n, p = NextPrime[p], If[smooth3Q[n - p], Return[p]]]; 0];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 14 2021 *)

a(n)=0 iff A081308(n)=0.

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A118955 Numbers of the form 2^k + prime.

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A081308 Number of ways to write n as sum of a prime and an 3-smooth number.

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A081310 Numbers having no representation as sum of a prime and an 3-smooth number.

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A081312 Numbers having a unique representation as sum of a prime and an 3-smooth number.

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A081313 Numbers having more than one representation as sum of a prime and a 3-smooth number.

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A081309 Smallest prime p such that n-p is a 3-smooth number, a(n)=0 if no such prime exists.

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