A081308 -id:A081308 - cp's OEIS frontend

A081311 Numbers that can be written as sum of a prime and an 3-smooth number.

3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75

Offset: 1

Reinhard Zumkeller, Mar 17 2003

nonn

A081308(a(n))>0; complement of A081310.

Up to 10^n this sequence has 8, 95, 916, 8871, 86974, 858055, 8494293, 84319349, 838308086, ... terms. The lower density is of this sequence is greater than 0.59368 (see Pintz), but seems to be less than 1; can this be proved? Charles R Greathouse IV, Sep 01 2015

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
J. Pintz, A note on Romanov's constant, Acta Mathematica Hungarica 112:1-2 (2006), pp. 1-14.

A118955 is a subsequence.
Union of A081312 and A081313.
Cf. A000040, A003586.

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 *)

is(n)=for(i=0, logint(n,3), my(k=3^i); while(kCharles R Greathouse IV, Sep 01 2015

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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A081311 Numbers that can be written 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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