A075226 -id:A075226 - cp's OEIS frontend

A075188 Number of times that the numerator of a sum generated from the set 1, 1/2, 1/3,..., 1/n is prime.

0, 1, 3, 9, 19, 43, 79, 162, 307, 607, 1075, 2186, 3872, 7573, 15101, 29139, 52295, 104953, 189915, 379275, 754081, 1462115, 2675851, 5351541, 10254019, 19987942, 38901233, 77620568, 144021667, 288428481, 537642772, 1056802340, 2113152353, 4138261885

Offset: 1

T. D. Noe, Sep 08 2002

Note that for each n there are only 2^(n-1) new sums to consider. Surprisingly, nearly half of the sums have a prime numerator. For the number of distinct primes, see A075189. For the largest generated prime, see A075226. For the smallest odd prime not generated, see A075227.

A217712(n) = number of primes occurring exactly once as numerators. - Reinhard Zumkeller, Jun 02 2013

			a(3) = 3 because 3 sums yield prime numerators: 1+1/2 = 3/2, 1/2+1/3 = 5/6 and 1+1/2+1/3 = 11/6.

Cf. A001008, A075135, A075189, A075226, A075227.

Cf. A010051.

import Data.Ratio (numerator)
a075188 n = a075188_list !! (n-1)
a075188_list = f 1 [] where
   f x hs = (length $ filter ((== 1) . a010051') (map numerator hs')) :
            f (x + 1) hs' where hs' = hs ++ map (+ recip x) (0 : hs)
-- Reinhard Zumkeller, May 28 2013

Needs["DiscreteMath`Combinatorica`"]; maxN=20; For[cnt=0; lst={}; i=0; n=1, n<=maxN, n++, While[i<2^n-1, i++; s=NthSubset[i, Range[n]]; k=Numerator[Plus@@(1/s)]; If[PrimeQ[k], cnt++ ]]; AppendTo[lst, cnt]]; lst

a(21)-a(25) by Reinhard Zumkeller, May 28 2013
a(26)-a(31) from Chai Wah Wu, Feb 14 2022
a(32)-a(34) from Sean A. Irvine, Feb 10 2025

A075189 Number of distinct primes in the numerator of the 2^n sums generated from the set 1, 1/2, 1/3, ..., 1/n.

Original entry on oeis.org

0, 1, 3, 6, 14, 20, 38, 74, 134, 232, 486, 526, 1078, 2036, 2505, 4762, 9929, 14598, 29831, 31521, 52223, 101123, 207892, 215796, 426772, 836665, 1640357, 1689653, 3401483, 3471770, 6868800, 13470379, 23182192, 45792615, 47136366

Offset: 1

T. D. Noe, Sep 08 2002

Every prime is generated eventually. For the largest generated prime, see A075226. For the smallest odd prime not generated, see A075227.

A217712(n) = number of primes occurring exactly once as numerators among the 2^n sums. - Reinhard Zumkeller, Jun 02 2013

			a(3) = 3 because 3 sums yield distinct prime numerators: 1+1/2 = 3/2, 1/2+1/3 = 5/6 and 1+1/2+1/3 = 11/6.

Cf. A001008, A075135, A075188, A075226, A075227.

Cf. A010051.

import Data.Ratio ((%), numerator)
import Data.Set (Set, empty, fromList, toList, union, size)
a075189 n = a075189_list !! (n-1)
a075189_list = f 1 empty empty where
   f x s s1 = size s1' : f (x + 1) (s `union` fromList hs) s1' where
     s1' = s1 `union` fromList
           (filter ((== 1) . a010051') $ map numerator hs)
     hs = map (+ 1 % x) $ 0 : toList s
-- Reinhard Zumkeller, May 28 2013

Needs["DiscreteMath`Combinatorica`"]; maxN=20; For[lst={}; prms={}; i=0; n=1, n<=maxN, n++, While[i<2^n-1, i++; s=NthSubset[i, Range[n]]; k=Numerator[Plus@@(1/s)]; If[PrimeQ[k], prms=Union[prms, {k}]]]; AppendTo[lst, Length[prms]]]; lst

a(21)-a(29) from Reinhard Zumkeller, May 28 2013

a(30)-a(35) from Sean A. Irvine, Feb 10 2025

A075227 Smallest odd prime not occurring in the numerator of any of the 2^n subset sums generated from the set 1/1, 1/2, 1/3, ..., 1/n.

Original entry on oeis.org

3, 5, 7, 17, 37, 43, 43, 151, 151, 409, 491, 491, 491, 1087, 2011, 3709, 3709, 7417, 7417, 7417, 19699, 30139, 35573, 35573, 40237, 40237, 132151, 132151, 158551, 158551, 245639, 245639, 961459, 1674769, 1674769, 1674769, 1674769, 4339207

Offset: 1

T. D. Noe, Sep 08 2002

The largest prime generated is given in A075226. For information about how often the numerator of these sums is prime, see A075188 and A075189.

			a(3) = 7 because 7 is the smallest prime not occurring in the numerator of any of the sums 1/1 + 1/2 = 3/2, 1/1 + 1/3 = 4/3, 1/2 + 1/3 = 5/6 and 1/1 + 1/2 + 1/3 = 11/6.

Cf. A001008, A075135, A075188, A075189, A075226.
Cf. A010051, A065091.
Cf. A217712.

import Data.Ratio ((%), numerator)
import Data.Set (Set, empty, fromList, toList, union)
a075227 n = a075227_list !! (n-1)
a075227_list = f 1 empty a065091_list where
   f x s ps = head qs : f (x + 1) (s `union` fromList hs) qs where
     qs = foldl (flip del)
          ps $ filter ((== 1) . a010051') $ map numerator hs
     hs = map (+ 1 % x) $ 0 : toList s
   del u vs'@(v:vs) = case compare u v
                      of LT -> vs'; EQ -> vs; GT -> v : del u vs
-- Reinhard Zumkeller, May 28 2013

Needs["DiscreteMath`Combinatorica`"]; maxN=20; For[lst={}; prms={}; i=0; n=1, n<=maxN, n++, While[i<2^n-1, i++; s=NthSubset[i, Range[n]]; k=Numerator[Plus@@(1/s)]; If[PrimeQ[k], AppendTo[prms, k]]]; prms=Union[prms]; j=2; While[MemberQ[prms, Prime[j]], j++ ]; AppendTo[lst, Prime[j]]]; lst
(* Second program; does not need Combinatorica *)
a[1] = 3; a[2] = 5; a[n_] := For[nums = (Total /@ Subsets[1/Range[n]]) // Numerator // Union // Select[#, PrimeQ]&; p = 3, p <= Last[nums], p = NextPrime[p], If[FreeQ[nums, p], Print[n, " ", p]; Return[p]]];
Table[a[n], {n, 1, 23}] (* Jean-François Alcover, Sep 10 2017 *)

from sympy import sieve
from fractions import Fraction
fracs, newnums, primeset = {0}, {0}, set(sieve.primerange(3, 10**6+1))
for n in range(1, 24):
  newfracs = set(Fraction(1, n) + f for f in fracs)
  fracs |= newfracs
  primeset -= set(f.numerator for f in newfracs)
  print(min(primeset), end=", ") # Michael S. Branicky, May 09 2021

a(21)-a(28) from Reinhard Zumkeller, May 28 2013
a(29)-a(33) from Jon E. Schoenfield, May 09 2021
a(34)-a(36) from Michael S. Branicky, May 10 2021
a(37)-a(38) from Michael S. Branicky, May 12 2021

A217712 Number of primes occurring exactly once as numerators in sums generated from the set 1, 1/2, 1/3,..., 1/n.

Original entry on oeis.org

0, 1, 3, 3, 11, 13, 27, 54, 106, 168, 378, 142, 733, 1597, 1283, 3418, 8204, 10112, 24644, 7829, 32866, 78136, 178741, 37002, 256392, 650596, 1402914, 286854, 2053463

Offset: 1

Reinhard Zumkeller, Jun 02 2013

nonn

For information about how often the numerator of the generated sums is prime, see A075188 and A075189; for the largest generated prime, see A075226; for the smallest odd prime not generated, see A075227.

			For n=3 there are the following fractions as sums of 1, 1/2 and 1/3:
{1/3, 1/2, 5/6, 1, 4/3, 3/2, 11/6}, three numerators are prime and they occur exactly once, therefore a(3) = A075188(3) = A075189(3) = #{3, 5, 11} = 3;
n=4: adding 1/4 to the previous fractions gives together: 1/4, 1/3, 1/2, 1/3+1/4=7/12, 1/2+1/4=3/4, 5/6, 1, 5/6+1/4=13/12, 1+1/4=5/4, 4/3, 3/2, 4/3+1/4=19/12, 3/2+1/4=7/4, 11/6 and 11/6+1/4=25/12:
A075188(4) = #{7/12, 3/4, 5/6, 13/12, 5/4, 3/2, 19/12, 7/4, 11/6} = 9,
A075189(4) = #{3, 5, 7, 11, 13, 19} = 6,
a(4) = #{11, 13, 19} = 3.

Cf. A010051.

import Data.Ratio ((%), numerator)
import Data.Set (Set, empty, fromList, toList, union, size)
import Data.Set (member, delete, insert)
a217712 n = a217712_list !! (n-1)
a217712_list = f 1 empty empty where
   f x s s1 = size s1' : f (x + 1) (s `union` fromList hs) s1' where
     s1' = g s1 $ filter ((== 1) . a010051') $ map numerator hs
     g v []                    = v
     g v (w:ws) | w `member` v = g (delete w v) ws
                | otherwise    = g (insert w v) ws
     hs = map (+ 1 % x) $ 0 : toList s

A256222 Largest Fibonacci number in the numerator of the 2^n sums generated from the set 1, 1/2, 1/3, ..., 1/n.

Original entry on oeis.org

0, 1, 3, 5, 13, 13, 13, 89, 89, 89, 1597, 1597, 1597, 1597, 1597, 1597, 17711, 17711, 17711, 28657, 28657, 28657, 28657, 1346269, 1346269, 1346269, 1346269, 24157817, 24157817, 24157817, 24157817, 24157817, 24157817, 39088169, 39088169, 39088169, 39088169

Offset: 0

Michel Lagneau, Mar 19 2015

nonn

The prime Fibonacci numbers in the sequence are 3, 5, 13, 89, 1597, 28657, ...

For information about how often the numerator of these sums is a Fibonacci number, see A256220 and A256221.

			a(3) = 5 because we obtain the 5 subsets {1}, {1/2}, {1/3}, {1,1/2} and {1/2, 1/3} having 5 sums with Fibonacci numerators: 1, 1, 1, 1+1/2 = 3/2 and 1/2+1/3 = 5/6 => the greatest Fibonacci number is 5.

Hiroaki Yamanouchi, Table of n, a(n) for n = 0..50

Cf. A000045, A005478, A075226, A256220, A256221, A256223.

<<"DiscreteMath`Combinatorica`"; maxN=24; For[t={}; mx=0; i=0; n=0, n<=maxN, n++, While[i<2^n-1, i++; s=NthSubset[i, Range[n]]; k=Numerator[Plus@@(1/s)]; If[IntegerQ[Sqrt[5*k^2+4]]||IntegerQ[Sqrt[5*k^2-4]], If[k>mx, t=s]; mx=Max[mx, k]]]; Print[mx]]

Corrected and extended by Alois P. Heinz, Mar 25 2015

a(30)-a(36) from Hiroaki Yamanouchi, Mar 30 2015

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A075188 Number of times that the numerator of a sum generated from the set 1, 1/2, 1/3,..., 1/n is prime.

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A075189 Number of distinct primes in the numerator of the 2^n sums generated from the set 1, 1/2, 1/3, ..., 1/n.

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A075227 Smallest odd prime not occurring in the numerator of any of the 2^n subset sums generated from the set 1/1, 1/2, 1/3, ..., 1/n.

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A217712 Number of primes occurring exactly once as numerators in sums generated from the set 1, 1/2, 1/3,..., 1/n.

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A256222 Largest Fibonacci number in the numerator of the 2^n sums generated from the set 1, 1/2, 1/3, ..., 1/n.

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