A075135 -id:A075135 - cp's OEIS frontend

A075226 Largest prime in the numerator of the 2^n sums generated from the set 1, 1/2, 1/3,..., 1/n.

3, 11, 19, 137, 137, 1019, 2143, 7129, 7129, 78167, 81401, 1085933, 1111673, 1165727, 2364487, 41325407, 41325407, 796326437, 809074601, 812400209, 822981689, 19174119571, 19652175721, 99554817251, 100483070801

Offset: 2

T. D. Noe, Sep 08 2002

For the smallest odd prime not generated, see A075227. For information about how often the numerator of these sums is prime, see A075188 and A075189. The Mathematica program also prints the subset that yields the largest prime. For n <=20, the largest prime occurs in a sum of n-2, n-1, or n reciprocals.

			a(3) =11 because 11 is largest prime numerator in the three sums that yield primes: 1+1/2 = 3/2, 1/2+1/3 = 5/6 and 1+1/2+1/3 = 11/6.

Chai Wah Wu, Table of n, a(n) for n = 2..441 (terms 2..100 from Martin Fuller)
Martin Fuller, PARI program

Cf. A001008, A075135, A075188, A075189, A075227, A010051, A217712.

import Data.Ratio (numerator)
a075226 n = a075226_list !! (n-1)
a075226_list = f 2 [recip 1] where
   f x hs = (maximum $ filter ((== 1) . a010051') (map numerator hs')) :
            f (x + 1) hs' where hs' = hs ++ map (+ recip x) hs
-- Reinhard Zumkeller, May 28 2013

Needs["DiscreteMath`Combinatorica`"]; maxN=20; For[t={}; lst={}; mx=0; i=0; n=2, n<=maxN, n++, While[i<2^n-1, i++; s=NthSubset[i, Range[n]]; k=Numerator[Plus@@(1/s)]; If[PrimeQ[k], If[k>mx, t=s]; mx=Max[mx, k]]]; Print[n, " ", t]; AppendTo[lst, mx]]; lst
Table[Max[Select[Numerator[Total/@Subsets[1/Range[n],{2,2^n}]],PrimeQ]],{n,2,30}] (* The program will take a long time to run. *) (* Harvey P. Dale, Jan 08 2019 *)

PARI
```
See Fuller link.
```

from math import gcd, lcm
from itertools import combinations
from sympy import isprime
def A075226(n):
    m = lcm(*range(1,n+1))
    c, mlist = 0, tuple(m//i for i in range(1,n+1))
    for l in range(n,-1,-1):
        if sum(mlist[:l]) < c:
            break
        for p in combinations(mlist,l):
            s = sum(p)
            s //= gcd(s,m)
            if s > c and isprime(s):
                c = s
    return c # Chai Wah Wu, Feb 14 2022

More terms from Martin Fuller, Jan 19 2008

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

A051540 Least common multiple of {2, 5, 8, 11, 14, ..., 3n+2} (A016789).

Original entry on oeis.org

2, 10, 40, 440, 3080, 52360, 52360, 1204280, 15655640, 454013560, 1816054240, 1816054240, 34505030560, 1414706252960, 1414706252960, 66491193889120, 332455969445600, 17620166380616800, 17620166380616800

Offset: 0

Asher Auel

Denominator of H(n,3,2), a generalized harmonic number. See A075135.

			a(3) = lcm{2, 5, 8, 11} = 440.

Robert Israel, Table of n, a(n) for n = 0..1026 (first 100 terms from Vincenzo Librandi)

Cf. A016789, A051552. The numerators are in A074597.
Cf. A075135.
Cf. A051536.

List([0..20],n->Lcm(List([0..n],k->3*k+2))); # Muniru A Asiru, Apr 14 2018

k:=56; [Lcm([h: h in [2..j by 3]]): j in [2..k by 3]];  // Bruno Berselli, May 03 2011

A[0]:= 2:
for n from 1 to 60 do A[n]:= ilcm(A[n-1],3*n+2) od:
seq(A[n],n=0..60); # Robert Israel, Apr 10 2018

Table[ Denominator[ Sum[1/i, {i, 2/3, n}]], {n, 1, 20}]
Table[ Apply[ LCM, Join[{1}, Table[2 + 3i, {i, 0, n}]]], {n, 0, 19}]

a(n) = lcm(vector(n+1, k, 3*k-1)); \\ Michel Marcus, Apr 10 2018

Edited by Robert G. Wilson v, Aug 27 2002

Offset corrected by Robert Israel, Apr 10 2018

A075136 Numerator of the generalized harmonic number H(n,4,1).

Original entry on oeis.org

1, 6, 59, 812, 14389, 104038, 534113, 15837352, 177575597, 6681333014, 278042982799, 93928709068, 665521987201, 35665695484178, 684591747070657, 42155877944972752, 42527303541794647, 986175536059084606

Offset: 1

T. D. Noe, Sep 04 2002

The denominators are in A051539. See A075135 for more details.

Numerators of the partial sums of the divergent series 1/3 + 1/7 + 1/11 + . . 1/(4n-1).

Cf. A051539, A075135.

a=4; b=1; maxN=20; s=0; Numerator[Table[s+=1/(a n + b), {n, 0, maxN-1}]]
Numerator[Accumulate[1/Range[1,69,4]]] (* Harvey P. Dale, Dec 15 2014 *)

sumrecip(n,a,b) = { s=0; default(realprecision,n); forstep(j=b,n,a, s=s+1/j; print1(numerator(s)",") ) }

Sum 1/a(n) = 1.111939597509272224249... - Cino Hilliard, Dec 21 2003

A074638 Denominator of 1/3 + 1/7 + 1/11 + ... + 1/(4n-1).

Original entry on oeis.org

3, 21, 231, 385, 7315, 168245, 4542615, 140821065, 28164213, 366134769, 15743795067, 739958368149, 12579292258533, 62896461292665, 3710891216267235, 3710891216267235, 248629711489904745, 17652709515783236895, 88263547578916184475, 6972820258734378573525

Offset: 1

Robert G. Wilson v, Aug 27 2002

This s(n) := Sum_{j=0..n-1} 1/(4*j + 3), for n >= 1, equals (Psi(n + 3/4) - Psi(3/4))/4, with the digamma function Psi(z). See Abramowitz-Stegun, p. 258, eqs. 6.3.7 and 6.3.5, with z -> 3/4. A200134 = -Psi(3/4). - Wolfdieter Lang, Apr 06 2022

Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions. p.258, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. p. 258.

The numerators times 4 are A074637.

Cf. A075135, A200134.

Table[ Denominator[ Sum[1/i, {i, 3/4, n}]], {n, 1, 20}]

a(n) = denominator(sum(i=1, n, 1/(4*i-1))); \\ Michel Marcus, Mar 21 2021

from fractions import Fraction
def a(n): return sum(Fraction(1, 4*i-1) for i in range(1, n+1)).denominator
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Mar 21 2021

Denominator( (Psi(n + 3/4) - Psi(3/4))/4 ). See the comment above. - Wolfdieter Lang, Apr 05 2022

A074637 Numerator of 4 * H(n,4,3), a generalized harmonic number.

Original entry on oeis.org

4, 40, 524, 976, 20084, 491192, 13935164, 450160544, 93250876, 1249813672, 55206526972, 2657681947952, 46167204272716, 235410309457592, 14140794103168588, 14376406243883968, 978062783205294796

Offset: 1

Robert G. Wilson v, Aug 27 2002

The denominators are in A074638.

Cf. A075135.

Table[ Numerator[ Sum[1/i, {i, 3/4, n}]], {n, 1, 20}]

a(n) = numerator(4 * Sum_{j=0..n-1} 1/(4*j + 3)) = numerator(Psi(n + 3/4) - Psi(3/4)), with the Digamma function Psi(z). See a comment in A074638 with the Abramowitz-Stegun link. - Wolfdieter Lang, Apr 05 2022

Better description from T. D. Noe, Sep 04 2002

A075137 Numerator of the generalized harmonic number H(n,5,1).

Original entry on oeis.org

1, 7, 83, 697, 1685, 22521, 714167, 6551627, 273085171, 6372562445, 109738148749, 111017326363, 6843690854527, 6909897986791, 494972427791585, 9482037783487391, 85993305141830183, 3724238207261666261

Offset: 1

T. D. Noe, Sep 04 2002

The denominators are in A075138. See A075135 for more details.

Cf. A075135, A075138.

a=5; b=1; maxN=20; s=0; Numerator[Table[s+=1/(a n + b), {n, 0, maxN-1}]]

A075144 Denominator of the generalized harmonic number H(n,5,4).

Original entry on oeis.org

4, 36, 252, 4788, 9576, 277704, 4720968, 61372584, 675098424, 4725688968, 14177066904, 836446947336, 6691575578688, 153906238309824, 5694530817463488, 449867934579615552, 449867934579615552

Offset: 1

T. D. Noe, Sep 04 2002

The numerators are in A075143. See A075135 for more details.

Cf. A075135, A075143.

a=5; b=4; maxN=20; s=0; Denominator[Table[s+=1/(a n + b), {n, 0, maxN-1}]]

A104174 Numerator of the fractional part of a harmonic number.

Original entry on oeis.org

0, 1, 5, 1, 17, 9, 83, 201, 2089, 2341, 551, 2861, 64913, 90653, 114677, 274399, 5385503, 2022061, 42503239, 9276623, 3338549, 3573693, 87368107, 276977179, 7281378067, 7624597867, 71595952403, 74464289303, 2239777822987

Offset: 1

Georg Haass (geha5001(AT)stud.uni-saarland.de), Mar 10 2005

nonn

Clark Kimberling, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Book Stacking Problem.

Cf. A064169, A075135.

h[n_] := Sum[1/k, {k, 1, n}]
Table[Numerator[FractionalPart[h[n]]], {n, 1, 30}]
(* Clark Kimberling, Aug 13 2012 *)
FractionalPart[HarmonicNumber[Range[30]]]//Numerator (* Harvey P. Dale, Jul 28 2019 *)

a(n) = numerator(frac(sum(k=1, n, 1/k))); \\ Michel Marcus, Sep 27 2021

from sympy import harmonic
def A104174(n): return (lambda x: x.p % x.q)(harmonic(n)) # Chai Wah Wu, Sep 26 2021

a(n) = numerator(frac(Sum_{k=1..n} 1/k)). [edited by Michel Marcus, Sep 27 2021]

A075138 Denominator of the generalized harmonic number H(n,5,1).

Original entry on oeis.org

1, 6, 66, 528, 1232, 16016, 496496, 4468464, 183207024, 4213761552, 71633946384, 71633946384, 4369670729424, 4369670729424, 310246621789104, 5894685813992976, 53052172325936784, 2281243410015281712

Offset: 1

T. D. Noe, Sep 04 2002

The numerators are in A075137. See A075135 for more details.

Cf. A075135, A075137.

a=5; b=1; maxN=20; s=0; Denominator[Table[s+=1/(a n + b), {n, 0, maxN-1}]]

cp's OEIS Frontend

A075226 Largest prime 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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A051540 Least common multiple of {2, 5, 8, 11, 14, ..., 3n+2} (A016789).

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A075136 Numerator of the generalized harmonic number H(n,4,1).

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A074638 Denominator of 1/3 + 1/7 + 1/11 + ... + 1/(4n-1).

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PARI

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A074637 Numerator of 4 * H(n,4,3), a generalized harmonic number.

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Mathematica

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A075137 Numerator of the generalized harmonic number H(n,5,1).

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