A024451 -id:A024451 - cp's OEIS frontend

A250130 Numerator of the harmonic mean of the first n primes.

2, 12, 90, 840, 11550, 180180, 3573570, 77597520, 2007835830, 64696932300, 2206165391430, 89048857617720, 3955253425853730, 183158658643380420, 9223346738827371150, 521426535635040715680, 32686925952621614864190, 2111190864469325477698860

Offset: 1

Colin Barker, Nov 13 2014

			a(3) = 90 because the first 3 primes are [2,3,5] and 3 / (1/2+1/3+1/5) = 90/31.
The first fractions are 2/1, 12/5, 90/31, 840/247, 11550/2927, 180180/40361, 3573570/716167, 77597520/14117683, ...

Colin Barker, Table of n, a(n) for n = 1..300

Cf. A024451 (denominators), A002110 (primorial numbers).

N:= 100: # to get a(1) to a(N)
B:= ListTools:-PartialSums([seq](1/ithprime(i),i=1..N)):
seq(numer(n/B[n]), n=1..N); # Robert Israel, Nov 13 2014

Table[n/Sum[1/Prime[k],{k,1,n}],{n,1,20}]//Numerator (* Vaclav Kotesovec, Nov 13 2014 *)
Table[n*Product[Prime[j], {j, n}], {n, 17}] (* L. Edson Jeffery, Jan 04 2015 *)

harmonicmean(v) = #v / sum(k=1, #v, 1/v[k])
s=vector(30); p=primes(#s); for(k=1, #p, s[k]=numerator( harmonicmean( vector(k, i, p[i])))); s

n=0; P=1; forprime(p=2, 100, n++; P *= p; print1(n*P, ", ")) \\ Jeppe Stig Nielsen, Aug 11 2019

from sympy import prime
from fractions import Fraction
def a(n):
  return (n/sum(Fraction(1, prime(k)) for k in range(1, n+1))).numerator
print([a(n) for n in range(1, 19)]) # Michael S. Branicky, Feb 12 2021

a(n) = n*A002110(n). - L. Edson Jeffery, Jan 04 2015

A354417 a(n) is the numerator of the sum of the reciprocals of the first n squarefree numbers.

Original entry on oeis.org

1, 3, 11, 61, 11, 82, 171, 1951, 26133, 13424, 41273, 716656, 13871719, 4700888, 9548741, 222854273, 112857219, 3310041496, 20075905417, 628822761157, 19239404599, 9709078632, 1959180271, 73097429088, 147378388979, 445594718515, 18404305970657, 3089336006908, 133763418792581

Offset: 1

Ilya Gutkovskiy, May 26 2022

			1, 3/2, 11/6, 61/30, 11/5, 82/35, 171/70, 1951/770, 26133/10010, 13424/5005, 41273/15015, ...

Robert Israel, Table of n, a(n) for n = 1..1433
Sebastian Zuniga Alterman, Explicit averages of square-free supported functions: to the edge of the convolution method, Colloquium Mathematicum, Vol. 168 (2022), pp. 1-23; arXiv preprint, arXiv:2003.05887 [math.NT], 2020.
Olivier Ramaré, Explicit average orders: news and problems, Banach Center Publications, Vol. 118 (2019), pp. 153-176.

Cf. A001008, A005117, A024451, A072980, A096795, A106830, A354418 (denominators).

Cf. A001620, A059956, A306016.

s:= 0: R:= NULL: count:= 0:
for x from 1 while count < 40 do
  if numtheory:-issqrfree(x) then
    s:= s + 1/x;
    v:= numer(s);
    R:= R, v;
    count:= count+1;
  fi;
od:
R; # Robert Israel, Mar 05 2023

Accumulate[1/Select[Range[43], SquareFreeQ]] // Numerator

a(n) = my(i=0, s=0); for(x=1, oo, if(core(x)==x, s+=1/x; i++; if(i==n, return(numerator(s))))) \\ Felix Fröhlich, May 26 2022

a(n)/A354418(n) ~ (6/Pi^2) * (log(n) + c) + O*(1.044/sqrt(n)), where f = O*(g) means |f| <= g and c = gamma + 2 * Sum_{p prime} log(p)/(p^2-1) = A001620 + 2 * A306016 = 1.71713765109059847340... (Ramaré, 2019; Alterman, 2022). - Amiram Eldar, Oct 29 2022

A369972 Numbers k such that (prime(k)#)' is a multiple of prime(1+k), where prime(k)# means the k-th primorial, A002110(k), and ' stands for taking the arithmetic derivative, A003415.

Original entry on oeis.org

0, 2, 7, 14, 21, 28, 261202

Offset: 1

Antti Karttunen, Feb 07 2024

Numbers k for which A024451(k) is a multiple of A000040(1+k).

			7 is included because the primorial prime(7)# = A002110(7) = 510510 has as its arithmetic derivative 510510' = A024451(7) = 716167 = 19*37693, which is divisible by the next larger prime not present in the primorial, in this case by prime(8) = 19.

Index entries for sequences related to primorial numbers

Cf. A000040, A000720, A024451, A293457 (corresponding primes), A369970, A369973 (corresponding primorials).

Cf. also A109628.

A024451(n) = numerator(sum(i=1, n, 1/prime(i)));
isA369972(n) = !(A024451(n)%prime(1+n));

a(n) = A000720(A293457(n)) - 1.

Found a(7) by computing it as A000720(A293457(7))-1. - Antti Karttunen, Feb 08 2024

A369973 Primorials whose arithmetic derivative is divisible by the next larger prime not present in that primorial.

Original entry on oeis.org

1, 6, 510510, 13082761331670030, 40729680599249024150621323470, 2566376117594999414479597815340071648394470

Offset: 1

Antti Karttunen, Feb 07 2024

nonn

Primorials A002110(k) such that A003415(A002110(k)) [= A024451(k)] is a multiple of A000040(1+k).

a(7) = A002110(261202), which is too large to include here, or even in a b-file.

			The zeroth primorial, 1 = A002110(0), is included, because its arithmetic derivative 1' = A024451(0) = 0 is divisible by the next larger prime not present in the primorial, in this case by prime(1) = 2.
The primorial 510510 = prime(7)# is included, because its arithmetic derivative 510510' = A024451(7) = 716167 = 19*37693 is divisible by the next larger prime not present in the primorial, in this case by prime(8) = 19.

Index entries for sequences related to primorial numbers

Cf. A000040, A002110, A003415, A024451, A293457 (the corresponding primes), A369972.

Subsequence of A369970.

A002110(n) = prod(i=1,n,prime(i));
A024451(n) = numerator(sum(i=1, n, 1/prime(i)));
isA369972(n) = !(A024451(n)%prime(1+n));
for(n=0,2^10,if(isA369972(n),print1(A002110(n),", ")));

a(n) = A002110(A369972(n)).

A370130 a(n) = A369669(A276086(n)), where A369669 is the greatest common divisor of the first and second arithmetic derivative of n, and A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 3, 1, 1, 16, 1, 5, 5, 5, 5, 5, 5, 100, 25, 25, 175, 25, 25, 1, 3, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4, 17, 1, 1, 5, 5, 5, 5, 20, 5, 25, 25, 25, 25, 325, 25, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 11, 1, 4, 1, 1, 1, 1, 1, 5, 5, 320, 95, 5, 5, 25, 25, 25, 25, 100, 25, 7, 7, 112, 7, 7, 7, 7, 7, 7, 7, 28

Offset: 0

Antti Karttunen, Feb 10 2024

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for sequences related to primorial base

Cf. A003415, A068346, A085731, A276086, A327860, A328242 (positions of 1's), A370131.

Cf. A000012, A024451, A143293.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A369669(n) = { my(d=A003415(n)); gcd(d, A003415(d)); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A370130(n) = A369669(A276086(n));

a(n) = A369669(A276086(n)).
a(n) = gcd(A327860(n), A370131(n)).
For n >= 1, a(n) = A085731(A327860(n)).

A074107 a(n) = Product of (prime + 1) for first n primes - primorial (n); Sum of proper divisors of the n-th primorial.

Original entry on oeis.org

0, 1, 6, 42, 366, 4602, 66738, 1231314, 25136790, 612982650, 18612572370, 602072009070, 23079296834790, 976751205195990, 43281303292150770, 2090585319354906990, 113506497027753468870, 6842978980142398176930, 426187457118982899608730, 29098035465450244144376910, 2102916875063497845451016610, 156173789584825539524342644530

Offset: 0

Amarnath Murthy, Aug 22 2002

nonn

			a(3) = (2+1)*(3+1)*(5+1) - 2*3*5 = 72 - 30 = 42.

Antti Karttunen, Table of n, a(n) for n = 0..349 (terms 1..349 from Harvey P. Dale)
Index entries for sequences related to primorial numbers

Cf. A001065, A002110, A003415, A024451, A051674, A054640, A070826, A348507.

for n from 1 to 25 do a[n] := product(ithprime(i)+1,i=1..n)-product(ithprime(i),i=1..n): od:seq(a[j],j=1..25);

Module[{nn=20,p,pr,pr1},p=Prime[Range[nn]];pr=FoldList[Times,1,p];pr1= FoldList[Times,1,p+1];#[[2]]-#[[1]]&/@Rest[Thread[{pr,pr1}]]](* Harvey P. Dale, Feb 07 2015 *)

A074107(n) = (prod(i=1,n,1+prime(i))-prod(i=1,n,prime(i))); \\ Antti Karttunen, Nov 19 2024

From Antti Karttunen, Nov 19 2024: (Start)
a(n) = A348507(A002110(n)) = A054640(n) - A002110(n) = A001065(A002110(n)).
a(n) >= A024451(n), because A348507(n) >= A003415(n).
For n >= 1, a(n) <= A070826(1+n) [= A002110(1+n)/2] < A051674(n).
(End)

More terms from Sascha Kurz, Feb 01 2003

Term a(0)=0 prepended, data section further extended, and secondary definition added by Antti Karttunen, Nov 19 2024

A223037 a(n) = largest prime p such that Sum_{primes q = 2, ..., p} 1/q does not exceed n.

Original entry on oeis.org

3, 271, 5195969, 1801241230056600467

Offset: 1

Jonathan Sondow, Apr 16 2013

Since Sum_{prime q} 1/q diverges, the sequence is infinite.
In fact, by the Prime Number Theorem Prime(k) ~ k log k as k -> infinity, and by integration Sum_{k <= n} 1/(k log k) ~ log log n, so a(n) ~ Prime(Floor(e^e^n)).
a(4) = A000040(A046024(4)-1) = Prime[43922730588128389], but Mathematica 7.0.0 cannot compute this prime on a Mac computer running OS X.
Instead, using a(4) = largest prime < A016088(4) = 1801241230056600523, Mathematica's PrimeQ function finds that a(4) = 1801241230056600467.
See A016088 for other relevant comments, references, links, and programs.

			a(1) = 3 because 1/2 + 1/3 < 1 < 1/2 + 1/3 + 1/5.

Cf. A016088, A046024, A024451, A096232.

a(n) = A000040(A046024(n)-1) = largest prime < A016088(n).

a(n) ~ Prime(Floor(e^e^n)) = A000040(A096232(n)) as n -> infinity.

A258566 Triangle in which n-th row contains all possible products of n-1 of the first n primes in descending order.

Original entry on oeis.org

1, 3, 2, 15, 10, 6, 105, 70, 42, 30, 1155, 770, 462, 330, 210, 15015, 10010, 6006, 4290, 2730, 2310, 255255, 170170, 102102, 72930, 46410, 39270, 30030, 4849845, 3233230, 1939938, 1385670, 881790, 746130, 570570, 510510

Offset: 1

Philippe Deléham, Jun 03 2015

Triangle read by rows, truncated rows of the array in A185973.

Reversal of A077011.

			Triangle begins:
      1;
      3,     2;
     15,    10,    6;
    105,    70,   42,   30;
   1155,   770,  462,  330,  210;
  15015, 10010, 6006, 4290, 2730, 2310;
  ...

Row sums: A024451.
T(n,1) = A070826(n).
T(n,n) = A002110(n-1).
For 2 <= n <= 9, T(n,2) = A118752(n-2). [corrected by Peter Munn, Jan 13 2018]
T(n,k) = A121281(n,k), but the latter has an extra column (0).
Cf. A077011, A185973, A286947.

T:= n-> (m-> seq(m/ithprime(j), j=1..n))(mul(ithprime(i), i=1..n)):
seq(T(n), n=1..10);  # Alois P. Heinz, Jun 18 2015

T[1, 1] = 1; T[n_, n_] := T[n, n] = Prime[n-1]*T[n-1, n-1];
T[n_, k_] := T[n, k] = Prime[n]*T[n-1, k];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 26 2016 *)

T(1,1) = 1, T(n,k) = A000040(n)*T(n-1,k) for k < n, T(n,n) = A000040(n-1) * T(n-1,n-1).

A260615 Irregular triangle read by rows: the n-th row is the continued fraction expansion of the sum of the reciprocals of the first n primes.

Original entry on oeis.org

0, 2, 0, 1, 5, 1, 30, 1, 5, 1, 2, 12, 1, 3, 1, 2, 1, 9, 1, 1, 7, 1, 2, 1, 9, 1, 2, 1, 2, 12, 7, 1, 2, 2, 13, 1, 1, 1, 8, 13, 5, 4, 1, 2, 5, 8, 1, 2, 6, 1, 1, 4, 10, 1, 2, 3, 1, 3, 1, 2, 238, 1, 28, 1, 42, 2, 2, 7, 1, 1, 4, 1, 1, 1, 6, 1, 41, 3, 1, 1, 51, 1, 9, 2, 3, 2, 5, 1, 2, 1, 6, 1, 1, 1, 3, 3, 3, 1, 1, 1, 3, 3, 1, 2, 19, 1, 13, 1, 1, 3, 4, 7, 1, 1, 3, 2, 1, 10

Offset: 1

Matthew Campbell, Aug 29 2015

			For row 3, the sum of the first three prime reciprocals equals 1/2 + 1/3 + 1/5 = 31/30. The continued fraction expansion of 31/30 is 1 + (1/30). Because of this, the terms in row 3 are 1 and 30.
From _Michael De Vlieger_, Aug 29 2015: (Start)
Triangle begins:
0,  2
0,  1,   5
1, 30
1,  5,   1,  2, 12
1,  3,   1,  2,  1,  9,  1,  1,  7
1,  2,   1,  9,  1,  2,  1,  2, 12,  7
1,  2,   2, 13,  1,  1,  1,  8, 13,  5,  4
1,  2,   5,  8,  1,  2,  6,  1,  1,  4, 10,  1,  2,  3,  1,  3
1,  2, 238,  1, 28,  1, 42,  2,  2,  7,  1,  1,  4
...
(End)

Matthew Campbell, Table of n, a(n) for n = 1..116505 The first 225 rows are in the b-file.

Cf. A000040.
For the continued fractions of the harmonic numbers, see A100398.
For the numerator of the sum, see A024451.
For the denominator of the sum, see A002110.

seq(op(numtheory:-cfrac(s,'quotients')),s=ListTools:-PartialSums(map2(`/`,1,[seq(ithprime(i),i=1..20)]))); # Robert Israel, Sep 06 2015

Table[ContinuedFraction[Sum[1/Prime@k, {k, n}]], {n, 11}] // Flatten (* Michael De Vlieger, Aug 29 2015 *)

row(n) = contfrac(sum(k=1, n, 1/prime(k)));
tabf(nn) = for(n=1, nn, print(row(n))); \\ Michel Marcus, Sep 18 2015

A293457 Primes that divide the numerator of the sum of the reciprocals of all smaller primes.

Original entry on oeis.org

2, 5, 19, 47, 79, 109, 3667387

Offset: 1

Logan J. Kleinwaks, Oct 09 2017

Exhaustive search finds no more terms among the first 10^7 primes.

Primes p that divide A024451(A000720(p)-1). - Antti Karttunen, Feb 08 2024

			Since 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + 1/13 + 1/17 = 716167/510510 and 19 divides 716167, 19 is in the sequence.
Since there are no primes less than 2, the sum of their reciprocals is 0/1, and as 2 divides 0, it is therefore included as the first term of this sequence. - _Antti Karttunen_, Feb 08 2024

Cf. A000040, A000720, A024451, A120289, A369972, A369973.

lista(nn) = my(s = 0); forprime(p=2, nn, if (!(numerator(s) % p), print1(p, ", ")); s += 1/p; ); \\ Michel Marcus, Oct 09 2017, edited for the new, more inclusive definition by Antti Karttunen, Feb 08 2024

a(n) = A000040(1+A369972(n)). - Antti Karttunen, Feb 08 2024

Relaxed the definition to include 2 as the first term - Antti Karttunen, Feb 08 2024

cp's OEIS Frontend

A250130 Numerator of the harmonic mean of the first n primes.

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A354417 a(n) is the numerator of the sum of the reciprocals of the first n squarefree numbers.

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A369972 Numbers k such that (prime(k)#)' is a multiple of prime(1+k), where prime(k)# means the k-th primorial, A002110(k), and ' stands for taking the arithmetic derivative, A003415.

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A369973 Primorials whose arithmetic derivative is divisible by the next larger prime not present in that primorial.

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A370130 a(n) = A369669(A276086(n)), where A369669 is the greatest common divisor of the first and second arithmetic derivative of n, and A276086 is the primorial base exp-function.

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A074107 a(n) = Product of (prime + 1) for first n primes - primorial (n); Sum of proper divisors of the n-th primorial.

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A223037 a(n) = largest prime p such that Sum_{primes q = 2, ..., p} 1/q does not exceed n.

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