A024451 -id:A024451 - cp's OEIS frontend

A070826 One half of product of first n primes A000040.

1, 3, 15, 105, 1155, 15015, 255255, 4849845, 111546435, 3234846615, 100280245065, 3710369067405, 152125131763605, 6541380665835015, 307444891294245705, 16294579238595022365, 961380175077106319535, 58644190679703485491635, 3929160775540133527939545, 278970415063349480483707695

Offset: 1

Wolfdieter Lang, May 10 2002

Also, with offset 0, product of first n odd primes. - N. J. A. Sloane, Feb 26 2017
Identical to A002110(n)/2, n>=1.
a(n+1) is the least odd number with exactly n distinct prime divisors. - Labos Elemer, Mar 24 2003
Also, odd numbers n for which sigma(n)*phi(n)/n^2 reaches a new record low, monotonically decreasing to the lower bound 8/Pi^2. - M. F. Hasler, Jul 08 2025

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A003266 (for Fibonacci), A070825 (for Lucas), A003046 (for Catalan).
Cf. also A002110, A024451, A060389, A091852, A276086, A203008 [= A003415(a(1+n))].
Range of A196529.

a:=n->mul(ithprime(j), j=2..n):seq(a(n), n=1..17); # Zerinvary Lajos, Aug 24 2008

Rest[ FoldList[ Times, 1, Prime[ Range[ 18]] ]]/2 (* Robert G. Wilson v, Feb 17 2004 *)
FoldList[Times, 1, Prime[Range[2, 18]]] (* Zak Seidov, Jan 26 2009 *)

a(n) = prod(k=2, n, prime(k)) \\ Michel Marcus, Mar 25 2017, simplified by M. F. Hasler, Jul 09 2025

from sympy import primorial
def A070826(n): return primorial(n)>>1 # Chai Wah Wu, Jul 21 2022

a(n) = A002110(n)/2.
From Antti Karttunen, Feb 06 2024: (Start)
a(1) = 1, and for n > 1, a(n) = A276086(A060389(n-1)).
a(n) = A024451(n) - 2*A203008(n-1).
(End)
a(n) = A000040(n)*a(n-1) for n > 1, a(1) = 1. - M. F. Hasler, Jul 09 2025

Formula corrected by Gary Detlefs, Dec 07 2011

A046024 a(n) = smallest k such that Sum_{ i = 1..k } 1/prime(i) exceeds n.

Original entry on oeis.org

1, 3, 59, 361139, 43922730588128390

Offset: 0

Eric W. Weisstein

The corresponding primes prime(a(n)) are in A016088.

Index m for which the prime harmonic number p[ m ] := Sum[ 1/Prime[ k ],{k,1,m} ] >= n.

E. Bach, D. Klyve, and J. P. Sorenson, Computing prime harmonic sums, Math. Comp. 78 (2009) 2283-2305.
Eric Weisstein's World of Mathematics, Prime Number.
Eric Weisstein's World of Mathematics, Harmonic Series of Primes.

Cf. A004080, A016088, A096232, A223037.

Cf. A024451/A002110 for Sum_{i = 1..n} 1/prime(i).

Table[m = 1; s = 0; While[(s = s + 1/Prime[m]) <= n, m++];
m, {n, 0, 4}] (* Robert Price, Mar 27 2019 *)

a(n)=my(t); forprime(p=2,, t+=1./p; if(t>n, return(primepi(p)))) \\ Charles R Greathouse IV, Apr 29 2015

From Jonathan Sondow, Apr 17 2013: (Start)
a(n) = A000720(A016088(n)) = A000720(A096232(n))+1.
a(n) = e^(e^(n + O(1))), see comment in A223037. [corrected by Charles R Greathouse IV, Aug 22 2013] (End)
a(n) = A103591(2*n). - Michel Marcus, Aug 22 2013

a(4) found by Tomás Oliveira e Silva (tos(AT)det.ua.pt), using the fourth term of A016088. - Dec 14 2005

a(0) from Jonathan Sondow, Apr 16 2013

A351228 Numbers k for which A003415(k) >= A276086(k), where A003415 is the arithmetic derivative and A276086 is the primorial base exp-function.

Original entry on oeis.org

6, 30, 32, 36, 60, 210, 212, 213, 214, 216, 240, 420, 2310, 2312, 2313, 2314, 2315, 2316, 2317, 2318, 2319, 2320, 2322, 2324, 2328, 2340, 2342, 2343, 2344, 2346, 2348, 2349, 2352, 2370, 2372, 2376, 2400, 2520, 2522, 2523, 2524, 2526, 2528, 2550, 2552, 2730, 4620, 4622, 4623, 4624, 4626, 4628, 4632, 4650, 4652, 4656

Offset: 1

Antti Karttunen, Feb 05 2022

Conjecture: Apart from the initial 6, the rest of terms are the numbers k for which A003415(k) > A276086(k), thus giving the positions of zeros in A351232. In other words, it seems that only k=6 satisfies A003415(k) = A276086(k). See also comments in A351088.

Antti Karttunen, Table of n, a(n) for n = 1..11643 (terms up to 3*A002110(9))
Victor Ufnarovski and Bo Åhlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003, #03.3.4.
Index entries for sequences related to primorial base

Cf. A003415, A024451, A048103, A060735, A235991, A276085, A276086, A327862, A351088, A351089, A351232, A369656, A369663, A369664, A371104.
Union of A370127 and A370128.
Subsequence of A328118.
Subsequences: A351229, A369959, A369960, A369970 (after its two initial terms).
Cf. also A369650.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA351228(n) = (A003415(n)>=A276086(n));

A327978 Numbers whose arithmetic derivative (A003415) is a primorial number (A002110) > 1.

Original entry on oeis.org

9, 161, 209, 221, 2189, 2561, 3281, 3629, 5249, 5549, 6401, 7181, 7661, 8321, 8909, 9089, 9869, 10001, 10349, 10541, 10961, 11009, 11021, 29861, 38981, 52601, 66149, 84101, 93029, 97481, 132809, 150281, 158969, 163301, 197669, 214661, 227321, 235721, 285449, 321989, 338021, 357881, 369701, 381449, 385349, 416261, 420089, 442889

Offset: 1

Antti Karttunen, Oct 09 2019

nonn

Numbers n such that A327859(n) = A276086(A003415(n)) is an odd prime.
Composite terms in A328232.
Although it first might seem that the numbers whose arithmetic derivative is A002110(k) all appear before any of those whose arithmetic derivative is A002110(k+1), that is not true, as for example, we have a(56) = 570149, and A003415(570149) = 2310, a(57) = 570209, and A003415(570209) = 30030, but then a(58) = 573641 with A003415(573641) = 2310 again.
Because this is a subsequence of A327862 (all primorials > 1 are of the form 4k+2), only odd numbers are present.
Conjecture: No multiples of 5 occur in this sequence, and no multiples of 3 after the initial 9.
Of the first 10000 terms, all others are semiprimes (with 9 the only square one), except 1547371 = 7^2 * 23 * 1373 and 79332523 = 17^2 * 277 * 991, the latter being the only known term whose decimal expansion ends with 3. If all solutions were semiprimes p*q such that p+q = A002110(k) for some k > 1 (see A002375), it would be a sufficient reason for the above conjecture to hold. - David A. Corneth and Antti Karttunen, Oct 11 2019
In any case, the solutions have to be of the form "odd numbers with an even number of prime factors with multiplicity" (see A235992), and terms must also be cubefree (A004709), as otherwise the arithmetic derivative would not be squarefree.
Sequence A366890 gives the non-Goldbachian solutions, i.e., numbers that are not semiprimes. See also A368702. - Antti Karttunen, Jan 17 2024

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1078 terms from Antti Karttunen)
Victor Ufnarovski and Bo Åhlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003, #03.3.4.
Index entries for sequences related to Goldbach conjecture
Index entries for sequences related to primorial numbers

Cf. A002110, A002375, A002620, A003415, A024451, A143293, A157037, A276150, A327859, A327969, A328233, A328243.
Cf. A351029 (number of k for which k' = A002110(n)).
Cf. A368703, A368704 (the least and the greatest k for which k' = A002110(n)).
Cf. A366890 (terms that are not semiprimes), A368702 (numbers k such that k' is one of the terms of this sequence).
Subsequence of following sequences: A004709, A189553, A327862, A328232, A328234.

ad[n_] := n * Total @ (Last[#]/First[#] & /@ FactorInteger[n]); primQ[n_] := Max[(f = FactorInteger[n])[[;;,2]]] == 1 && PrimePi[f[[-1,1]]] == Length[f]; Select[Range[10^4], primQ[ad[#]] &] (* Amiram Eldar, Oct 11 2019 *)

A002620(n) = ((n^2)>>2);
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); };
isA327978flat(n) = { my(u=A003415(n)); ((u>1)&&(1==A276150(u))); }; \\ Slow!
k=0; for(n=1,A002620(30030),if(isA327978flat(n), k++; write("b327978.txt", k, " ", n)));

A327969(a(n)) = 4 for all n.

A106830 Numerator of Sum_{ primes p <= n} 1/p.

Original entry on oeis.org

0, 1, 5, 5, 31, 31, 247, 247, 247, 247, 2927, 2927, 40361, 40361, 40361, 40361, 716167, 716167, 14117683, 14117683, 14117683, 14117683, 334406399, 334406399, 334406399, 334406399, 334406399, 334406399, 9920878441, 9920878441, 314016924901, 314016924901

Offset: 1

N. J. A. Sloane, May 22 2005

Very similar to A024451, which see for further information.

			0, 1/2, 5/6, 5/6, 31/30, 31/30, 247/210, 247/210, 247/210, 247/210, 2927/2310, 2927/2310, 40361/30030, 40361/30030, 40361/30030, ...

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Denominators are in A034386.

The following fractions are all related to each other: Sum 1/n: A001008/A002805, Sum 1/prime(n): A024451/A002110 and A106830/A034386, Sum 1/nonprime(n): A282511/A282512, Sum 1/composite(n): A250133/A296358.

Accumulate[Table[If[PrimeQ[n],1/n,0],{n,40}]]//Numerator (* Harvey P. Dale, Feb 17 2018 *)

A115964 Denominator of Sum_{i=1..n} 1/prime(i)^3.

Original entry on oeis.org

8, 216, 27000, 9261000, 12326391000, 27081081027000, 133049351085651000, 912585499096480209000, 11103427767506874702903000, 270801499821725167129101267000, 8067447481189014453943055845197000

Offset: 1

Jonathan Vos Post, Mar 14 2006

Numerators are in A115963.

Also the primorials cubed. - Reikku Kulon, Sep 18 2008

			1/8, 35/216, 4591/27000, 1601713/9261000, 2141141003/12326391000, 4716413174591/27081081027000.

Cf. A115963 (numerators).
Cf. A024451 (numerator of sum_{i=1..n} 1/prime(i)), A002110 (primorial, also denominator of sum_{i=1..n} 1/prime(i)), A061015 (numerator of sum_{i=1..n} 1/prime(i)^2).
Cf. A000040, A075986/A075987, A106830/A034386.
Cf. A061742, A100778. - Reikku Kulon, Sep 18 2008

a[n_]:=Product[Prime[i]^3, {i, 1, n}]; (* Vladimir Joseph Stephan Orlovsky, Dec 05 2008 *)

a(n) = denominator of Sum_{i=1..n} 1/A000040(i)^3.

a(n) = A002110(n)^3. - Reikku Kulon, Sep 18 2008

A024530 Numerator of -Sum_{k=1..n} (-1)^k / prime(k).

Original entry on oeis.org

0, 1, 1, 11, 47, 727, 7141, 151427, 2366603, 64131559, 1636722341, 57208085801, 1916138684507, 85982424199597, 3392993977055461, 172553478253276697, 8530444564835173531, 535885387802465283059, 30766248305796169627529

Offset: 0

Clark Kimberling

Limit of the fractional values is A078437. - Charles R Greathouse IV, Apr 22 2011

			0/1, 1/2, 1/6, 11/30, 47/210, 727/2310, 7141/30030, 151427/510510, ...

Seiichi Manyama, Table of n, a(n) for n = 0..350 (terms n = 1..100 from T. D. Noe)

Denominators are A002110.

Cf. A024451, A078437.

A024530 := n->numer(add((-1)^(k+1)/ithprime(k),k=1..n));

a[n_] := Numerator[ -Total[(-1)^Range[n] / Prime[ Range[n]]]]; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Dec 12 2011 *)

a(n)=numerator(-sum(k=1,n,(-1)^k/prime(k)));
concat(0, vector(33,n,a(n)))  /* show terms */

a(n+1) = a(n)*prime(n+1) + (-1)^n*primorial(prime(n)) for n >= 2. - Alexandre Herrera, Sep 03 2023

a(0)=0 prepended by Seiichi Manyama, Jul 25 2021

A250133 Denominator of the harmonic mean of the first n composite numbers.

Original entry on oeis.org

1, 5, 13, 47, 271, 301, 2287, 491, 1045, 367, 1919, 1999, 22829, 23599, 121691, 1628183, 15054047, 15440147, 15800507, 32276689, 32931889, 570652913, 83022119, 84480719, 1631388461, 1656970061, 1681912121, 11939665247, 12098387447, 12253582487, 285324285601

Offset: 1

Colin Barker, Nov 14 2014

nonn

Also numerator of the sum of the reciprocals of the first n composite numbers (A250133/A296358).

			a(3) = 13 because the first 3 composite numbers are [4,6,8] and 3 / (1/4+1/6+1/8) = 72/13.
1/4, 5/12, 13/24, 47/72, 271/360, 301/360, 2287/2520, 491/504, 1045/1008, 367/336, 1919/1680, 1999/1680, 22829/18480, ... = A250133/A296358

Colin Barker, Table of n, a(n) for n = 1..500

Cf. A250132 (numerators).

The following fractions are all related to each other: Sum 1/n: A001008/A002805, Sum 1/prime(n): A024451/A002110 and A106830/A034386, Sum 1/nonprime(n): A282511/A282512, Sum 1/composite(n): A250133/A296358.

harmonicmean(v) = #v / sum(k=1, #v, 1/v[k])
composite(n) = for(k=0, primepi(n), isprime(n++)&&k--); n \\ from A002808
s=vector(100); for(k=1, #s, s[k]=denominator(harmonicmean(vector(k, i, composite(i))))); s

A296358 Denominator of the sum of the reciprocals of the first n composite numbers.

Original entry on oeis.org

4, 12, 24, 72, 360, 360, 2520, 504, 1008, 336, 1680, 1680, 18480, 18480, 92400, 1201200, 10810800, 10810800, 10810800, 21621600, 21621600, 367567200, 52509600, 52509600, 997682400, 997682400, 997682400, 6983776800, 6983776800, 6983776800

Offset: 1

N. J. A. Sloane, Dec 15 2017

Same as A282512 without the initial 1.

			1/4, 5/12, 13/24, 47/72, 271/360, 301/360, 2287/2520, 491/504, 1045/1008, 367/336, 1919/1680, 1999/1680, 22829/18480, ... = A250133/A296358

Amiram Eldar, Table of n, a(n) for n = 1..3953

Numerators are in A250133.

The following fractions are all related to each other: Sum 1/n: A001008/A002805, Sum 1/prime(n): A024451/A002110 and A106830/A034386, Sum 1/nonprime(n): A282511/A282512, Sum 1/composite(n): A250133/A296358.

Accumulate[1/Select[Range[100],CompositeQ]]//Denominator (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 19 2018 *)

Gerry Felderman (Personal communication, Dec 15 2017) observes that Sum_{k=1..n} 1/composite(k) (= A250133(n)/A296358(n)) ~ log(n) - loglog(n) ~ log pi(n) as n -> oo.

A024528 a(n) = n-th elementary symmetric function of {1, prime(1), prime(2), ..., prime(n)}.

Original entry on oeis.org

1, 3, 11, 61, 457, 5237, 70391, 1226677, 23817373, 557499269, 16390571671, 514577415031, 19239924846277, 796257656832167, 34543329507310391, 1636619248175258407, 87355709935877186981, 5186576044693944076609

Offset: 0

Clark Kimberling

nonn

a(0) through a(12) are squarefree. - Gerald McGarvey, Sep 03 2004

For n>0 a(n) is the determinant of the n X n matrix with elements M[i,j] = 1+Prime[i] if i=j and 1 otherwise. - Alexander Adamchuk, Jun 02 2006

			a(0) = 1
a(1) = 1*2 + A002110(0) = 2 + 1 = 3
a(2) = 3*3 + A002110(1) = 9 + 2 = 11
a(3) = 11*5 + A002110(2) = 55 + 6 = 61
a(4) = 61*7 + A002110(3) = 427 + 30 = 457
a(5) = 457*11 + A002110(4) = 5027 + 210 = 5237
a(6) = 5237*13 + A002110(5) = 68081 + 2310 = 70391
a(7) = 70391*17 + A002110(6) = 1196647 + 30030 = 1226677 - _Philippe Deléham_, Jun 03 2015

Clark Kimberling, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Harmonic Series of Primes

Cf. A000040, A002110, A024451.

Cf. A125707 (indices of primes).

N:= 30: # to get a(0) to a(N)
Primes:= [seq(ithprime(i),i=1..N)]:
seq(mul(Primes[i],i=1..n)*(1+add(1/Primes[i],i=1..n)),n=0..N); # Robert Israel, Jun 03 2015

Table[ Det[ DiagonalMatrix[ Table[ Prime[i], {i, 1, n} ] ] + 1 ], {n, 1, 20} ] (* Alexander Adamchuk, Jun 02 2006 *)
p[0] = 1; p[n_] := Prime[n];
t[n_] := Table[p[k], {k, 0, n}]
a[n_] := SymmetricPolynomial[n, t[n]]
Table[a[n], {n, 0, 20}]
(* Clark Kimberling, Aug 18 2012 *)

This sequence is the numerators of the prime harmonic numbers + 1, i.e. a(n)/A002110(n) = Sum_{i=0...n} 1/p(i) where p(0) = 1, p(i) is the i-th prime for n > 0 and A002110 are the primorial numbers. - Gerald McGarvey, Sep 03 2004
a(n) = Det[ DiagonalMatrix[ Table[ Prime[i], {i, 1, n} ] ] + 1 ]. - Alexander Adamchuk, Jun 02 2006
a(0) = 1, a(n) = A000040(n)*a(n-1) + A002110(n-1) for n>=1. - Philippe Deléham, Jun 03 2015

More terms from T. D. Noe, Sep 09 2004

cp's OEIS Frontend

A070826 One half of product of first n primes A000040.

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A046024 a(n) = smallest k such that Sum_{ i = 1..k } 1/prime(i) exceeds n.

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A351228 Numbers k for which A003415(k) >= A276086(k), where A003415 is the arithmetic derivative and A276086 is the primorial base exp-function.

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A327978 Numbers whose arithmetic derivative (A003415) is a primorial number (A002110) > 1.

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A106830 Numerator of Sum_{ primes p <= n} 1/p.

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A115964 Denominator of Sum_{i=1..n} 1/prime(i)^3.

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A024530 Numerator of -Sum_{k=1..n} (-1)^k / prime(k).

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