A024451 -id:A024451 - cp's OEIS frontend

A282511 Numerator of the sum of the reciprocals of the first n nonprimes.

1, 5, 17, 37, 119, 631, 661, 4807, 995, 2053, 703, 3599, 3679, 41309, 42079, 214091, 2829383, 25864847, 26250947, 26611307, 53898289, 54553489, 938220113, 135531719, 136990319, 2629070861, 2654652461, 2679594521, 18923442047, 19082164247, 19237359287

Offset: 1

Ralf Steiner, Feb 17 2017

			The first few fractions are 1, 5/4, 17/12, 37/24, 119/72, 631/360, 661/360, 4807/2520, 995/504, 2053/
1008, 703/336, 3599/1680, ... - _N. J. A. Sloane_, Dec 15 2017

Cf. A018252 (nonprime numbers), A282512 (denominators).

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.

With[{nn = 45}, Numerator@ Accumulate[1/Complement[Range@ nn, Prime@ Range@ PrimePi@ nn]]] (* Michael De Vlieger, Feb 18 2017 *)

a018252(n) = if(n==1, 1, my(i=1); forcomposite(c=1, , i++; if(i==n, return(c))))
a(n) = numerator(sum(k=1, n, 1/a018252(k))) \\ Felix Fröhlich, Feb 17 2017

More terms from Felix Fröhlich, Feb 17 2017

A282512 Denominator of the sum of the reciprocals of the first n nonprimes.

Original entry on oeis.org

1, 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, 160626866400, 160626866400, 1124388064800, 1124388064800

Offset: 1

Ralf Steiner, Feb 17 2017

			The first few fractions are 1, 5/4, 17/12, 37/24, 119/72, 631/360, 661/360, 4807/2520, 995/504, 2053/1008, 703/336, 3599/1680, ... - _N. J. A. Sloane_, Dec 15 2017

Robert Israel, Table of n, a(n) for n = 1..3954

Cf. A018252 (nonprime numbers), A282511 (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.

NP:= remove(isprime, [$1..100]):
S:= ListTools:-PartialSums(map(t -> 1/t, NP)):
map(denom, S); # Robert Israel, May 01 2017

With[{nn=50}, Denominator@Accumulate[1/Complement[Range@nn, Prime@Range@PrimePi@nn]]] (* Vincenzo Librandi, May 02 2017 *)

Corrected by Robert Israel, May 01 2017

A077011 Triangle read by rows: T(n,k) = A002110(n)/prime(n+1-k), k = 1..n.

Original entry on oeis.org

1, 2, 3, 6, 10, 15, 30, 42, 70, 105, 210, 330, 462, 770, 1155, 2310, 2730, 4290, 6006, 10010, 15015, 30030, 39270, 46410, 72930, 102102, 170170, 255255, 510510, 570570, 746130, 881790, 1385670, 1939938, 3233230, 4849845, 9699690, 11741730

Offset: 1

Amarnath Murthy, Oct 26 2002

Original Name was: "Triangle in which the n-th row contains all possible products of n-1 of the first n primes in ascending order."
A024451(n) gives the sum of the n-th row.
When the triangle is parsed in blocks of ascending length, as shown in the example, there is the following interpretation: The integers Z regarded as a module over themselves contain unshortenable generating sets of different lengths, in fact, infinitely many of each desired length. Each of the blocks is the minimal example of an unshortenable generating set of the respective length. For example, {6,10,15} generates Z as 1=6+10-15. However, removing one of the numbers leaves two numbers that are not relatively prime, precluding generation of Z. An analogous argument succeeds for all other blocks alike. Each block contains numbers such that there is no prime factor common to all. Taking differences sufficiently often one ends up with two coprime numbers whence the generating property follows from Bézout's theorem. If just one number is removed from the set, relative primality is lost. The minimality of the numbers used in each block is evident from the construction. - Peter C. Heinig (algorithms(AT)gmx.de), Oct 04 2006

			Triangle begins:
      1;
      2,     3;
      6,    10,    15;
     30,    42,    70,   105;
    210,   330,   462,   770,   1155;
   2310,  2730,  4290,  6006,  10010,  15015;
  30030, 39270, 46410, 72930, 102102, 170170, 255255;
  ...

Alois P. Heinz, Rows n = 1..130, flattened

Row sums give A024451.
Reversal of A258566.
Cf. A002110, A089633, A372666.

T:= proc(n) local t;
      t:= mul(ithprime(i), i=1..n);
      seq(t/ithprime(n-i), i=0..n-1)
    end:
seq(T(n), n=1..10);  # Alois P. Heinz, Jun 04 2012

T[n_] := Module[{t = Product[Prime[i], {i, 1, n}]}, Table[t/Prime[n - i], {i, 0, n - 1}]];
Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, May 19 2016, translated from Maple *)

T(n,k) = vecprod(primes(n))/prime(n+1-k); \\ Michel Marcus, May 19 2024

A089633(n-1) = Sum_{p | n} 2^(pi(p) - 1) for n > 1, pi(x) = A000720(x). - Michael De Vlieger, May 19 2024

More terms from Sascha Kurz, Jan 26 2003

Name changed by David James Sycamore, May 19 2024

A373842 a(n) = A003415(A276085(n)), where A003415 is the arithmetic derivative and A276085 is the primorial base log-function.

Original entry on oeis.org

0, 0, 1, 1, 5, 1, 31, 1, 4, 1, 247, 4, 2927, 1, 12, 4, 40361, 1, 716167, 12, 80, 1, 14117683, 1, 16, 1, 5, 80, 334406399, 6, 9920878441, 1, 216, 568, 60, 5, 314016924901, 33975, 3740, 6, 11819186711467, 14, 492007393304957, 216, 7, 28300, 21460568175640361, 5, 92, 1, 60080, 3740, 1021729465586766997, 1, 540, 14

Offset: 1

Antti Karttunen, Jun 20 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..481

Cf. A003415, A024451, A276085, A373843 [= gcd(n, a(n))], A373846 (positions of 1's), A373847 [k such that a(n)<=k], A373848.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*prod(i=1,primepi(f[k, 1]-1),prime(i))); };
A373842(n) = A003415(A276085(n));

For n >= 1, a(A000040(n)) = A024451(n-1).

A109628 Numbers k such that the numerator of Sum_{i=1..k} 1/prime(i), in reduced form, is prime.

Original entry on oeis.org

2, 3, 5, 6, 18, 19, 22, 47, 57, 58, 63, 70, 73, 112, 632, 1382, 4621

Offset: 1

Ryan Propper, Aug 02 2005

Terms <= 112 correspond to certified primes.

Numbers k such that the arithmetic derivative of the k-th primorial, A024451(k) [= A003415(A002110(k))] is prime. - Antti Karttunen, Jan 09 2024

			Sum_{i=1..6} 1/prime(i) = 40361/30030 and 40361 is prime, hence 6 is a term.

Cf. A002110, A003415.

Positions of primes in A024451.

s = 0; Do[s += 1/Prime[n]; k = Numerator[s]; If[PrimeQ[k], Print[n]], {n, 1, 1500}]
Position[Accumulate[1/Prime[Range[120]]],?(PrimeQ[ Numerator[ #]]&)] //Flatten (* To generate terms greater than 120, increase the Range constant, but the program may take much longer to run. *) (* _Harvey P. Dale, Jan 01 2019 *)

a(17) from Michael S. Branicky, Sep 16 2024

A373848 Numbers k such that k is not divisible by p^p for any prime p, and for which 1 < A373842(k) <= k, where A373842 is the arithmetic derivative of the primorial base log-function.

Original entry on oeis.org

5, 9, 15, 25, 30, 42, 45, 63, 75, 105, 110, 125, 126, 147, 150, 165, 175, 198, 210, 225, 231, 245, 275, 294, 315, 330, 343, 363, 375, 385, 441, 462, 495, 525, 539, 605, 625, 650, 686, 693, 726, 735, 750, 770, 825, 847, 875, 882, 990, 1029, 1050, 1089, 1125, 1155, 1170, 1190, 1210, 1225, 1250, 1331, 1375, 1386, 1430

Offset: 1

Antti Karttunen, Jun 20 2024

nonn

The initial 5 is the only prime in this sequence (for a proof, consider Henry Bottomley's Sep 27 2006 formula for A024451), the next three terms 9, 15, 25 are only semiprimes (see A087112 and A370129), and there are 21 terms with three prime factors in total: 30, 42, 45, 63, 75, 105, 110, 125, 147, 165, 175, 231, 245, 275, 343, 363, 385, 539, 605, 847, 1331 (see A369979, A370138 and A373844). In general, there should be only a finite amount of terms x such that A001222(x) = k, for any k >= 1.

It is conjectured that 5 is the only fixed point of A373842, which would imply that x=6 is the only number for which A003415(x) = A276086(x). See A351228.

Antti Karttunen, Table of n, a(n) for n = 1..1565

Intersection of A048103 with the setwise difference A373847\(A373846 U {1, 2}).
Subsequence of A373847.
Cf. A001222, A003415, A024451, A087112, A276085, A276086, A359550, A369979, A370129, A370138, A373842, A373844, A373845.
Cf. also A351228, A373603.

\\ Uses the code from A373842, or its precomputed data:
A359550(n) = { my(f = factor(n)); prod(k=1, #f~, (f[k, 2]A373848(n) = if(!A359550(n), 0, my(u=A373842(n)); ((1

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]));
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*prod(i=1,primepi(f[k, 1]-1),prime(i))); };
\\ The following routine checks that n is not a prime larger than five, is in A048103, and in case n is odd, rules out cases that certainly cannot give A373842(n) <= n:
prefilter_for_A373848(n) = if(n < 3 || (isprime(n) && n > 5), 0, my(f=factor(n), k=#f~, lpf=f[1,1], p=f[k,1], m=f[k,2]); for(i=1, k, if(f[i, 2]>=f[i, 1], return(0))); if(2==lpf, return(1)); while(p>lpf, p = precprime(p-1); m *= p; if(m>n, return(0))); (1));
isA373848(n) = if(!prefilter_for_A373848(n), 0, my(x=A276085(n)); if(x>A002620(n), 0, (!isprime(x) && A003415(x)<=n)));

A301277 Nearest integer to mean of first n primes.

Original entry on oeis.org

2, 3, 3, 4, 6, 7, 8, 10, 11, 13, 15, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 45, 47, 49, 51, 53, 55, 58, 60, 63, 65, 67, 70, 72, 75, 77, 80, 82, 85, 87, 90, 92, 94, 97, 100, 102, 105, 107, 110, 113, 115, 118, 121, 123, 126, 128, 131, 133

Offset: 1

N. J. A. Sloane, Mar 18 2018

nonn

Differs from A075465 where ties are involved. - R. J. Mathar, Mar 20 2018

			The means are 2, 5/2, 10/3, 17/4, 28/5, 41/6, 58/7, 77/8, 100/9, 129/10, 160/11, 197/12, 238/13, 281/14, 328/15, 381/16, 440/17, 167/6, 568/19, 639/20, 712/21, 791/22, 38, 321/8, 212/5, ...

Altug Alkan, Table of n, a(n) for n = 1..10000
Joel E. Cohen, Statistics of Primes (and Probably Twin Primes) Satisfy Taylor's Law from Ecology, The American Statistician, 70 (2016), 399-404.

Mean and variance of primes: A301273/A301274, A301275/A301276, A301277, A273462.

Cf. A007504, A060620, A062049, A075465, A225804, A250130/A024451.

m := n -> add(ithprime(j),j=1..n)/n;
m1:=[seq(m(n),n=1..100)];
m2:=map(numer,m1); # A301273
m3:=map(denom,m1); # A301274
m4:=map(round,m1); # A301277

Rest@ FoldList[{Append[First@ #1, #2], If[And[EvenQ@ #1, #2 == 1/2] & @@ {IntegerPart@ #, FractionalPart@ #}, Round@ # + 1, Round@ #] &@ Mean@ First@ #1} &, {{2}, 2}, Prime@ Range[2, 63]][[All, -1]] (* Michael De Vlieger, Apr 05 2018 *)

a(n) = round(sum(i=1, n, prime(i))/n); \\ Altug Alkan, Mar 22 2018

a(n) = round(A007504(n) / n). - David A. Corneth, Mar 22 2018

a(n) ~ prime(n)/2 ~ n*log(n)/2. - Daniel Forgues, Mar 22 2018

A370129 Triangle read by rows: T(n,k) = A003415(A002110(n)+A002110(k)), 0 <= k <= n; arithmetic derivatives of the sums of two primorial numbers.

Original entry on oeis.org

1, 1, 4, 1, 12, 16, 1, 80, 60, 92, 1, 216, 540, 608, 704, 1, 3740, 3100, 4548, 6324, 8164, 568, 60080, 40060, 56292, 116208, 61768, 110752, 33975, 1021040, 1041768, 794468, 2415104, 1091004, 1357128, 1942844, 28300, 9789116, 29099520, 19722884, 18576860, 35347200, 35779644, 26575580, 37935056, 704080, 335024060

Offset: 0

Antti Karttunen, Feb 29 2024

Apart from those positions (A014545) at the left edge where a(n) = 1, a(n) <= A087112(1+n) only at n=2, 4 and 5, i.e., never after the third row.

			Triangle begins as:
      1;
      1,       4;
      1,      12,       16;
      1,      80,       60,       92;
      1,     216,      540,      608,      704;
      1,    3740,     3100,     4548,     6324,     8164;
    568,   60080,    40060,    56292,   116208,    61768,   110752;
  33975, 1021040,  1041768,   794468,  2415104,  1091004,  1357128,  1942844;
  28300, 9789116, 29099520, 19722884, 18576860, 35347200, 35779644, 26575580, 37935056;

Cf. A002110, A003415, A024451, A370121, A370135, A370136.
Cf. A014545 (positions of 1's at the left edge), A087112.
Cf. also A024451 (arithmetic derivatives of primorials).

A002110(n) = prod(i=1,n,prime(i));
A370121(n) = { my(c = (sqrtint(8*n + 1) - 1) \ 2); (A002110(c) + A002110(n - binomial(c + 1, 2))); };
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A370129(n) = A003415(A370121(n));

a(n) = A003415(A370121(n)).

For n, k >= 1, T(n,k) = A002110(k)*A370136(n,k) + A024451(k)*A370135(n,k).

A111392 a(n) = Product_{i=1..n-1} (Product_{k=1..i} p_k + Product_{k=i+1..n} p_k).

Original entry on oeis.org

2, 5, 187, 162319, 10697595389, 63619487169453143, 74365399061678006800073593, 11864736003419293844093922527852416537, 601642845102734414280661105098046392912578705726003

Offset: 1

Yasutoshi Kohmoto, Nov 08 2005

This sequence gives another proof that there are infinitely many primes. Let N = Product_{1<=i

a(1) could also be chosen to be 1.

Vincenzo Librandi, Table of n, a(n) for n = 1..29

Cf. A024451, A112404.

a:=n->mul(mul(ithprime(k),k=1..i)+mul(ithprime(k),k=i+1..n),i=1..n-1): 2,seq(a(n),n=2..10); # Muniru A Asiru, Dec 06 2018

Join[{2}, Rest[f[n_]:=Product[(Product[Prime[k], {k, i}] + Product[Prime[k], {k, i + 1, n}]), {i, n - 1}]; Array[f, 10] ]] (* Robert G. Wilson v, Nov 12 2005 *)

t=10; for(n=2, t, print1(prod(i=1, n-1, prod(k=1,i,prime(k)) + prod(k=i+1,n,prime(k))), ", ")); \\ Gerald McGarvey, Nov 12 2005

Corrected and extended by Gerald McGarvey and Robert G. Wilson v, Nov 12 2005

A115963 Numerator of Sum_{i=1..n} 1/prime(i)^3.

Original entry on oeis.org

1, 35, 4591, 1601713, 2141141003, 4716413174591, 23198819007792583, 159253748925534977797, 1938552948676080555065099, 47290471293028435532185602511, 1409101231790431848106470385672201

Offset: 1

Jonathan Vos Post, Mar 14 2006

Denominators = A115964. See also: A024451 Numerator of Sum_{i=1..n} 1/prime(i). A002110 Primorial [denominator of Numerator of Sum_{i=1..n} 1/prime(i)]. A061015 Numerator of Sum_{i=1..n} 1/prime(i)^2.

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

Cf. A000040, A075986/A075987, A106830/A034386.

Accumulate[1/Prime[Range[20]]^3]//Numerator (* Harvey P. Dale, Dec 30 2024 *)

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

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A282511 Numerator of the sum of the reciprocals of the first n nonprimes.

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A282512 Denominator of the sum of the reciprocals of the first n nonprimes.

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A077011 Triangle read by rows: T(n,k) = A002110(n)/prime(n+1-k), k = 1..n.

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A373842 a(n) = A003415(A276085(n)), where A003415 is the arithmetic derivative and A276085 is the primorial base log-function.

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A109628 Numbers k such that the numerator of Sum_{i=1..k} 1/prime(i), in reduced form, is prime.

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A373848 Numbers k such that k is not divisible by p^p for any prime p, and for which 1 < A373842(k) <= k, where A373842 is the arithmetic derivative of the primorial base log-function.

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A301277 Nearest integer to mean of first n primes.

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A370129 Triangle read by rows: T(n,k) = A003415(A002110(n)+A002110(k)), 0 <= k <= n; arithmetic derivatives of the sums of two primorial numbers.

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