A005867 -id:A005867 - cp's OEIS frontend

A059861 a(n) = Product_{i=2..n} (prime(i) - 2).

1, 1, 3, 15, 135, 1485, 22275, 378675, 7952175, 214708725, 6226553025, 217929355875, 8499244879125, 348469040044125, 15681106801985625, 799736446901266875, 45584977473372211875, 2689513670928960500625

Offset: 1

Labos Elemer, Feb 28 2001

nonn

Arises in Hardy-Littlewood k-tuple conjecture. Also a(n) is the exact number of d=2 and also d=4 differences in dRRS[modulus=n-th primorial]; see A049296 (dRRS[m]=set of first differences of reduced residue system modulo m).
For n>1 this is the determinant of the (n-1) X (n-1) matrix whose diagonal is A006093(n) = {1, 2, 4, 6, 10, 12, 16, 18..} = the first primes minus 1 and all other elements are 1's. The determinant begins: / (2-1) 1 1 1 1 1 1 ... / 1 (3-1) 1 1 1 1 1 ... / 1 1 (5-1) 1 1 1 1 ... / 1 1 1 (7-1) 1 1 1 ... / 1 1 1 1 (11-1) 1 1 ... / 1 1 1 1 1 (13-1) 1 ... - Alexander Adamchuk, May 21 2006
From Gary W. Adamson, Apr 21 2009: (Start)
Equals (-1)^n * (1, 1, 1, 3, 15, ...) dot (1, -2, 4, -6, 10, ...).
a(6) = 135 = (1, 1, 1, 3, 15) dot (1, -2, 4, -6, 10) = (1, -2, 4, -18, 150). (End)

			n=4, a(4) = 1*(3-2)*(5-2)*(7-2) = 15. 48 first terms of A049296 give one complete period of dRRS[210], in which 15 d=2, 15 d=4 and 18 larger differences occur. For n=1, 2, ..., 5 in the periods of length {1, 2, 8, 48, 480, ...} [see A005867] the number of d=2 and also d=4 differences is {1, 1, 3, 15, 135, ..}

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 84-94.
R. K. Guy, Unsolved Problems in Number Theory, Sections A8, A1.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979.
G. Polya, Mathematics and Plausible Reasoning, Vol. II, Appendix Princeton UP, 1954.

A.H.M. Smeets, Table of n, a(n) for n = 1..100
Steven Brown, Distance between consecutive elements of the multiplicative group of integers modulo n, arXiv:2311.06873 [math.NT], 2023. See Table 1 p. 25.
C. K. Caldwell, Prime k-tuple Conjecture
Steven R. Finch, Hardy-Littlewood Constants [Broken link]
Steven R. Finch, Hardy-Littlewood Constants [From the Wayback machine]
G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: on the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, pp. 1-70, 1923.
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]
G. Polya, Heuristic reasoning in the theory of numbers, Am. Math. Monthly, 66 (1959), 375-384.

Cf. A049296, A002110, A005867, A000847, A022008, A051160-A051168, A048298, A059861-A059865, A040976.

Cf. A067549, A006093.

Table[ Det[ DiagonalMatrix[ Table[ Prime[i-1] - 2, {i, 2, n} ] ] + 1 ], {n, 2, 20} ] (* Alexander Adamchuk, May 21 2006 *)
Table[Product[Prime@k - 2, {k, 2, n}], {n, 1, 18}] (* Harlan J. Brothers, Jul 02 2018 *)
a[1] = 1; a[n_] := a[n] = a[n - 1] (Prime[n] - 2);
Table[a[n], {n, 18}]  (* Harlan J. Brothers, Jul 02 2018 *)
Join[{1},FoldList[Times,Prime[Range[2,20]]-2]] (* Harvey P. Dale, Apr 19 2023 *)

a(n) = prod(i=2, n, prime(i)-2); \\ Michel Marcus, Apr 16 2017

a(n) = Det[ DiagonalMatrix[ Table[ Prime[i-1] - 2, {i, 2, n} ] ] + 1 ] for n>1. - Alexander Adamchuk, May 21 2006
a(n) = a(n-1) * (A000040(n) - 2) for n > 1. - A.H.M. Smeets, Dec 14 2019
a(n) = |{r | 0 <= r < primorial(n) and gcd(r, primorial(n)) = 1 and gcd(r + 2, primorial(n)) = 1}|. - Greg Tener, Oct 22 2021

Offset corrected by A.H.M. Smeets, Dec 14 2019

A048863 Number of nonprimes (1 and composites) in the reduced residue system of n-th primorial number (A002110).

Original entry on oeis.org

1, 1, 1, 1, 6, 142, 2518, 49836, 1012859, 24211838, 721500294, 22627459401, 844130935668, 34729870646918, 1491483322755274, 69890000837179157, 3692723747920861125, 217158823263305180123, 13182405032836651359192, 879055475442725460400606

Offset: 0

Labos Elemer

			For n = 3, the 3rd primorial is 30, phi(30) = 8, a(3) = 1 since 1 is nonprime. See A048597.
For n = 4, the 4th primorial is 210, the size of its reduced residue system (RRS) is 48 of which 6 are either composite numbers or 1: {1, 121, 143, 169, 187, 209}.

Kim Walisch, Fast C++ prime counting function implementation (primecount).

Cf. A000010 (phi), A000720, A002110, A005867, A007625, A048597, A048862.

Table[Function[P, EulerPhi@ P - # &[PrimePi@ P - n]]@ Product[Prime@ i, {i, n}], {n, 0, 12}] (* Michael De Vlieger, May 08 2017 *)

a(n) = A005867(n) - A000849(n) + n.

a(n) = A000010(A002110(n)) - A000720(A002110(n)) + A001221(A002110(n)).

a(14)-a(15) from Max Alekseyev, Aug 21 2013
a(0) prepended, a(15) corrected, a(16)-a(17) computed from A000849 by Max Alekseyev, Feb 21 2016
a(18)-a(19) calculated using Kim Walisch's primecount and added by Amiram Eldar, Sep 03 2024

A281890 Square array A(n,k): number of integers having prime(n) as k-th factor when written as product of primes in nondecreasing order, in any interval of primorial(n)^k positive integers.

Original entry on oeis.org

1, 1, 1, 1, 5, 2, 1, 19, 62, 8, 1, 65, 1322, 1976, 48, 1, 211, 24182, 318392, 140496, 480, 1, 665, 408842, 42729464, 260656752, 19373280, 5760, 1, 2059, 6609302, 5208402488, 395975417424, 485262187680, 4125121920, 92160, 1, 6305, 103999562, 600582229496

Offset: 1

Peter Munn, Feb 08 2017

Square array read by descending antidiagonals: A(n,k) with rows n >= 1, columns k >= 1. Primorial(n) = A002110(n): product of first n primes.
Visualize the prime factorizations of the positive integers as a table with row headings giving each successive integer, and the primes of which the row heading is the product listed across the columns in nondecreasing order, repeated when necessary. Except for 1, which lacks prime factors, column 1 has the row heading's least prime factor, column 2 has a value for composite numbers but is blank for primes, and so on. This sequence measures precisely how frequently the various primes occur in each column. This is possible because any given prime occurs cyclically in any given column, for the reason following.
The occurrence pattern of up to k factors of prime(n) in such prime factorizations has a fundamental period over the positive integers of prime(n)^k. The least common period for up to k factors of each of the first n primes is Primorial(n)^k, and this covers everything that can affect the occurrence of prime(n) in the least k factors. Thus prime(n) is k-th least prime factor of integer m if and only if it is k-th least prime factor of m+Primorial(n)^k.
Intermediate values in the calculation of this sequence appear in A281891.
A(n,1) = A005867(n-1) in accordance with the comment on A005867 dated Jul 16 2006.
A(2,k) = A001047(k) = 3^k - 2^k.

			Prime(2)=3 occurs as second least factor five times in the prime factorizations of every interval of 36=Primorial(2)^2 positive integers. See A014673. So A(2,2) = 5.

A079474 re-read as a square array gives values of primorial(n)^k = A002110(n)^k.
The values in the body of the factorization table described in the author's comments are in the irregular array A027746.
Cf. A001047, A005867, A014673, A281891.

A(n,k) = primorial(n-1) * A281891(n,k-1) - prime(n)^(k-1) * A281891(n-1,k).

A286941 Irregular triangle read by rows: the n-th row corresponds to the totatives of the n-th primorial, A002110(n).

Original entry on oeis.org

1, 1, 5, 1, 7, 11, 13, 17, 19, 23, 29, 1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 209

Offset: 1

Jamie Morken and Michael De Vlieger, May 16 2017

Values in row n of a(n) are those of row n of A286942 complement those of row n of A279864.
From Michael De Vlieger, May 18 2017: (Start)
Numbers t < p_n# such that gcd(t, p_n#) = 0, where p_n# = A002110(n).
Numbers in the reduced residue system of A002110(n).
A005867(n) = number of terms of a(n) in row n; local minimum of Euler's totient function.
A048862(n) = number of primes in row n of a(n).
A048863(n) = number of nonprimes in row n of a(n).
Since 1 is coprime to all n, it delimits the rows of a(n).
The prime A000040(n+1) is the second term in row n since it is the smallest prime coprime to A002110(n) by definition of primorial.
The smallest composite in row n is A001248(n+1) = A000040(n+1)^2.
The Kummer numbers A057588(n) = A002110(n) - 1 are the last terms of rows n, since (n - 1) is less than and coprime to all positive n. (End)

			The triangle starts
1;
1, 5;
1, 7, 11, 13, 17, 19, 23, 29;
1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 209;

Michael De Vlieger, Table of n, a(n) for n = 1..6299 (rows 1 <= n <= 6 flattened).
Mathoverflow, Lower bound for Euler's totient for almost all integers. - _Michael De Vlieger_, May 18 2017
Eric Weisstein's World of Mathematics, Totative. - _Michael De Vlieger_, May 18 2017

Cf. A002110, A005867, A048862, A057588, A279864, A286941, A286942, A309497, A038110, A058250, A329815.

Cf. A285784 (nonprimes that appear), A335334 (row sums).

Table[Function[P, Select[Range@ P, CoprimeQ[#, P] &]]@ Product[Prime@ i, {i, n}], {n, 4}] // Flatten (* Michael De Vlieger, May 18 2017 *)

row(n) = my(P=factorback(primes(n))); select(x->(gcd(x, P) == 1), [1..P]); \\ Michel Marcus, Jun 02 2020

More terms from Michael De Vlieger, May 18 2017

A053144 Cototient of the n-th primorial number.

Original entry on oeis.org

1, 4, 22, 162, 1830, 24270, 418350, 8040810, 186597510, 5447823150, 169904387730, 6317118448410, 260105476071210, 11228680258518030, 529602053223499410, 28154196550210460730, 1665532558389396767070

Offset: 1

Labos Elemer, Feb 28 2000

nonn

a(n) > A005367(n), a(n) > A002110(n)/2.

Limit_{n->oo} a(n)/A002110(n) = 1 because (in the limit) the quotient is the probability that a randomly selected integer contains at least one of the first n primes in its factorization. - Geoffrey Critzer, Apr 08 2010

			In the reduced residue system of q(4) = 2*3*5*7 - 210 the number of coprimes to 210 is 48, while a(4) = 210 - 48 = 162 is the number of values divisible by one of the prime factors of q(4).

Michael De Vlieger, Table of n, a(n) for n = 1..350

Cf. A002110, A051953, A005867, A038110, A038111, A174909, A293558.
Cf. A000040 (prime numbers).
Cf. A161527, A060753.
Column 1 of A281891.

Abs[Table[ Total[Table[(-1)^(k + 1)* Total[Apply[Times, Subsets[Table[Prime[n], {n, 1, m}], {k}], 2]], {k, 0, m - 1}]], {m, 1, 22}]] (* Geoffrey Critzer, Apr 08 2010 *)
Array[# - EulerPhi@ # &@ Product[Prime@ i, {i, #}] &, 17] (* Michael De Vlieger, Feb 17 2019 *)

a(n) = prod(k=1, n, prime(k)) - prod(k=1, n, prime(k)-1); \\ Michel Marcus, Feb 08 2019

a(n) = A051953(A002110(n)) = A002110(n) - A005867(n).
a(n) = a(n-1)*A000040(n) + A005867(n-1). - Bob Selcoe, Feb 21 2016
a(n) = (1/A000040(n+1) - A038110(n+1)/A038111(n+1))*A002110(n+1). - Jamie Morken, Feb 08 2019
a(n) = A161527(n)*A002110(n)/A060753(n+1). - Jamie Morken, May 13 2022

A070918 Triangle of T(n,k) coefficients of polynomials with first n prime numbers as roots.

Original entry on oeis.org

1, -2, 1, 6, -5, 1, -30, 31, -10, 1, 210, -247, 101, -17, 1, -2310, 2927, -1358, 288, -28, 1, 30030, -40361, 20581, -5102, 652, -41, 1, -510510, 716167, -390238, 107315, -16186, 1349, -58, 1, 9699690, -14117683, 8130689, -2429223, 414849, -41817, 2451, -77, 1

Offset: 0

Rick L. Shepherd, May 20 2002

Analog of the Stirling numbers of the first kind (A008275): The Stirling numbers (beginning with the 2nd row) are the coefficients of the polynomials having exactly the first n natural numbers as roots. This sequence (beginning with first row) consists of the coefficients of the polynomials having exactly the first n prime numbers as roots.

			Row 4 of this sequence is 210, -247, 101, -17, 1 because (x-prime(1))(x-prime(2))(x-prime(3))(x-prime(4)) = (x-2)(x-3)(x-5)(x-7) = x^4 - 17*x^3 + 101*x^2 - 247*x + 210.
Triangle begins:
        1;
       -2,      1;
        6,     -5,       1;
      -30,     31,     -10,      1;
      210,   -247,     101,    -17,      1;
    -2310,   2927,   -1358,    288,    -28,    1;
    30030, -40361,   20581,  -5102,    652,  -41,   1;
  -510510, 716167, -390238, 107315, -16186, 1349, -58, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A008275 (Stirling numbers of first kind).
Cf. A005867 (absolute values of row sums).
Cf. A054640 (sum of absolute values of terms in rows).
Cf. A000040, A078456.

T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(mul(x-ithprime(i), i=1..n)):
seq(T(n), n=0..10);  # Alois P. Heinz, Aug 18 2019

Table[CoefficientList[Expand[Times@@(x-Prime[Range[n]])],x],{n,0,10}]// Flatten (* Harvey P. Dale, Feb 12 2020 *)

p=1; for(k=1,10,p=p*(x-prime(k)); for(n=0,k,print1(polcoeff(p,n),",")))

From Alois P. Heinz, Aug 18 2019: (Start)
T(n,k) = [x^k] Product_{i=1..n} (x-prime(i)).
Sum_{k=0..n} |T(n,k)| = A054640(n).
|Sum_{k=0..n} T(n,k)| = A005867(n).
|Sum_{k=0..n} k * T(n,k)| = A078456(n). (End)

First term T(0,0)=1 prepended by Alois P. Heinz, Aug 18 2019

A059865 Product_{i=4..n} (prime(i) - 6).

Original entry on oeis.org

1, 1, 1, 1, 5, 35, 385, 5005, 85085, 1956955, 48923875, 1516640125, 53082404375, 1964048961875, 80526007436875, 3784722349533125, 200590284525255625, 11032465648889059375, 672980404582232621875, 43743726297845120421875

Offset: 1

Labos Elemer, Feb 28 2001

nonn

Arises in Hardy-Littlewood prime k-tuplet conjectural formulas. Also the sequence gives the exact numbers of X42424Y difference-pattern in dRRS[m], where m=modulus=A002110(n). See A049296 (=dRRS[210]=list of first differences of reduced residue system modulo 210=4th primorial). A pattern X42424Y corresponds to a residue-sextuple or it is their difference-quintuple, X,Y > 4. Analogous pattern for primes is in A022008.

a(352) has 1001 decimal digits. - Michael De Vlieger, Mar 06 2017

			a(7) = (prime(4)-6) * (prime(5)-6) * (prime(6)-6) * (prime(7)-6) = 1 * 5* 7 *11 = 385
 Also in one period of dRRS with 2,6,30,210,2310,... modulus [A002110(n)] 1,2,8,48,480,... differences occur [A005867(n)]. The number of X42424Y residue-difference-patterns are 0,1,1,1,5,... respectively starting at suitable residues coprime to A002110(n).

See A059862 for references.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 84-94.

Michael De Vlieger, Table of n, a(n) for n = 1..351
C. K. Caldwell, Prime k-tuple Conjecture
Steven R. Finch, Hardy-Littlewood Constants [Broken link]
Steven R. Finch, Hardy-Littlewood Constants [From the Wayback machine]
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]

Cf. A049296, A002110, A005867, A000847, A022008, A051160-A051168, A048298, A059861-A059865.

Table[Product[Prime@ i - 6, {i, 4, n}], {n, 19}] (* Michael De Vlieger, Mar 06 2017 *)

a(n) = prod(k=4, n, prime(k) - 6); \\ Michel Marcus, Mar 06 2017

A078456 Number of numbers less than prime(1)...prime(n) having exactly one prime factor among (prime(1),...,prime(n)) where prime(n) is the n-th prime.

Original entry on oeis.org

1, 3, 14, 92, 968, 12096, 199296, 3679488, 82607616, 2349508608, 71507128320, 2604912721920, 105300128563200, 4466750187110400, 207324589680230400, 10866166392736972800, 634672612705724006400, 38337584554108256256000

Offset: 1

Benoit Cloitre, Dec 31 2002

nonn

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

Second column of A096294. - Eric Desbiaux, Jun 20 2013

			a(2)=3 since 2*3=6 and 2,3,4 have 1 prime factor among (2,3)
3 1 1 1 1 ...
1 5 1 1 1 ...
1 1 7 1 1 ...
1 1 1 11 1 ...
1 1 1 1 13 ...
and so a(2) = 3, a(3) = 3*5 - 1*1 = 14, a(4) = 3*5*7 + 1*1*1 + 1*1*1 - 7*1*1 - 5*1*1 - 3*1*1 = 92, etc.

Charles R Greathouse IV, Table of n, a(n) for n = 1..350

Cf. A135212, A120271.

Cf. A117494, A002110.

Table[ Det[ DiagonalMatrix[ Table[ Prime[i+1]-1, {i, 1, n-1} ] ] + 1 ], {n, 1, 20} ] (* Alexander Adamchuk, Jun 02 2006 *)

a(n)=sum(k=1,prod(i=1,n, prime(i)),if(isprime(gcd(k,prod(i=1,n, prime(i)))),1,0))

a(n) = matdet(matrix(n-1, n-1, j, k, if (j==k, prime(j+1), 1))); \\ after Mathematica; Michel Marcus, Oct 02 2016

a(n) = (prime(n)-1)*a(n-1) + A005867(n). - Matthew Vandermast, Jun 06 2004
a(n) = A120071(n) * A135212(n). - Alexander Adamchuk, Nov 23 2007
a(n) = A117494(A002110(n)). - Ridouane Oudra, Sep 18 2022

a(7) from Ralf Stephan, Mar 25 2003
a(8)-a(12) from Matthew Vandermast, Jun 06 2004
More terms from Alexander Adamchuk, Jun 02 2006

A101301 The sum of the first n primes, minus n.

Original entry on oeis.org

1, 3, 7, 13, 23, 35, 51, 69, 91, 119, 149, 185, 225, 267, 313, 365, 423, 483, 549, 619, 691, 769, 851, 939, 1035, 1135, 1237, 1343, 1451, 1563, 1689, 1819, 1955, 2093, 2241, 2391, 2547, 2709, 2875, 3047, 3225, 3405, 3595, 3787, 3983, 4181, 4391, 4613, 4839

Offset: 1

Jorge Coveiro, Dec 22 2004

nonn

Also: a(n) = sum_{k=1..n} phi(prime(k)).
Partial sums of A006093. - Omar E. Pol, Oct 31 2013
Difference minus n, between the constant term prime(n) for a polynomial P(x) built from the first n primes took as coefficients and the value that such term should have in order to make P(x) divisible by (x-1). See links. - R. J. Cano, Jan 14 2014
Sum of all deficiencies of the first n primes. - Omar E. Pol, Feb 21 2014

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
R. J. Cano, Additional information
R. J. Cano, PARI/GP code: alternative sequencer

Cf. A000010, A000040, A006093, A005867.

a101301 n = a101301_list !! (n-1)
a101301_list = scanl1 (+) a006093_list
-- Reinhard Zumkeller, May 01 2013

seq((sum(phi(ithprime(x)),k=1..n)),n=1..100);

f[n_]:=Plus@@Prime[Range[n]]-n; Table[f[n],{n,1,50}] (* Enrique Pérez Herrero, Jun 10 2012 *)

a(n)=my(s);forprime(p=2,prime(n),s+=p); s-n \\ Charles R Greathouse IV, Oct 31 2013

PARI
```
\\ See links.
```

a(n)=sum_{k=1..n} (prime(k)-1)
a(n)=A007504(n)-n. - Juri-Stepan Gerasimov, Nov 23 2009
A027424(A000040(n)) < a(n). - Charles R Greathouse IV, Apr 07 2021

Name simplified by Juri-Stepan Gerasimov, Nov 23 2009

A309497 Irregular triangle read by rows: T(n,k) = A060753(n)k-A038110(n)A286941(n,k).

Original entry on oeis.org

0, 1, 2, 1, 11, 2, 1, 8, 7, 14, 13, 4, 27, -18, 1, 4, 23, 26, 13, 32, 19, 22, 41, 44, 31, 18, 37, 24, 27, 46, 33, 36, 23, -6, -3, 16, 19, 38, 41, 12, -1, 2, -11, 8, 11, -2, 17, 4, -9, -6, 13, 16, 3, 22, 9, 12, 31, 34, 53, 8

Offset: 0

Jamie Morken, Aug 05 2019

The sequence is Primorial rows of A308121.
Row n has length A005867(n).
Row n > 1 average value = A060753(n)/2.
Row n > 1 has sum = A002110(n-1)*A038110(n)/2.
First value on row(n) = A161527(n-1).
Last value on row(n) = A038110(n) for n > 2.
For n > 1, A060753(n) = Max(row) + Min(row).
For values x and y on row n > 1 at positions a and b on the row:
x + y = A060753(n), where a = A005867(n-1) - (b-1).
For n > 2 the penultimate value on row A002110(n) is given by
  A038110(n)*A000040(n)-A060753(n).
Related identity:
  A038110(n)/A038111(n)*(Prime(n)^2) - (A038110(n)/A038111(n)*((A038110(n)*Prime(n) - A060753(n))*Prime(n)/A038110(n))) = 1.

			The triangle starts:
row1: 0;
row2: 1;
row3: 2, 1;
row4: 11, 2, 1, 8, 7, 14, 13, 4;
row5: 27, -18, 1, 4, 23, 26, 13, 32, 19, 22, 41, 44, 31, 18, 37, 24, 27, 46, 33, 36, 23, -6, -3, 16, 19, 38, 41, 12, -1, 2, -11, 8, 11, -2, 17, 4, -9, -6, 13, 16, 3, 22, 9, 12, 31, 34, 53, 8;

Michael De Vlieger, Table of n, a(n) for n = 0..6299 (rows n = 0..6, flattened)
Eric Weisstein's World of Mathematics, Totative

Cf. A058250, A005867, A002110, A038110, A038111, A060753, A286941, A058262, A161527, A083140, A308121.

row[0] = 0; row[n_] := -(v = Numerator[Product[1 - 1/Prime[i], {i, 1, n}] / Prime[n]] * Select[Range[(p = Product[Prime[i], {i, 1, n}])], CoprimeQ[p, #] &]) + Denominator[Product[((pr = Prime[i]) - 1)/pr, {i, 1, n}]] * Range[Length[v]]; Table[row[n], {n, 0, 4}] // Flatten (* Amiram Eldar, Aug 10 2019 *)

cp's OEIS Frontend

A059861 a(n) = Product_{i=2..n} (prime(i) - 2).

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A048863 Number of nonprimes (1 and composites) in the reduced residue system of n-th primorial number (A002110).

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A281890 Square array A(n,k): number of integers having prime(n) as k-th factor when written as product of primes in nondecreasing order, in any interval of primorial(n)^k positive integers.

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A286941 Irregular triangle read by rows: the n-th row corresponds to the totatives of the n-th primorial, A002110(n).

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A053144 Cototient of the n-th primorial number.

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A070918 Triangle of T(n,k) coefficients of polynomials with first n prime numbers as roots.

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A059865 Product_{i=4..n} (prime(i) - 6).

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A078456 Number of numbers less than prime(1)...prime(n) having exactly one prime factor among (prime(1),...,prime(n)) where prime(n) is the n-th prime.

A309497 Irregular triangle read by rows: T(n,k) = A060753(n)k-A038110(n)A286941(n,k).