A005867 -id:A005867 - cp's OEIS frontend

A051854 Table of solutions to all possible Chinese Remainder Equations x = a1 mod p1, x = a2 mod p2, ..., x = an mod pn, where p1 - pn are the first n primes and each a1 - an varies between 1 and (its respective) p-1, with the rightmost a (an) varying fastest.

1, 1, 5, 1, 7, 13, 19, 11, 17, 23, 29, 1, 121, 31, 151, 61, 181, 127, 37, 157, 67, 187, 97, 43, 163, 73, 193, 103, 13, 169, 79, 199, 109, 19, 139, 71, 191, 101, 11, 131, 41, 197, 107, 17, 137, 47, 167, 113, 23, 143, 53, 173, 83, 29, 149, 59, 179, 89, 209, 1, 211, 421

Offset: 1

Antti Karttunen, Dec 13 1999

			Rows have lengths 1,2,8,48,480,5760,92160,... (A005867(n)) and terms 1; 1,5; 1,7,13,19,11,17,23,29;

Cf. A051853.

with(numtheory); incr_plist_from_right := proc(aa) local i,n,a; a := aa; n := nops(a); for i from n by -1 to 1 do if(a[i] < (ithprime(i)-1)) then a[i] := a[i]+1; RETURN(a); else a[i] := 1; fi; od; RETURN([op(a),1]); end;
incr_plist_from_right_n_times := proc(aa,n) local a,i; a := aa; for i from 1 to n do a := incr_plist_from_right(a); od; RETURN(a); end; prim_chrem_right := proc(n) local r,m; r := incr_plist_from_right_n_times([],n); m := form_modlist(r); RETURN(chrem(r,m)); end; # For form_modlist see A051853.

row[n_] := Module[{i}, pp = Prime[Range[n]]; iter = Sequence @@ Table[{ i[k], 1, pp[[k]] - 1}, {k, 1, n}]; Table[ChineseRemainder[Array[i, n], pp], iter // Evaluate] // Flatten]; Table[row[n], {n, 1, 5}] // Flatten (* Jean-François Alcover, Mar 06 2016 *)

a(n) = prim_chrem_right(n) (see Maple code)

A058251 LCM of n-th primorial number and its Euler totient.

Original entry on oeis.org

1, 2, 6, 120, 1680, 36960, 5765760, 1568286720, 536354058240, 24672286679040, 2861985254768640, 2661646286934835200, 3545312854197200486400, 5814313080883408797696000, 10500649424075436288638976000

Offset: 0

Labos Elemer, Dec 05 2000

nonn

			a(6) = LCM(30030,5760) = 5765760.

Cf. A002110, A000010, A005867, A009262.

[seq(ilcm(product(ithprime(k), k=1..m), product(ithprime(k)-1, k=1..m)), m=1..20)];

LCM[#,EulerPhi[#]]&/@Rest[FoldList[Times,1,Prime[Range[15]]]] (* Harvey P. Dale, Nov 29 2011 *)

P(n) = prod(k=1, n, prime(k)); \\ A002110
a(n) = my(p= P(n)); lcm(p, eulerphi(p)); \\ Michel Marcus, Apr 27 2022

a(n) = LCM(A002110(n), A000010(A002110(n))) = LCM(A002110(n), A005867(n)).

a(n) = A009262(A002110(n)). - Michel Marcus, Apr 27 2022

a(0)=1 inserted by Jamie Morken, Apr 27 2022

A058256 a(n) = A058254(n+1)/A058254(n).

Original entry on oeis.org

2, 2, 3, 5, 1, 4, 3, 11, 7, 1, 1, 1, 1, 23, 13, 29, 1, 1, 1, 1, 1, 41, 1, 2, 5, 17, 53, 3, 1, 1, 1, 1, 1, 37, 1, 1, 3, 83, 43, 89, 1, 19, 2, 7, 1, 1, 1, 113, 1, 1, 1, 1, 5, 4, 131, 67, 1, 1, 1, 47, 73, 1, 31, 1, 79, 1, 1, 173, 1, 1, 179, 61, 1, 1, 191, 97, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Labos Elemer, Dec 06 2000

nonn

a(n) = 1 if in prime(n+1)-1 no new prime divisor or new power of a prime appear, like LCM[{1, 2, 4, 6, 10, 12, 16, 22}]= LCM[{1, 2, 4, 6, 10, 12, 16, 22, 28}].

a(n) > 1 if in prime(n+1)-1 new prime divisor(s) or new power(s) of a prime arise, like in A058254(15) compared with A058254(14), where the new prime divisor is 23 only, so a(14)=23. Such sites of increase do not correspond to the natural order of primes and prime-powers like in A054451.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A058254, A002110, A005867, A003418, A054451, A000142, A000010, A003418, A000961.

f(n) = lcm(apply(p->p-1, primes(n))); \\ A058254
a(n) = f(n+1)/f(n); \\ Michel Marcus, Mar 22 2020

a(n) = lcm{i=1..n+1} (prime(i)-1) / lcm{i=1..n} (prime(i)-1).

Offset corrected by Amiram Eldar, Sep 24 2019

A059863 a(n) = Product_{i=3..n} (prime(i)-4).

Original entry on oeis.org

1, 1, 1, 3, 21, 189, 2457, 36855, 700245, 17506125, 472665375, 15597957375, 577124422875, 22507852492125, 967837657161375, 47424045200907375, 2608322486049905625, 148674381704844620625, 9366486047405211099375, 627554565176149143658125, 43301264997154290912410625

Offset: 1

Labos Elemer, Feb 28 2001

nonn

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

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.

a(n) = prod(i=3, n, prime(i)-4); \\ Michel Marcus, Aug 25 2019

More terms from Michel Marcus, Aug 25 2019

A059864 a(n) = Product_{i=4..n} (prime(i)-5), where prime(i) is i-th prime.

Original entry on oeis.org

1, 1, 1, 2, 12, 96, 1152, 16128, 290304, 6967296, 181149696, 5796790272, 208684449792, 7930009092096, 333060381868032, 15986898329665536, 863292509801938944, 48344380548908580864, 2997351594032332013568

Offset: 1

Labos Elemer, Feb 28 2001

nonn

Such products arise in Hardy-Littlewood prime k-tuplet conjectural formulas.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 84-94.
R. K. Guy, Unsolved Problems in Number Theory, 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

G. C. Greubel, Table of n, a(n) for n = 1..350
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. A000847, A002110, A005867, A022008, A048298, A049296, A051160, A051161, A051162.
Cf. A051163, A051164, A051165, A051166, A051167, A051168, A059861, A059862, A059863.
Cf. A059864, A059865.

[n le 3 select 1 else (&*[NthPrime(j) -5: j in [4..n]]): n in [1..30]]; // G. C. Greubel, Feb 02 2023

Join[{1,1,1},FoldList[Times,Prime[Range[4,20]]-5]] (* Harvey P. Dale, Dec 29 2018 *)

a(n) = prod(k=4, n, prime(k)-5); \\ Michel Marcus, Dec 12 2017

def A059864(n): return product(nth_prime(j) -5 for j in range(4,n+1))
[A059864(n) for n in range(1,31)] # G. C. Greubel, Feb 02 2023

A112228 Product of the first n (semiprimes - 1).

Original entry on oeis.org

3, 15, 120, 1080, 14040, 196560, 3931200, 82555200, 1981324800, 49533120000, 1585059840000, 52306974720000, 1778437140480000, 65802174197760000, 2500482619514880000, 112521717878169600000, 5401042458152140800000

Offset: 1

Jonathan Vos Post, Nov 28 2005

Semiprime analog of A005867.

			a(10) = 3*5*8*9*13*14*20*21*24*25 =
(4-1)*(6-1)*(9-1)*(10-1)*(14-1)*(15-1)*(21-1)*(22-1)*(25-1)*(26-1) = 49533120000, the product of the first 10 (semiprimes minus one).

Cf. A001358, A005867.

FoldList[Times,Select[Range[50],PrimeOmega[#]==2&]-1] (* Harvey P. Dale, Jan 21 2023 *)

A001358(n)={
    if(n==1,
        return(4),
        for(a=A001358(n-1)+1,100000000000000000,
            if (bigomega(a)==2,
                return(a)
            ) ;
        ) ;
    ) ;
}
A112228(n)={
    prod(i=1,n,A001358(i)-1)
} /* R. J. Mathar, Mar 09 2012 */

a(n) = Prod[from i = 1 to n] (A001358(i)-1).

A280050 a(n) = Sum_{k=2..n} k/lpf(k), where lpf(k) is the least prime dividing k (A020639).

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 13, 16, 21, 22, 28, 29, 36, 41, 49, 50, 59, 60, 70, 77, 88, 89, 101, 106, 119, 128, 142, 143, 158, 159, 175, 186, 203, 210, 228, 229, 248, 261, 281, 282, 303, 304, 326, 341, 364, 365, 389, 396, 421, 438, 464, 465, 492, 503, 531, 550, 579, 580, 610, 611, 642, 663, 695, 708, 741, 742, 776, 799, 834, 835

Offset: 1

Ilya Gutkovskiy, Jan 02 2017

Sum of the largest proper divisors of all positive integers <= n.

			For n = 8 the divisors of the first eight positive integers are {1}, {1, 2}, {1, 3}, {1, 2, 4}, {1, 5}, {1, 2, 3, 6}, {1, 7}, {1, 2, 4, 8}, so a(8) = 1 + 1 + 2 + 1 + 3 + 1 + 4 = 13.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Least Prime Factor.
Eric Weisstein's World of Mathematics, Proper Divisor.

Cf. A001065, A002110, A005867, A020639, A032742, A046669, A046670, A153485.

Table[Sum[k/FactorInteger[k][[1, 1]], {k, 2, n}], {n, 71}]
Join[{0}, Accumulate[Table[k/FactorInteger[k][[1, 1]], {k, 2, 71}]]] (* Amiram Eldar, Jul 03 2025 *)

list(kmax) = {my(s = 0); print1(s, ", "); for(k = 2, kmax, s += k/factor(k)[1,1]; print1(s, ", "));} \\ Amiram Eldar, Jul 03 2025

a(n) = Sum_{k=2..n} k/A020639(k).
a(n) + 1 = Sum_{k=1..n} A032742(k).
a(p^k) = a(p^k-1) + p^(k-1), when p is prime.
a(n) ~ c * n^2, where c = (1/2) * Sum_{k>=1} A005867(k-1)/(prime(k)*A002110(k)) = 0.165049... . - Amiram Eldar, Jul 03 2025

A307826 The number of integers r such that all primes above a certain value have the form primorial(n)*q +- r.

Original entry on oeis.org

1, 1, 4, 24, 240, 2880, 46080, 829440, 18247680, 510935040, 15328051200, 551809843200, 22072393728000, 927040536576000, 42643864682496000, 2217480963489792000, 128613895882407936000, 7716833752944476160000, 509311027694335426560000

Offset: 1

William Boyles, Apr 30 2019

nonn

			For n=3, the third primorial is 2*3*5=30, and all primes at least 17 have the form 30n +- (1,7,11,13). So, a(3) = 4.

William Boyles, Table of n, a(n) for n = 1..350

Cf. A000010, A002110, A005867, A156037.

a[1]=1; a[n_] := EulerPhi[Product[Prime[i], {i, 1, n}]]/2; Array[a, 20] (* Amiram Eldar, Jul 08 2019 *)

import sympy
def A307826(n):
    sympy.sieve.extend_to_no(n)
    s = list(sympy.sieve._list)
    prod = s[0]
    print("1")
    for i in range(1,n):
        prod*=s[i]
        print(sympy.ntheory.factor_.totient(prod)//2)

a(n) = Product_{k=1..n} A156037(k).
a(n) = A000010(A002110(n))/2 for n > 1.
a(n) = A005867(n)/2 for n > 1. - Alexandre Herrera, Apr 16 2023

A308665 Cardinality of the shortcut set of the multiplicative group of integers modulo the n-th primorial.

Original entry on oeis.org

1, 2, 4, 48, 384, 2304, 4608, 18432, 552960, 19906560, 796262400, 33443020800, 66886041600, 267544166400, 535088332800, 32105299968000, 2118949797888000, 148326485852160000, 10679506981355520000, 833001544545730560000, 1666003089091461120000, 146608271840048578560000

Offset: 3

Alexey V. Bazhin, Jun 15 2019

{R(1),...,R(phi(prime(n)#))} = {(prime(n)#/2 +- {R(1),...,R(k)}*2^{1,2,...,z}) mod prime(n)#}.

The formula is used for the compressed representation of the set of the multiplicative group of integers modulo prime(n)#. This is a new formula in number theory. {R(1),...,R(k)} is the shortcut set of the multiplicative group of integers modulo prime(n)# (or compressed representation of the set of the multiplicative group of integers modulo prime(n)#).

			First example:
Let R = {1, 7, 11, 13, 17, 19, 23, 29} be the set of the multiplicative group of integers modulo 30 (or it is the set of positive integers less than 2*3*5 and prime to 2*3*5, or it is the reduced residue system for the 3rd primorial number 30 = A002110(3)).
The cardinality of |R| is equal to 8 elements, 8 = A005867(3).
The set {1} is the shortcut set of the multiplicative group of integers modulo 30.
The cardinality of |{1}| = 1, i.e., a(1) = 1.
R = { (30/2 +- {1}*2^{1, ..., 4}) mod 30 } = {(30/2 +- {1}*2^{1,2,3,4}) mod 30}={(15+2^1) mod 30, (15-2^1) mod 30,(15+2^2) mod 30, (15-2^2) mod 30, (15+2^3) mod 30, (15-2^3) mod 30, (15+2^4) mod 30, (15-2^4) mod 30} = {1,7,11,13,17,19,23,29},
where 30 = A002110(3), 4 = A155747(2), |R| = A005867(3).
4 is the smallest number m with the property that 2^m-1 is divisible by the first n odd primes.
Second example:
Let R = {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} be the set of the multiplicative group of integers modulo 210 (or it is the set of positive integers less than 2*3*5*7 and prime to 2*3*5*7, or it is the reduced residue system for the 4th primorial number 210 = A002110(4)).
The cardinality of |R| is equal to 48 elements, 48 = A005867(5).
The set {1, 11} is the shortcut set of the multiplicative group of integers modulo 210.
The cardinality of |{1, 11}| = 2, i.e., a(2) = 2.
R = { ( 210/2 +- {1, 11}*2 ^{1..12}) mod 210 }, where 210 = A002110(4), 12 = A155747(3), |R| = A005867(4).
12 is the smallest number m with the property that 2^m-1 is divisible by the first n odd primes.
General case:
R = {R(1), ..., R(phi(prime(n)#))},
R = {(prime(n)#/2 +- {R(1),R(2),...,R(k)}*2^{1,2,...,z}) mod prime(n)#}, where prime(n)# is the product of the first n primes, i.e., it is A002110(n); phi is Euler's totient function, which counts the positive integers up to a given argument that are relatively prime to the argument, in our case phi(prime(n)#) is A005867;
R is the set of the multiplicative group of integers modulo prime(n)#;
z is a term of A155747.
k is a term of a(n).
R(1) <= R(k) < R(phi(prime(n)#)).
Table:
+-----+-----------+----------------+---------+------+
|  n  | A002110   | A005867        | A155747 | a(n) |
|     | prime(n)# | phi(prime(n)#) | z       | k    |
+-----+-----------+----------------+---------+------+
|  0  |        1  |          1     |   -     |    - |
|  1  |        2  |          1     |   -     |    - |
|  2  |        6  |          2     |   2     |    - |
|  3  |       30  |          8     |   4     |    1 |
|  4  |      210  |         48     |  12     |    2 |
|  5  |     2310  |        480     |  60     |    4 |
|  6  |    30030  |       5760     |  60     |   48 |
|  7  |   510510  |      92160     | 120     |  384 |
|  8  |  9699690  |    1658880     | 360     | 2304 |
| ..  |      ...  |        ...     | ...     |  ... |

Alexey V. Bazhin, Table of n, a(n) for n = 3..24

Cf. A000040, A002110, A005867, A155747.

f(n) = prod(k=1, n, prime(k)-1); \\ A005867
g(n) = lcm(vector(n, k, znorder(Mod(2, prime(k+1))))); \\ A155747
a(n) = f(n+2)/(g(n+1)*2); \\ Michel Marcus, Jun 25 2019

a(n) = A005867(n+2)/(2*A155747(n+1)).

If {R(1),...,R(phi(prime(n)#))} = {(prime(n)#/2 +- {R(1),...,R(k)}*2^{1,2,...,z}) mod prime(n)#} then a(n) = k.

A318766 a(0) = 1; for n > 0, a(n) = (prime(n)^2 - 1) * a(n-1).

Original entry on oeis.org

1, 3, 24, 576, 27648, 3317760, 557383680, 160526499840, 57789539942400, 30512877089587200, 25630816755253248000, 24605584085043118080000, 33660439028338985533440000, 56549537567609495696179200000, 104503545424942348046539161600000

Offset: 0

Seiichi Manyama, Sep 03 2018

nonn

The limit of A061742(n)/a(n) is zeta(2) (cf. A013661).

Seiichi Manyama, Table of n, a(n) for n = 0..195

Product_{k=1..n} (prime(k)^m - 1): A005867 (m=1), this sequence (m=2).

Cf. A000040, A013661, A061742, A084920.

a[n_]:=Product[Prime[k]^2-1, {k, 1, n}]; Join[{1},Array[a, nmax]] (* Stefano Spezia, Sep 03 2018 *)

PARI
```
{a(n) = prod(k=1, n, prime(k)^2-1)}
```

a(n) = A084920(n) * a(n-1) for n > 0.

a(n) = Product_{k=1..n} (prime(k)^2 - 1).

cp's OEIS Frontend

A051854 Table of solutions to all possible Chinese Remainder Equations x = a1 mod p1, x = a2 mod p2, ..., x = an mod pn, where p1 - pn are the first n primes and each a1 - an varies between 1 and (its respective) p-1, with the rightmost a (an) varying fastest.

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A058251 LCM of n-th primorial number and its Euler totient.

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A058256 a(n) = A058254(n+1)/A058254(n).

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A059863 a(n) = Product_{i=3..n} (prime(i)-4).

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A059864 a(n) = Product_{i=4..n} (prime(i)-5), where prime(i) is i-th prime.

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A112228 Product of the first n (semiprimes - 1).

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A280050 a(n) = Sum_{k=2..n} k/lpf(k), where lpf(k) is the least prime dividing k (A020639).

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