cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

A008364 11-rough numbers: not divisible by 2, 3, 5 or 7.

Original entry on oeis.org

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, 211, 221, 223, 227, 229, 233, 239, 241, 247
Offset: 1

Views

Author

Keywords

Comments

The first A005867(4) = 48 terms give the reduced residue system for the 4th primorial number 210 = A002110(4).
This sequence is closed under multiplication: any product of terms is also a term. - Labos Elemer, Feb 26 2003
Conjecture: these are numbers n such that (Sum_{k=1..n} k^4) mod n = 0 and (Sum_{k=1..n} k^6) mod n = 0. - Gary Detlefs, Dec 20 2011
From Peter Bala, May 03 2018: (Start)
The above conjecture is true. Let m be even and let the m-th Bernoulli number be written in reduced form as Bernoulli(m) = N(m)/D(m). Apply Ireland and Rosen, Proposition 15.2.2, to show the congruence D(m)*( Sum_{k = 1..n} k^m )/n = N(m) (mod n) holds for all n >= 1. It follows easily from this congruence that ( Sum_{k = 1..n} k^m )/n is integral iff n is coprime to D(m). Now Bernoulli(4) = -1/(2*3*5) and Bernoulli(6) = 1/(2*3*7) so the numbers n such that both (Sum_{k=1..n} k^4) mod n = 0 and (Sum_{k=1..n} k^6) mod n = 0 are exactly those numbers coprime to the primes 2, 3, 5 and 7, that is, the 11-rough numbers. (End)
Conjecture: these are numbers n such that (n^6 mod 210 = 1) or (n^6 mod 210 = 169). - Gary Detlefs, Dec 30 2011
The second Detlefs conjecture above is true and extremely easy to verify with some basic properties of congruences: take the terms of this sequence up to 209 and compute their sixth powers modulo 210: there should only be 1's and 169's there. Then take the complement of this sequence up to 210, where you will see no instances of 1 or 169. - Alonso del Arte, Jan 12 2014
It is well-known that the product of 7 consecutive integers is divisible by 7!. Conjecture: This sequence is exactly the set of positive values of r such that ( Product_{k = 0..6} n + k*r )/7! is an integer for all n. - Peter Bala, Nov 14 2015
From Ruediger Jehn, Nov 05 2020: (Start)
This conjecture is true. The first part of the proof deals with numbers not in A008364, i.e., numbers which are divisible by p (p either 2, 3, 5, 7). Let r = p*s and n = 1, then (Product_{k = 0..6} n + k*r) is not divisible by p, because none of the factors 1 + k*p*s are divisible by p. Hence dividing the product by 7! does not return an integer.
The second part deals with numbers in A008364. If r and q are coprime, then for any i < q there exists k < q with (k*r mod q) = i. From this, it also follows that for any n there exists k < q with ((n + k*r) mod q) = 0. But this means that Product_{k = 0..6} n + k*r is divisible by all numbers from 2 to 7 because there is always a factor that is divisible. We still have to show that the product is also divisible by 2 times 3 times 4 times 6. If the k_1 with ((n + k_1*r) mod 4) = 0 is even, then (n mod 2) = ((n + 2*r) mod 2) = ((n + 4*r) mod 2) = ((n + 6*r) mod 2) = 0. If this k_1 is odd, then ((n + r) mod 2) = ((n + 3*r) mod 2) = ((n + 5*r) mod 2) = 0. In both cases there are at least 2 other factors divisible by 2. If the k_2 with ((n + k_2*r) mod 6) = 0 is smaller than 4, then ((n + (k_2 + 3)*r) mod 3) = 0. Otherwise, ((n + (k_2 - 3)*r) mod 3) = 0. In both cases there is at least 1 other factor divisible by 3. And therefore Product_{k = 0..6} n + k*r is divisible by 7! for any n.
(End)

References

  • Diatomic sequence of 4th prime: A. de Polignac (1849), J. Dechamps (1907).
  • Dickson L. E., History of the Theory of Numbers, Vol. 1, p. 439, Chelsea, 1952.
  • K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, 1980.

Crossrefs

First differences give A049296. Cf. A002110, A048597.
For k-rough numbers with other values of k, see A000027, A005408, A007310, A007775, A008364, A008365, A008366, A166061, A166063. - Michael B. Porter, Oct 10 2009
Cf. A005867, A092695, A210679, A080672 (complement).

Programs

  • Haskell
    a008364 n = a008364_list !! (n-1)
    a008364_list = 1 : filter ((> 7) . a020639) [1..]
    -- Reinhard Zumkeller, Mar 26 2012
  • Maple
    for i from 1 to 500 do if gcd(i,210) = 1 then print(i); fi; od;
    t1:=[]; for i from 1 to 1000 do if gcd(i,210) = 1 then t1:=[op(t1),i]; fi; od: t1;
    S:= (j,n)-> sum(k^j,k=1..n): for n from 1 to 247 do if (S(4,n) mod n = 0) and (S(6,n) mod n = 0) then print(n) fi od; # Gary Detlefs, Dec 20 2011
  • Mathematica
    Select[ Range[ 300 ], GCD[ #1, 210 ] == 1 & ]
    Select[Range[250], Mod[#, 2]>0 && Mod[#, 3]>0 && Mod[#, 5]>0 && Mod[#, 7]>0 &] (* Vincenzo Librandi, Nov 16 2015 *)
    Cases[Range@1000, x_ /; NoneTrue[Array[Prime, 4], Divisible[x, #] &]] (* Mikk Heidemaa, Dec 07 2017 *)
    Select[Range[250],Union[Divisible[#,{2,3,5,7}]]=={False}&] (* Harvey P. Dale, Sep 24 2021 *)
  • PARI
    isA008364(n) = gcd(n,210)==1 \\ Michael B. Porter, Oct 10 2009
    

Formula

Starting with a(49) = 211, a(n) = a(n-48) + 210. - Zak Seidov, Apr 11 2011
a(n) = a(n-1) + a(n-48) - a(n-49). - Charles R Greathouse IV, Dec 21 2011
A020639(a(n)) > 7. - Reinhard Zumkeller, Mar 26 2012
G.f.: x*(x^48 + 10*x^47 + 2*x^46 + 4*x^45 + 2*x^44 + 4*x^43 + 6*x^42 + 2*x^41 + 6*x^40 + 4*x^39 + 2*x^38 + 4*x^37 + 6*x^36 + 6*x^35 + 2*x^34 + 6*x^33 + 4*x^32 + 2*x^31 + 6*x^30 + 4*x^29 + 6*x^28 + 8*x^27 + 4*x^26 + 2*x^25 + 4*x^24 + 2*x^23 + 4*x^22 + 8*x^21 + 6*x^20 + 4*x^19 + 6*x^18 + 2*x^17 + 4*x^16 + 6*x^15 + 2*x^14 + 6*x^13 + 6*x^12 + 4*x^11 + 2*x^10 + 4*x^9 + 6*x^8 + 2*x^7 + 6*x^6 + 4*x^5 + 2*x^4 + 4*x^3 + 2*x^2 + 10*x + 1) / (x^49 - x^48 - x + 1). - Colin Barker, Sep 27 2013
a(n) = 35*n/8 + O(1). - Charles R Greathouse IV, Sep 14 2015
A007775 INTERSECT A206547. - R. J. Mathar, Apr 10 2024

Extensions

New name from Charles R Greathouse IV, Dec 21 2011 based on comment from Michael B. Porter, Oct 10 2009

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

Original entry on oeis.org

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

Views

Author

Labos Elemer, Feb 28 2001

Keywords

Comments

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)

Examples

			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, ..}
		

References

  • 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.

Crossrefs

Programs

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

Formula

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

Extensions

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

Views

Author

Labos Elemer, Feb 28 2001

Keywords

Comments

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

Examples

			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).
		

References

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

Crossrefs

Programs

  • Mathematica
    Table[Product[Prime@ i - 6, {i, 4, n}], {n, 19}] (* Michael De Vlieger, Mar 06 2017 *)
  • PARI
    a(n) = prod(k=4, n, prime(k) - 6); \\ Michel Marcus, Mar 06 2017

A059862 a(n) = Product_{i=3..n} (prime(i) - 3).

Original entry on oeis.org

1, 1, 2, 8, 64, 640, 8960, 143360, 2867200, 74547200, 2087321600, 70968934400, 2696819507200, 107872780288000, 4746402332672000, 237320116633600000, 13289926531481600000, 770815738825932800000, 49332207284859699200000, 3354590095370459545600000, 234821306675932168192000000
Offset: 1

Views

Author

Labos Elemer, Feb 28 2001

Keywords

Examples

			For n = 6, a(6) = 640 because:
prime(1..6)-3 = (-1,0,2,4,8,10) -> (1,1,2,4,8,10)
and
1*1*2*4*8*10 = 640. [Example generalized and reformatted per observation of _Jon E. Schoenfield_ by _Harlan J. Brothers_, Jul 15 2018]
		

References

  • 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.

Crossrefs

Programs

  • Maple
    a:= proc(n) option remember;
          `if`(n<3, 1, a(n-1)*(ithprime(n)-3))
        end:
    seq(a(n), n=1..21);  # Alois P. Heinz, Nov 19 2021
  • Mathematica
    Join[{1, 1}, Table[Product[Prime[i] - 3, {i, 3, n}], {n, 3, 19}]] (* Harlan J. Brothers, Jul 02 2018 *)
    a[1] = 1; a[2] = 1; a[n_] := a[n] = a[n - 1] (Prime[n] - 3);
    Table[a[n], {n, 19}] (* Harlan J. Brothers, Jul 02 2018 *)
  • PARI
    a(n) = prod(i=3, n, prime(i) - 3); \\ Michel Marcus, Jul 15 2018

Formula

a(1) = a(2) = 1; a(n) = a(n-1) * (prime(n) - 3) for n >= 3. - David A. Corneth, Jul 15 2018

Extensions

Name clarified, offset corrected by David A. Corneth, Jul 15 2018

A271564 Number of 6's found in the first differences of a reduced residue system modulo a primorial p#.

Original entry on oeis.org

0, 0, 2, 14, 142, 1690, 26630, 470630, 10169950, 280323050, 8278462850, 293920842950, 11604850743850, 481192519512250, 21869408938627250, 1124832660535333750, 64590101883781223750, 3837395864206055401250, 250972362651045466681250, 17415757437491856599406250, 1243227958252662737649043750
Offset: 1

Views

Author

Logan W. Wilbur, Apr 09 2016

Keywords

Comments

Technically, the formula is undefined modulo 2# or 3#, but their values are listed as "0", since there are no 6's in the first differences of their reduced residue systems. For our purposes, by "6's", we mean n such that n,n+6 are relatively prime to the primorial modulus, while n+1,n+2,n+3,n+4,n+5 all share a factor (or factors) with p#. The values of this sequence are tied to actual distribution of sexy primes over N (conjecture).

Examples

			Modulo 5# (=30), there are (2*(5-2)-2*(5-3))=2 occurrences where n,n+6 are relatively prime, but n+1,n+2,n+3,n+4,n+5 share a factor with 30; they are n=1,n=23(mod30). Modulo 7# (=210), there are (2*(7-2)*(5-2)-2*(7-3)*(5-3))=30-16=14 such occurrences.
		

Crossrefs

Cf. A059861 (d=2,4 values), A049296, A271565.

Programs

  • Mathematica
    Table[2 Product[Prime@ k - 2, {k, 3, n}] - 2 Product[Prime@ k - 3, {k, 3, n}], {n, 21}] (* Michael De Vlieger, Apr 11 2016 *)
  • PARI
    a(n) = 2*prod(k=3, n, prime(k)-2) - 2*prod(k=3, n, prime(k)-3); \\ Michel Marcus, Apr 10 2016

Formula

a(n) = 2*product(p-2)-2*product(p-3), where p runs over the primes greater than 3.

Extensions

Corrected and extended by Michel Marcus, Apr 10 2016

A271565 Number of 8's found in the first differences of a reduced residue system modulo a primorial p#.

Original entry on oeis.org

0, 0, 0, 2, 28, 394, 6812, 128810, 2918020, 83120450, 2524575200, 91589444450, 3682730287600, 155231331960250, 7156139793803000, 372520258834974250, 21613446896458917500, 1296556574981939521250, 85520460088068245240000, 5980843188551617897761250, 430093937447553491544932500
Offset: 1

Views

Author

Logan W. Wilbur, Apr 10 2016

Keywords

Comments

Technically, the formula is undefined modulo 2# or 3#, but I have listed their values as "0", since there are no 8's in the first differences of their reduced residue systems. For our purposes, by "8's", we mean n such that n,n+8 are relatively prime to the primorial modulus, while n+1,n+2,n+3,n+4,n+5,n+6,n+7 all share a factor (or factors) with p#.

Examples

			Modulo 5# (=30), there are (5-2)-2*(5-3)+(5-4)=0 occurrences where n, n+8 are relatively prime but n+1, n+2, n+3, n+4, n+5, n+6, n+7 share a factor with 30.
Modulo 7# (=210), there are (7-2)(5-2)-2*(7-3)(5-3)+(7-4)(5-4)=15-16+3=2 such occurrences; i.e when n=89,113 (mod210).
		

Crossrefs

Programs

  • Mathematica
    Table[Product[Prime@ k - 2, {k, 3, n}] - 2 Product[Prime@ k - 3, {k, 3, n}] + Product[Prime@ k - 4, {k, 3, n}], {n, 21}] (* Michael De Vlieger, Apr 11 2016 *)
  • PARI
    a(n) = prod(k=3, n, prime(k)-2) - 2*prod(k=3, n, prime(k)-3) + prod(k=3, n, prime(k)-4); \\ Michel Marcus, Apr 11 2016

Formula

a(n) = product(p-2) - 2*product(p-3) + product(p-4), where p runs through the primes > 3 and <= prime(n).

Extensions

More terms from Michel Marcus, Apr 11 2016

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

Views

Author

Labos Elemer, Feb 28 2001

Keywords

References

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

Crossrefs

Programs

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

Extensions

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

Views

Author

Labos Elemer, Feb 28 2001

Keywords

Comments

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

References

  • 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

Crossrefs

Programs

  • Magma
    [n le 3 select 1 else (&*[NthPrime(j) -5: j in [4..n]]): n in [1..30]]; // G. C. Greubel, Feb 02 2023
    
  • Mathematica
    Join[{1,1,1},FoldList[Times,Prime[Range[4,20]]-5]] (* Harvey P. Dale, Dec 29 2018 *)
  • PARI
    a(n) = prod(k=4, n, prime(k)-5); \\ Michel Marcus, Dec 12 2017
    
  • SageMath
    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

A236186 Differences between terms of compacting Eratosthenes sieve for prime(5) = 11.

Original entry on oeis.org

2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 8, 6, 4, 6, 2, 4, 6, 2, 6, 6, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 10, 2, 10, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 8, 6, 4, 6, 2, 4, 6, 2, 6, 6, 4, 2, 4
Offset: 1

Views

Author

Christopher J. Hanson, Jan 21 2014

Keywords

Comments

P(x) is a function which represents a prime number at a particular ordinal x. This pattern, dp(x), describes the difference between consecutive prime numbers as described by p(x) (see A236175) and therefore the length of dp(x) is len(p(x)) - 1 and each value in dp(x) times P(x) is the difference between values determined not primed when running one pass of a reductive sieve, starting at P(x)^2. See A236185.

Crossrefs

Essentially the same as A049296.

Programs

  • PARI
    {a(n) = my(A); if( n<1, 0, A = vector( n*50 + 148, k, k+1); for( i = 1, 4, A = select( k -> k%prime(i), A) ); polcoeff( (1 - x) * Ser( select( k -> k>11 && (k%11) == 0, A) / 11), n))}; /* Michael Somos, Mar 10 2014 */

Formula

a(n + 48) = a(n). - Michael Somos, Mar 10 2014

Extensions

Made sequence periodic. - Michael Somos, Mar 10 2014
Showing 1-9 of 9 results.