A216975 -id:A216975 - cp's OEIS frontend

A272083 Irregular triangle read by rows: strictly decreasing positive integer sequences in lexicographic order with the property that the sum of inverses equals one.

1, 6, 3, 2, 12, 6, 4, 2, 15, 10, 3, 2, 15, 12, 10, 4, 2, 15, 12, 10, 6, 4, 3, 18, 9, 3, 2, 18, 12, 9, 4, 2, 18, 12, 9, 6, 4, 3, 18, 15, 10, 9, 6, 2, 18, 15, 12, 10, 9, 4, 3, 20, 5, 4, 2, 20, 6, 5, 4, 3, 20, 12, 6, 5, 2, 20, 15, 10, 5, 4, 3, 20, 15, 12, 10, 5

Offset: 1

Peter Kagey, Apr 19 2016

			First six rows:
[1]                   because 1/1 = 1.
[6, 3, 2]             because 1/6 + 1/3 + 1/2 = 1.
[12, 6, 4, 2]         because 1/12 + 1/6 + 1/4 + 1/2 = 1.
[15, 10, 3, 2]        because 1/15 + 1/10 + 1/3 + 1/2 = 1.
[15, 12, 10, 4, 2]    because 1/15 + 1/12 + 1/10 + 1/4 + 1/2 = 1.
[15, 12, 10, 6, 4, 3] because 1/15 + 1/12 + 1/10 + 1/6 + 1/4 + 1/3 = 1.

Peter Kagey, Table of n, a(n) for n = 1..1578 (All rows with first term less than or equal to 30.)
Wikipedia, Erdős-Graham problem.

Cf. A073546, A216975, A216993, A272020, A272036, A272081, A272082.

A216993 Triangle read by rows in which row n gives the lexicographically earliest denominators with the least possible maximum value among all n-term Egyptian fractions with unit sum.

Original entry on oeis.org

1, 0, 0, 2, 3, 6, 2, 4, 6, 12, 2, 4, 10, 12, 15, 3, 4, 6, 10, 12, 15, 3, 4, 9, 10, 12, 15, 18, 3, 5, 9, 10, 12, 15, 18, 20, 4, 5, 8, 9, 10, 15, 18, 20, 24, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 5, 6, 8, 9, 10, 15, 18, 20, 21, 24, 28, 4, 8, 9, 10, 12, 15, 18, 20, 21, 24, 28, 30, 4, 8, 9, 11, 12, 18, 20, 21, 22, 24, 28, 30, 33

Offset: 1

Robert Price, Sep 21 2012

This sequence is the lexicographically earliest Egyptian fraction (denominators only) describing the minimum largest denominator given in A030659.

Row 2 = [0,0] corresponds to the fact that 1 cannot be written as an Egyptian fraction with 2 (distinct) terms.

			Row 5 = [2,4,10,12,15]: lexicographically earliest denominators with the least possible maximum value (15) among 72 possible 5-term Egyptian fractions equal to 1. 1 = 1/2 + 1/4 + 1/10 + 1/12 + 1/15.
Triangle begins:
  1;
  0, 0;
  2, 3,  6;
  2, 4,  6, 12;
  2, 4, 10, 12, 15;
  3, 4,  6, 10, 12, 15;

Robert Price, Rows n = 1..24, flattened
Mohammad K. Azarian, Sylvester's Sequence and the Infinite Egyptian Fraction Decomposition of 1, Problem 958, College Mathematics Journal, Vol. 42, No. 4, September 2011, p. 330.
Mohammad K. Azarian, Sylvester's Sequence and the Infinite Egyptian Fraction Decomposition of 1, Solution College Mathematics Journal, Vol. 43, No. 4, September 2012, pp. 340-342.
Harry Ruderman and Paul Erdős, Problem E2427: Bounds for Egyptian fraction partitions of unity (comments), Amer. Math. Monthly, 1974 (Vol. 81), pp. 780-782.
Eric Weisstein's World of Mathematics, Egyptian Fraction
Wikipedia, Egyptian fraction
Index entries for sequences related to Egyptian fractions

Cf. A030659, A073546, A213062, A216975, A378723.

A216979 Primes of the form n^6+2.

Original entry on oeis.org

2, 3, 3518743763, 17596287803, 282429536483, 54980371265627, 93385106978411, 110322650964683, 151939915084883, 1363532208525371, 1870004703089603, 3684302682180851, 5257948522194371, 15813440003753003, 22416464978706683, 33227552537453171, 80425212553252451

Offset: 1

Michel Lagneau, Sep 21 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A056899, A144953, A216975, A216977.

[a: n in [0..700] | IsPrime(a) where a is n^6 + 2 ]; // Vincenzo Librandi, Oct 12 2012

lst={}; Do[p=n^6+2; If[PrimeQ[p], AppendTo[lst, p]], {n, 6!}]; lst
Select[Table[n^6 + 2, {n, 0, 700}], PrimeQ] (* Vincenzo Librandi, Oct 12 2012 *)

v=select(n->isprime(n^6+2),vector(2000,n,n-1)); /* A216978 */
vector(#v, n, v[n]^6+2)
/* Joerg Arndt, Sep 21 2012 */

A216981 Primes of the form n^7+2.

Original entry on oeis.org

2, 3, 4782971, 1801088543, 1174711139839, 3938980639169, 93206534790701, 425927596977749, 1107984764452583, 2149422977421877, 7416552901015627, 19891027786401119, 307732862434921877, 830512886046548069, 1042842864990234377, 3678954248903875651

Offset: 1

Michel Lagneau, Sep 21 2012

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A056899, A144953, A216975, A216977, A216979.

[a: n in [0..500] | IsPrime(a) where a is n^7+2]; // Vincenzo Librandi, Mar 15 2013

lst={}; Do[p=n^7+2; If[PrimeQ[p], AppendTo[lst, p]], {n, 6!}]; lst
Select[Table[n^7 + 2, {n, 0, 400}], PrimeQ] (* Vincenzo Librandi, Mar 15 2013 *)

v=select(n->isprime(n^7+2),vector(2000,n,n-1)); /* A216980 */
vector(#v, n, v[n]^7+2)
/* Joerg Arndt, Sep 21 2012 */

select(isprime, vector(2000,n,(n-1)^7+2)) \\ Charles R Greathouse IV, Sep 21 2012

cp's OEIS Frontend

A272083 Irregular triangle read by rows: strictly decreasing positive integer sequences in lexicographic order with the property that the sum of inverses equals one.

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A216993 Triangle read by rows in which row n gives the lexicographically earliest denominators with the least possible maximum value among all n-term Egyptian fractions with unit sum.

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A216979 Primes of the form n^6+2.

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Magma

Mathematica

PARI

A216981 Primes of the form n^7+2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI