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 1-6 of 6 results.

A193869 Smallest product of n distinct primes of the form n*k + 1.

Original entry on oeis.org

2, 15, 1729, 32045, 60551711, 85276009, 52814801041129, 1312422595226609, 1130308388231798179, 4182230628909121261, 100166053986652515419641469, 1898732717895963155960377, 1011844196551535741726366525322443
Offset: 1

Views

Author

Omar E. Pol, Sep 01 2011

Keywords

Comments

Also the row products of triangle A077316.
Note that a(3) = 1729 is known as the Hardy-Ramanujan number.

Examples

			a(1) = 2
a(2) = 3*5 = 15
a(3) = 7*13*19 = 1729
a(4) = 5*13*17*29 = 32045
a(5) = 11*31*41*61*71 = 60551711
a(6) = 7*13*19*31*37*43 = 85276009

Links

Crossrefs

Cf. A001235, A077316, A092609, A121940, A154729.

Programs

  • Maple
    Tj := proc(n,k) option remember: local j,p: if(k=0)then return 0:fi: for j from procname(n,k-1)+1 do if(isprime(n*j+1))then return j: fi: od: end: A193869 := proc(n) return mul(n*Tj(n,k)+1,k=1..n): end: seq(A193869(n),n=1..15); # Nathaniel Johnston, Sep 02 2011

Extensions

a(7)-a(14) from Nathaniel Johnston, Sep 02 2011

A154728 Products of three consecutive primes of the form 6n+1 (see A002476).

Original entry on oeis.org

1729, 7657, 21793, 49321, 97051, 175741, 298351, 386389, 559399, 789289, 1089019, 1425829, 1924177, 2665603, 3295273, 3864241, 4631971, 5694079, 6951667, 8103877, 9363547, 10775137, 12307147, 14956219, 18091147, 21243961, 24066037
Offset: 1

Views

Author

Omar E. Pol, Jan 18 2009, Jan 21 2009

Keywords

Comments

Note that a(1)=1729 is the Hardy-Ramanujan number (see taxicab numbers in A001235, A011541).

Examples

			13, 19, 31 are three consecutive primes of the form 6n+1 and 13*19*31 = 7657. - _Emeric Deutsch_, Jan 21 2009

Links

Crossrefs

Cf. A001235, A002476, A011541, A154716, A154717, A154729.

Programs

  • Maple
    a := proc (n) if `mod`(ithprime(n), 6) = 1 then ithprime(n) else end if end proc: A := [seq(a(n), n = 1 .. 100)]: seq(A[j]*A[j+1]*A[j+2], j = 1 .. 30); # Emeric Deutsch, Jan 21 2009
  • Mathematica
    Times@@@Partition[Select[Prime[Range[100]],IntegerQ[(#-1)/6]&],3,1] (* Harvey P. Dale, Jan 13 2019 *)

Extensions

Extended by Emeric Deutsch, Jan 21 2009

A154716 Products of three consecutive happy primes A035497.

Original entry on oeis.org

1729, 5681, 13547, 56327, 237553, 789289, 1089019, 1560553, 2530217, 4480109, 7703209, 12131401, 18417101, 24119467, 30355679, 38022301, 46039783, 53272619, 57627329, 62188859, 79075651, 112140029, 169169677, 226833263, 271152373, 300157327, 325898231
Offset: 1

Views

Author

Omar E. Pol, Jan 18 2009

Keywords

Comments

Note that a(1) = 1729 is the Hardy-Ramanujan number (see taxicab numbers in A001235, A011541).

Crossrefs

Cf. A001235, A011541, A035497, A154717, A154728, A154729.

Programs

  • Mathematica
    happyQ[n_, b_] := NestWhile[Total[IntegerDigits[#, b]^2] &, n, UnsameQ, All] == 1; Times @@@ Partition[Select[Prime[Range[150]], happyQ[#, 10] &], 3, 1] (* Amiram Eldar, Jan 17 2025 *)

Extensions

a(5)-a(27) from Nathaniel Johnston, Apr 30 2011

A154717 Products of three distinct happy primes A035497.

Original entry on oeis.org

1729, 2093, 2821, 3059, 4123, 4991, 5681, 7189, 7657, 8827, 9269, 9373, 9919, 10507, 12649, 12719, 12901, 13547, 13699, 14497, 15197, 15617, 16583, 17143, 17549, 17563, 18487, 19513, 21049, 21749, 22211, 22351, 22379, 23621, 23653, 23933, 23959, 25441
Offset: 1

Views

Author

Omar E. Pol, Jan 18 2009

Keywords

Comments

Note that a(1)=1729 is the Hardy-Ramanujan number (see taxicab numbers in A001235, A011541).

Crossrefs

Cf. A001235, A011541, A035497, A154716, A154728, A154729.

Extensions

a(5) - a(38) from Nathaniel Johnston, Apr 30 2011

A193873 Smallest product of three distinct primes of the form n*k+1.

Original entry on oeis.org

30, 105, 1729, 1105, 13981, 1729, 88537, 50881, 51319, 13981, 137149, 29341, 548497, 88537, 285541, 186337, 3372529, 51319, 18326641, 252601, 1152271, 137149, 1809641, 1366633, 3828001, 548497, 4814857, 645569, 4797703, 285541, 79230049, 4811297
Offset: 1

Views

Author

Omar E. Pol, Sep 02 2011

Keywords

Comments

Note that the Hardy-Ramanujan number is the first and the smallest repeated number: a(3) = a(6) = 1729.

Examples

			a(1) =  2*3*5 = 30
a(2) =  3*5*7 = 105
a(3) =  7*13*19 = 1729
a(4) =  5*13*17 = 1105
a(5) = 11*31*41 = 13981

Links

Crossrefs

Cf. A001235, A077316, A092609, A121940, A154729, A193869.

Programs

  • Mathematica
    a[n_] := Module[{s = {}, c = 0, m = n + 1}, While[c < 3, While[!PrimeQ[m], m += n]; c++; AppendTo[s, m]; m += n]; Times @@ s]; Array[a, 100] (* Amiram Eldar, Jan 17 2025 *)
  • PARI
    a(n)=my(p,q,k=1);while(!isprime(k+=n),);p=k;while(!isprime(k+=n),);q=k;while(!isprime(k+=n),);p*q*k \\ Charles R Greathouse IV, Sep 03 2011

A194263 Numbers that occur more than once in A193873, in order of appearance.

Original entry on oeis.org

1729, 13981, 88537, 51319, 137149, 548497, 285541, 3372529, 18326641, 1152271, 1809641, 3828001, 4814857, 4797703, 79230049, 4413223, 4209661, 19703611, 3882139, 50357677, 70611161, 26536591, 175493677, 85409941, 12932989, 84350561, 192754871, 59756863
Offset: 1

Views

Author

Omar E. Pol, Sep 02 2011

Keywords

Comments

Note that a(1) = 1729 is known as the Hardy-Ramanujan number (see A001235).
Apparently these numbers occur only twice in A193873.
Many of these numbers occur more than twice. For example, 812405017 occurs 3 times, a(67), a(134), and a(268). Robert Price, Mar 17 2020

Examples

			1729 is in the sequence because the Hardy-Ramanujan number is also the smallest product of three distinct primes of the form 3*k+1 and also of the form 6*k+1, so 1729 occurs more than once in A193873.

Links

Crossrefs

Cf. A001235, A154729, A193869, A193873.

Formula

It appears that a(n) = A193873(2*n+1).
This conjecture only holds for n<62: a(62)=A193873(124)=A193873(2n)=620378449
, a(63)=A193873(125)=A193873(2n-1)=424315751
. Robert Price, Mar 17 2020
Showing 1-6 of 6 results.

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