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.

A077318 Sum of terms in n-th row of A077316.

Original entry on oeis.org

2, 8, 39, 64, 215, 150, 791, 760, 1179, 970, 3619, 1668, 6227, 3360, 5595, 6144, 13821, 5436, 24985, 10280, 17115, 14850, 35765, 18600, 42425, 26390, 43389, 34580, 92481, 24180, 115909, 57440, 79431, 61200, 115605, 55332, 197913, 106020
Offset: 1

Views

Author

Amarnath Murthy, Nov 04 2002

Keywords

Comments

By definition a(n) == 0 (mod n).
It appears that a(n) is also the smallest sum of n distinct primes of the form n*k+1. - Omar E. Pol, Sep 03 2011

Crossrefs

Cf. A034694, A077316, A077317, A077319.

Extensions

Edited, corrected and extended by Franklin T. Adams-Watters, Aug 29 2006

A077319 Average of terms in n-th row of A077316.

Original entry on oeis.org

2, 4, 13, 16, 43, 25, 113, 95, 131, 97, 329, 139, 479, 240, 373, 384, 813, 302, 1315, 514, 815, 675, 1555, 775, 1697, 1015, 1607, 1235, 3189, 806, 3739, 1795, 2407, 1800, 3303, 1537, 5349, 2790, 3787, 2652, 6705, 1869, 7255, 3645, 4663, 3504, 8411, 3097
Offset: 1

Views

Author

Amarnath Murthy, Nov 04 2002

Keywords

Crossrefs

Cf. A034694, A077316, A077317, A077318.

Formula

a(n) = A077318(n)/n.

Extensions

Edited, corrected and extended by Franklin T. Adams-Watters, Aug 29 2006

A077317 a(n) is the n-th prime == 1 (mod n).

Original entry on oeis.org

2, 5, 19, 29, 71, 43, 211, 193, 271, 191, 661, 277, 937, 463, 691, 769, 1531, 613, 2357, 1021, 1723, 1409, 3313, 1609, 3701, 2029, 3187, 2437, 6961, 1741, 7193, 3617, 4951, 3877, 7001, 3169, 10657, 6271, 7879, 5521, 13613, 3823, 15137, 7349, 9091, 7499
Offset: 1

Views

Author

Amarnath Murthy, Nov 04 2002, Jul 23 2005 (submitted incorrectly on two separate occasions; each time it has had to be corrected)

Keywords

Comments

a(n) is the n-th prime in the arithmetic progression with first term 1 and common difference n.
Main diagonal of A077316.

Examples

			a(5) = 71, the fifth prime in the sequence 1, 6, 11, 16, 21, 26, 31, 36, 41, 46, 51, 56, 61, 66, 71, ...

Links

Crossrefs

Cf. A034694, A077316, A077318, A077319, A093871.

Programs

  • Maple
    a:=proc(n) local N, B, j: N:=[seq(n*x+1,x=0..500)]: B:={}: for j from 1 to nops(N) do if isprime(N[j])=true then B:=B union {N[j]} else fi od: B[n]: end: seq(a(n),n=1..55); # Emeric Deutsch, Sep 03 2005
  • Mathematica
    Module[{nn=50,prs=Prime[Range[3000]]},Join[{2},Table[Select[prs,Mod[#,n] == 1&][[n]],{n,2,nn}]]] (* Harvey P. Dale, Nov 13 2020 *)

Extensions

Edited, corrected and extended by Emeric Deutsch, Sep 03 2005 and by Franklin T. Adams-Watters, Aug 29 2006
Edited by N. J. A. Sloane, Aug 23 2008 at the suggestion of R. J. Mathar

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

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

A093870 Triangle read by rows: T(n,k) is the k-th prime = -1 (mod n), 1 <= k <= n.

Original entry on oeis.org

2, 3, 5, 2, 5, 11, 3, 7, 11, 19, 19, 29, 59, 79, 89, 5, 11, 17, 23, 29, 41, 13, 41, 83, 97, 139, 167, 181, 7, 23, 31, 47, 71, 79, 103, 127, 17, 53, 71, 89, 107, 179, 197, 233, 251, 19, 29, 59, 79, 89, 109, 139, 149, 179, 199, 43, 109, 131, 197, 241, 263, 307, 373, 439, 461, 571
Offset: 1

Views

Author

Amarnath Murthy, Apr 20 2004

Keywords

Examples

			2;
3,5;
2,5,11;
3,7,11,19;
...

Crossrefs

Cf. A077316, A038700 (first column), A093871 (main diagonal).

Programs

  • Mathematica
    row[n_] := Reap[Module[{k, p}, For[k = 0; p = n - 1, k < n, p += n, If[PrimeQ[p], k++; Sow[p]]]]][[2, 1]];
Array[row, 12] // Flatten (* Jean-François Alcover, Jun 08 2020 *)

Extensions

Edited and extended by Franklin T. Adams-Watters, Aug 29 2006
Definition clarified by Zak Seidov, Apr 16 2013
Duplicate term 139 removed by Jean-François Alcover, Jun 08 2020
Showing 1-6 of 6 results.

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