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 1-4 of 4 results.

A209396 Each entry is the first of three consecutive primes with equal digital sum.

Original entry on oeis.org

22193, 25373, 69539, 107509, 111373, 167917, 200807, 202291, 208591, 217253, 221873, 236573, 238573, 250073, 250307, 274591, 290539, 355573, 373073, 382373, 404273, 407083, 415391, 417383, 439009, 441193, 447907, 515173, 542837, 581873, 582083, 591673
Offset: 1

Views

Author

Antonio Roldán, Mar 13 2012

Keywords

Comments

Subsequence of A066540.
The differences between the three primes of the triple are multiples of 18.

Examples

			200807 is in the sequence because 200807, 200843, 200861 are consecutive primes and sum_of_digits(200807)= sum_of_digits(200843)= sum_of_digits(200861)=17

Links

Crossrefs

Cf. A066540, A209663, A210629.

Programs

  • Mathematica
    Select[Prime[Range[100000]], Total[IntegerDigits[#]] == Total[IntegerDigits[NextPrime[#, 1]]] == Total[IntegerDigits[NextPrime[#, 2]]] &] (* T. D. Noe, Mar 13 2012 *)
Transpose[Select[Partition[Prime[Range[50000]],3,1],Differences[ Total/@ (IntegerDigits/@#)]=={0,0}&]][[1]] (* Harvey P. Dale, Jul 22 2016 *)

A210629 Each entry is the first of four consecutive primes with equal digital sum.

Original entry on oeis.org

1442173, 2288509, 2660183, 2805773, 3830891, 4137473, 4951073, 5216137, 5517173, 5521819, 5521891, 5914591, 6474119, 6518173, 7118519, 7570273, 8508473, 8584273, 8689573, 8912591, 9383053, 9958519, 10116373, 10204391, 11418193, 11878873, 11890873, 12948773, 13738163, 13873073, 14377157, 14436391, 14677573, 14732191
Offset: 1

Views

Author

Harvey P. Dale, Mar 25 2012

Keywords

Comments

The differences between each of the 4-consecutive primes are multiples of 18. - Harvey P. Dale, Jul 22 2016

Examples

			2288509 is in the sequence because 2288509, 2288527, 2288563, and 2288581 are consecutive primes and the sum of the digits of each = 34

Crossrefs

Cf. A066540, A209663, A209396.

Programs

  • Mathematica
    Transpose[Select[Partition[Prime[Range[1000000]],4,1],Total[ IntegerDigits[#[[1]]]]==Total[IntegerDigits[#[[2]]]] == Total[ IntegerDigits[#[[3]]]]==Total[IntegerDigits[#[[4]]]]&]][[1]]
Transpose[Select[Partition[Prime[Range[10^6]],4,1], Differences[ Total/@ (IntegerDigits/@#)]=={0,0,0}&]][[1]] (* Harvey P. Dale, Jul 22 2016 *)

A209764 Triangle of coefficients of polynomials v(n,x) jointly generated with A209763; see the Formula section.

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 4, 8, 14, 8, 5, 14, 32, 34, 16, 6, 22, 62, 96, 86, 32, 7, 32, 108, 218, 280, 202, 64, 8, 44, 174, 432, 718, 760, 470, 128, 9, 58, 264, 778, 1584, 2194, 1992, 1066, 256, 10, 74, 382, 1304, 3142, 5360, 6382, 5048, 2390, 512, 11, 92, 532
Offset: 1

Views

Author

Clark Kimberling, Mar 14 2012

Keywords

Comments

Row n begins with n and ends with 2^(n-1).
Row sums: 1,4,11,34,101,304,911,2734,... A060925.
Alternating row sums: 1,0,3,2,5,4,7,6,... A060925.
For a discussion and guide to related arrays, see A208510.

Examples

			First five rows:
1
2...2
3...4....4
4...8....14...8
5...14...32...34...16
First three polynomials v(n,x): 1, 2 + 2x , 3 + 4x + 4x^2.

Crossrefs

Cf. A209663, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A209763 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209764 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A081250 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A060925 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A033999 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A004442*)

Formula

u(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x),
v(n,x)=2x*u(n-1,x)+v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A209875 Primes p such that p and p+18 are consecutive primes with equal digital sum.

Original entry on oeis.org

523, 1069, 1259, 1759, 1913, 2503, 3803, 4159, 4373, 4423, 4463, 4603, 4703, 4733, 5059, 5209, 6229, 6529, 6619, 7159, 7433, 7459, 8191, 9109, 9749, 9949, 10691, 10753, 12619, 12763, 12923, 13763, 14033, 14303, 14369, 15859, 15973, 16529, 16673, 16903, 17239, 17359
Offset: 1

Views

Author

Antonio Roldán, Mar 14 2012

Keywords

Comments

Subsequence of A066540 and A209663 (A066540 contains some consecutive primes with differences greater than 18; A209663 allows nonconsecutive primes).

Examples

			19013 is in the sequence because 19013 is prime, 19013 + 18 = 19031 is the next prime, and sum_of_digits(19013) = sum_of_digits(19031) = 14.

Links

Crossrefs

Cf. A066540, A209663.

Programs

  • PARI
    {forprime(n=3, 20000, my(m=nextprime(n+1)); if(m-n==18 && sumdigits(n) == sumdigits(m), print1(n, ", ")))} \\ Antonio Roldán, Dec 21 2012

Extensions

"Correction" of early 2012 undone by R. J. Mathar, Feb 20 2023
Showing 1-4 of 4 results.

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