A005017 -id:A005017 - cp's OEIS frontend

A115261 Prime numbers such that the absolute difference of the sum of their digits in odd positions and the sum of their digits in even positions is also a prime.

2, 3, 5, 7, 13, 29, 31, 41, 47, 53, 61, 79, 83, 97, 101, 113, 137, 139, 151, 157, 163, 167, 173, 179, 191, 193, 211, 223, 227, 233, 251, 269, 277, 281, 283, 311, 313, 337, 359, 379, 383, 401, 409, 421, 431, 443, 467, 487, 541, 557, 563, 577, 599, 601, 607, 641

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jan 20 2006

			1237 is in the sequence because it is prime and abs((7+2)-(3+1)) = 5 is prime

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A040997, A005017, A063792, A087593, A042939, A041000, A040164, A115259, A115260.

Df:=proc(N) j:=1; for n from 1 while j<=N do B:= convert(ithprime(n),base,10); ap:=-(sum(B[2*i],i=1..nops(B)/2)-sum(B[2*n+1],i=0..(nops(B)-1)/2)); if (isprime(abs(ap)) = true) then a[j]:=ithprime(n); j:=j+1; fi; od; end:

A004936 Numerator of (binomial(2*n-2,n-1)/n!)^2.

Original entry on oeis.org

1, 1, 1, 25, 49, 49, 121, 20449, 20449, 5909761, 17631601, 17631601, 55190041, 55190041, 55190041, 46414824481, 154341336769, 154341336769, 427538329, 585299972401, 585299972401, 983889253606081, 3438962627443561, 3438962627443561, 7596668444022826249

Offset: 1

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 1..1000
Pavel Valtr, The probability that $n$ random points in a triangle are in convex position, Combinatorica 16 (1996), no. 4, 567-573.
Eric Weisstein's World of Mathematics, Sylvester's Four-Point Problem.

Cf. A000108, A005017 (denominators).

[Numerator((Catalan(n-1)/Factorial(n-1))^2): n in [1..40]]; // G. C. Greubel, Sep 12 2023

Numerator[Table[(Binomial[2n-2,n-1]/n!)^2,{n,30}]] (* Harvey P. Dale, May 30 2012 *)

a(n) = numerator((binomial(2*n-2,n-1)/n!)^2); \\ Michel Marcus, Jul 14 2022

[numerator((catalan_number(n-1)/factorial(n-1))^2) for n in range(1,41)] # G. C. Greubel, Sep 12 2023

a(n) = numerator( (A000108(n-1)/(n-1)!)^2 ). - G. C. Greubel, Sep 12 2023

A115260 Prime numbers in the sequence of the absolute difference of the sum of digits in odd positions and the sum of digits in even positions of prime numbers.

Original entry on oeis.org

2, 3, 5, 7, 2, 7, 2, 3, 3, 2, 5, 2, 5, 2, 2, 3, 5, 7, 3, 3, 2, 2, 3, 3, 7, 5, 2, 3, 7, 2, 2, 5, 2, 5, 3, 3, 5, 7, 7, 5, 2, 5, 13, 3, 2, 3, 5, 3, 2, 7, 2, 5, 5, 7, 13, 3, 5, 2, 2, 7, 13, 3, 2, 3, 5, 17, 7, 13, 5, 3, 7, 17, 13, 7, 3, 7, 7, 2, 3, 5, 5, 2, 2, 7, 3, 3, 7, 2, 3, 7, 2, 3, 7, 2, 5, 5, 3, 2, 7, 3, 5, 7

Offset: 1

Giorgio Balzarotti, Paolo P. Lava, Jan 20 2006

Primes in the sequence A115259.

			a(37) = 3 because 37th prime = 157, (7+1) - 5 = 3, 3 is prime.

Cf. A040997, A005017, A063792, A087593, A042939, A041000, A040164, A115259, A115261.

select(isprime,[seq(abs(sum(convert(ithprime(a),base,10)[2*i],i=1..nops(convert (ithprime(a),base,10))/2)-sum(convert(ithprime(a),base,10)[2*i+1],i=0..(nops (convert(ithprime(a),base,10))-1)/2)),a=1..N)]);

A242780 Decimal expansion of the maximum probability that the convex hull of four points, chosen at random inside a convex planar region, is a quadrilateral (Sylvester's four-point problem).

Original entry on oeis.org

7, 0, 4, 4, 7, 9, 8, 8, 1, 0, 4, 3, 1, 8, 1, 4, 9, 9, 9, 5, 5, 3, 5, 1, 5, 6, 5, 6, 3, 8, 2, 9, 4, 3, 8, 6, 5, 2, 8, 9, 5, 3, 5, 7, 3, 8, 7, 2, 6, 1, 4, 2, 3, 2, 5, 3, 3, 6, 4, 0, 3, 2, 3, 6, 4, 1, 9, 9, 5, 0, 6, 3, 8, 6, 0, 1, 4, 6, 6, 2, 9, 8, 5, 8, 9, 7, 2, 9, 5, 1, 0, 5, 0, 2, 6, 9, 6, 4, 0, 2, 9, 3, 6

Offset: 0

Jean-François Alcover, May 22 2014

It is proved that this maximum probability is achieved when the region is an ellipse (or a disk). [after Steven Finch]

			0.70447988104318149995535...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 8.18, p. 533.

Eric Weisstein's MathWorld, Sylvester's Four-Point Problem.

Cf. A004677, A004824, A004936, A005017, A051050, A051051.

RealDigits[1 - 35/(12*Pi^2), 10, 103] // First

1 - 35/(12*Pi^2) \\ Stefano Spezia, Dec 26 2024

Equals 1 - 35/(12*Pi^2).

cp's OEIS Frontend

A115261 Prime numbers such that the absolute difference of the sum of their digits in odd positions and the sum of their digits in even positions is also a prime.

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A004936 Numerator of (binomial(2*n-2,n-1)/n!)^2.

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A115260 Prime numbers in the sequence of the absolute difference of the sum of digits in odd positions and the sum of digits in even positions of prime numbers.

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Maple

A242780 Decimal expansion of the maximum probability that the convex hull of four points, chosen at random inside a convex planar region, is a quadrilateral (Sylvester's four-point problem).

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