A249573 -id:A249573 - cp's OEIS frontend

A280720 For p = prime(n), number of iterations of the function f(x) = 5x + 2 that leave p prime.

0, 1, 0, 1, 0, 2, 0, 3, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 2, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Felix Fröhlich, Jan 07 2017

nonn

Records are a(1) = 0 [p = 2], a(2) = 1 [p = 3], a(6) = 2 [p = 13], a(8) = 3 [p = 19], a(74) = 4 [p = 373], a(12656) = 6 [p = 135859], a(1165346) = 7 [p = 18235423], a(1659004) = 8 [p = 26588257], a(5386789) = 9 [p = 93112729], .... - Charles R Greathouse IV, Jan 12 2017

John Cerkan, Table of n, a(n) for n = 1..10001

Cf. A016873, A023217, A023252, A023283, A023313, A023341, A084958, A249573.

Table[Length@ NestWhileList[5 # + 2 &, Prime@ n, PrimeQ] - 2, {n, 120}] (* Michael De Vlieger, Jan 09 2017 *)

a016873(n) = 5*n+2
a(n) = my(p=prime(n), i=0); while(1, if(!ispseudoprime(a016873(p)), return(i), p=a016873(p); i++))

A259576 Number of distinct differences in row n of the reciprocity array of 1.

Original entry on oeis.org

1, 2, 1, 2, 3, 4, 3, 4, 3, 6, 3, 6, 3, 6, 5, 6, 3, 8, 3, 8, 5, 6, 3, 10, 5, 6, 5, 10, 3, 10, 3, 8, 5, 6, 7, 14, 3, 6, 5, 12, 3, 12, 3, 10, 11, 6, 3, 14, 5, 10, 5, 10, 3, 12, 9, 12, 5, 6, 3, 18, 3, 6, 11, 10, 9, 14, 3, 10, 5, 16, 3, 18, 3, 6, 9, 10, 7, 14, 3

Offset: 1

Clark Kimberling, Jul 01 2015

The "reciprocity law" that Sum_{k=0..m} [(n*k+x)/m] = Sum_{k=0..n} [(m*k+x)/n] where x is a real number and m and n are positive integers, is proved in Section 3.5 of Concrete Mathematics (see References). See A259572 for a guide to related sequences.

			In the array at A259575, row 6 is (1,3,6,8,11,15,16,18,...), with differences (2,3,2,3,4,1,2,...), and distinct differences {1,2,3,4}, so that a(6) = 4.

R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, Addison-Wesley, 1989, pages 90-94.

Antti Karttunen, Table of n, a(n) for n = 1..5000

Cf. A249572, A249573, A259575.

x = 1;  s[m_, n_] := Sum[Floor[(n*k + x)/m], {k, 0, m - 1}];
t[m_] := Table[s[m, n], {n, 1, 1000}];
u = Table[Length[Union[Differences[t[m]]]], {m, 1, 120}]  (* A259576 *)

A259575sq(m,n) = sum(k=0,m-1,(1+(n*k))\m);
A259576(n) = #Set(vector(n,k,A259575sq(n,1+k)-A259575sq(n,k))); \\ Antti Karttunen, Mar 02 2023

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A280720 For p = prime(n), number of iterations of the function f(x) = 5x + 2 that leave p prime.

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