A184859 -id:A184859 - cp's OEIS frontend

A385538 First differences of A184859.

1, 2, 6, 2, 6, 4, 6, 2, 6, 10, 6, 8, 10, 2, 6, 4, 6, 8, 10, 6, 18, 8, 10, 8, 6, 4, 6, 8, 10, 2, 4, 2, 24, 4, 6, 6, 2, 10, 6, 12, 8, 6, 10, 14, 4, 6, 20, 12, 4, 6, 8, 12, 4, 18, 8, 10, 2, 22, 18, 2, 16, 8, 16, 18, 2, 24, 10, 6, 8, 16, 12, 8, 6, 18, 10, 6, 12, 14

Offset: 1

Guido Heinig, Jul 02 2025

Cf. A001622, A007067, A184859.

Differences[Select[Floor[Range[420]*GoldenRatio + 1/2], PrimeQ]] (* Amiram Eldar, Jul 10 2025 *)

lista(nn) = my(v=select(isprime,vector(nn, k, round(k*quadgen(5))))); vector(#v-1, k, v[k+1]-v[k]) \\ Michel Marcus, Jul 08 2025

a(n) = A184859(n+1) - A184859(n).

A184861 Numbers m such that prime(m) is of the form floor(nr+h), where r=(1+sqrt(5))/2 and h=1/2; complement of A184864.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 9, 10, 11, 12, 15, 16, 18, 20, 21, 22, 23, 24, 25, 28, 30, 32, 34, 35, 37, 38, 39, 40, 42, 43, 44, 45, 46, 48, 49, 51, 52, 53, 54, 55, 57, 59, 61, 62, 63, 64, 66, 68, 70, 71, 72, 73, 75, 76, 79, 80, 81, 82, 86, 89, 90, 92, 93, 96, 98, 99, 101, 102, 103, 105, 107, 109, 111, 112, 115, 116, 118, 120, 122, 124, 125, 126, 127, 130, 131, 132, 133, 134, 136, 137, 140, 141, 144, 147, 149, 151, 153, 154, 156, 157, 158, 159, 161, 162, 164

Offset: 1

Clark Kimberling, Jan 23 2011

nonn

			See A184859.

Cf. A184859, A184864.

r=(1+5^(1/2))/2; h=1/2; s=r/(r-1);
a[n_]:=Floor [n*r+h];
Table[a[n], {n, 1, 120}]  (* A007067 *)
t1={}; Do[If[PrimeQ[a[n]], AppendTo[t1, a[n]]], {n, 1, 600}]; t1
t2={}; Do[If[PrimeQ[a[n]], AppendTo[t2, n]], {n, 1, 600}]; t2
t3={}; Do[If[MemberQ[t1, Prime[n]], AppendTo[t3, n]], {n, 1, 300}]; t3
(* Lists t1, t2, t3 match A184859, A184860, A184861. *)

A184860 Numbers k such that floor(nr+h) is prime, where r=(1+sqrt(5))/2 and h=1/2.

Original entry on oeis.org

1, 2, 3, 7, 8, 12, 14, 18, 19, 23, 29, 33, 38, 44, 45, 49, 51, 55, 60, 66, 70, 81, 86, 92, 97, 101, 103, 107, 112, 118, 119, 122, 123, 138, 140, 144, 148, 149, 155, 159, 166, 171, 175, 181, 190, 192, 196, 208, 216, 218, 222, 227, 234, 237, 248, 253, 259, 260, 274, 285, 286, 296, 301, 311, 322, 323, 338, 344, 348, 353, 363, 370, 375, 379, 390, 396, 400, 407, 416, 422, 427, 433, 438, 453, 457, 459, 464, 468, 475, 478, 500, 501, 511, 527, 531, 542, 546, 548, 563, 568, 574, 579, 585, 589, 600

Offset: 1

Clark Kimberling, Jan 23 2011

nonn

			See A184859.

Cf. A184859, A184861.

r=(1+5^(1/2))/2; h=1/2; s=r/(r-1);
a[n_]:=Floor [n*r+h];
Table[a[n], {n, 1, 120}]  (* A007067 *)
t1={}; Do[If[PrimeQ[a[n]], AppendTo[t1, a[n]]], {n, 1, 600}]; t1
t2={}; Do[If[PrimeQ[a[n]], AppendTo[t2, n]], {n, 1, 600}]; t2
t3={}; Do[If[MemberQ[t1, Prime[n]], AppendTo[t3, n]], {n, 1, 300}]; t3
(* Lists t1, t2, t3 match A184859, A184860, A184861. *)

A184862 Primes of the form floor(n+nr-r/2), where r=(1+sqrt(5))/2.

Original entry on oeis.org

7, 17, 41, 43, 59, 67, 101, 103, 109, 127, 137, 151, 179, 211, 229, 263, 271, 281, 313, 331, 347, 373, 389, 397, 431, 433, 439, 449, 457, 467, 491, 499, 509, 541, 569, 577, 593, 601, 617, 619, 643, 653, 661, 677, 719, 727, 761, 787, 797, 821, 823, 829, 839, 857, 863, 881, 907, 941, 967, 983, 991, 1009, 1033, 1049, 1051, 1069, 1093, 1109, 1117, 1151, 1153, 1187, 1193, 1213, 1229, 1237, 1279, 1289, 1297, 1321, 1373, 1381, 1399, 1423, 1433, 1439, 1483, 1499, 1543, 1549, 1559, 1567

Offset: 1

Clark Kimberling, Jan 23 2011

nonn

See "conjecture generalized" at A184774.

Cf. A007064, A184774, A184859, A184863, A184864.

r=(1+5^(1/2))/2;
a[n_]:=Floor [n+n*r-r/2];
Table[a[n],{n,1,120}]  (* A007064 *)
t1={}; Do[If[PrimeQ[a[n]], AppendTo[t1,a[n]]],{n,1,600}];t1
t2={}; Do[If[PrimeQ[a[n]], AppendTo[t2,n]],{n,1,600}];t2
t3={}; Do[If[MemberQ[t1,Prime[n]],AppendTo[t3,n]],{n,1,300}];t3
*( Lists t1, t2, t3 match A184862, A184863, A184864.)
With[{gr=GoldenRatio},Select[Table[Floor[n+n*gr-gr/2],{n,2000}],PrimeQ]] (* Harvey P. Dale, Sep 18 2024 *)

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A385538 First differences of A184859.

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A184861 Numbers m such that prime(m) is of the form floor(nr+h), where r=(1+sqrt(5))/2 and h=1/2; complement of A184864.

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A184860 Numbers k such that floor(nr+h) is prime, where r=(1+sqrt(5))/2 and h=1/2.

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A184862 Primes of the form floor(n+nr-r/2), where r=(1+sqrt(5))/2.

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