A047253 -id:A047253 - cp's OEIS frontend

A055678 Integers not congruent to 0 (mod 6) that are divisible by the number of their divisors.

1, 2, 8, 9, 40, 56, 80, 88, 104, 128, 136, 152, 184, 225, 232, 248, 296, 328, 344, 376, 424, 441, 448, 472, 488, 536, 560, 568, 584, 625, 632, 640, 664, 712, 776, 808, 824, 856, 872, 880, 896, 904, 1016, 1040, 1048, 1089, 1096, 1112, 1192, 1208, 1250, 1256

Offset: 1

Robert G. Wilson v, Jun 09 2000

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A033950 and A047253.

Do[If[Mod[n,6]!=0,If[IntegerQ[n/DivisorSigma[0,n]],Print[n]]],{n,1500}]

A108612 Beatty-2 (or nested Beatty) sequence for 1/sin(1).

Original entry on oeis.org

1, 4, 9, 16, 25, 42, 56, 72, 90, 110, 143, 168, 195, 224, 255, 304, 340, 378, 418, 460, 504, 572, 621, 672, 725, 780, 864, 924, 986, 1050, 1116, 1216, 1287, 1360, 1435, 1512, 1591, 1710, 1794, 1880, 1968, 2058, 2193, 2288, 2385, 2484, 2585, 2736, 2842, 2950

Offset: 1

Zak Seidov, Jun 13 2005

nonn

Cf. A023800, A031943, A037918, A039215, A043091, A047253, A108120, A108611, A108613.

a(n) = floor(n*floor(n/sin(1))).

A108613 Excess of Beatty-2 function of 1/sin(1) over n^2.

Original entry on oeis.org

0, 0, 0, 0, 0, 6, 7, 8, 9, 10, 22, 24, 26, 28, 30, 48, 51, 54, 57, 60, 63, 88, 92, 96, 100, 104, 135, 140, 145, 150, 155, 192, 198, 204, 210, 216, 222, 266, 273, 280, 287, 294, 344, 352, 360, 368, 376, 432, 441, 450, 459, 468, 477, 540, 550, 560, 570, 580, 649, 660

Offset: 0

Zak Seidov, Jun 13 2005

nonn

Cf. A108612 Beatty-2 (or nested Beatty) function of 1/sin(1).

Cf. A023800, A031943, A037918, A039215, A043091, A047253, A108120, A108611, A108612.

a(n) = A108612[n] - n^2 = floor(n*floor(n/sin(1))) - n^2.

A215898 a(4n) = 1+4n, a(1+4n) = -2-6n, a(2+4n) = 4+6n, a(3+4n) = -3-4n.

Original entry on oeis.org

1, -2, 4, -3, 5, -8, 10, -7, 9, -14, 16, -11, 13, -20, 22, -15, 17, -26, 28, -19, 21, -32, 34, -23, 25, -38, 40, -27, 29, -44, 46, -31, 33, -50, 52, -35, 37, -56, 58, -39, 41, -62, 64, -43, 45, -68, 70, -47, 49, -74, 76, -51, 53, -80, 82, -55, 57, -86, 88, -59

Offset: 0

Paul Curtz, Aug 25 2012

A permutation of A047253, numbers that are not divisible by 6.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1,1,1,1,1).

Cf. A016813, A016933, A016957, A004767.

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x+3*x^2+3*x^4-x^5+x^6)/((1-x)*(1+x+x^2+x^3)^2))); // Bruno Berselli, Sep 06 2012

a[n_ /; Mod[n, 4] == 0] := n+1; a[n_ /; Mod[n, 4] == 1] := -(3n+1)/2; a[n_ /; Mod[n, 4] == 2] := (3n+2)/2; a[n_ /; Mod[n, 4] == 3] := -n; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Sep 03 2012 *)
LinearRecurrence[{-1,-1,-1,1,1,1,1},{1,-2,4,-3,5,-8,10},60] (* Harvey P. Dale, Mar 24 2023 *)

makelist(expand(1+(5-%i^(n*(n+1)))*((2*n+1)*(-1)^n-1)/8), n, 0, 60); /* Bruno Berselli, Sep 07 2012 */

a(n) = 2*a(n-4) - a(n-8).
a(2*n) + a(1+2*n) = -A109613(n)*(-1)^n.
a(3*n) + a(1+3*n) + a(2+3*n) = 3*a(n).
a(4*n) + a(1+4*n) + a(2+4*n) + a(3+4*n) = 0.
a(5*n) + a(1+5*n) + a(2+5*n) + a(3+5*n) + a(4+5*n) = 5*a(n).
From Bruno Berselli, Sep 07 2012: (Start)
G.f.: (1-x+3*x^2+3*x^4-x^5+x^6)/((1-x)*(1+x+x^2+x^3)^2).
a(n) = 1+(5-i^(n*(n+1)))*((2*n+1)*(-1)^n-1)/8, where i=sqrt(-1).
a(2*n) = 1+(5-(-1)^n)*n/2; a(2*n+1) = 1-(5+(-1)^n)*(n+1)/2.
a(n) = a(-n-1) = -a(n-1)-a(n-2)-a(n-3)+a(n-4)+a(n-5)+a(n-6)+a(n-7). (End)

A274677 Numbers k such that 7*10^k + 19 is prime.

Original entry on oeis.org

1, 2, 3, 4, 27, 32, 63, 69, 107, 145, 154, 173, 190, 271, 412, 1219, 1509, 2392, 4444, 5567, 7424, 32174, 51573

Offset: 1

Vincenzo Librandi, Jul 04 2016

No term is divisible by 6 (A047253) because 7*1000000^k + 19 = 7*(76923*13 + 1)^k + 19 is divisible by 13 and is therefore not prime. - Bruno Berselli, Jul 05 2016

			3 is in this sequence because 7*10^3 + 19 = 7019 is prime.
5 is not in the sequence because 7*10^5 + 19 = 79*8861.
Initial terms and associated primes:
a(1) = 1: 89;
a(2) = 2: 719;
a(3) = 3: 7019;
a(4) = 4: 70019, etc.

Makoto Kamada, Search for 70w19.

Subsequence of A047253.

Cf. similar sequences listed in A274676.

[n: n in [1..800] | IsPrime(7*10^n+19)];

Select[Range[0, 3000], PrimeQ[7 10^# + 19] &]

lista(nn) = for(n=1, nn, if(ispseudoprime(7*10^n+19), print1(n, ", "))); \\ Altug Alkan, Jul 05 2016

from sympy import isprime
def afind(limit, startk=0):
    sevenpow10 = 7*10**startk
    for k in range(startk, limit+1):
        if isprime(sevenpow10 + 19):
            print(k, end=", ")
        k += 1
        sevenpow10 *= 10
afind(500) # Michael S. Branicky, Dec 31 2021

a(20)-a(21) from Michael S. Branicky, Dec 31 2021

a(22)-a(23) from Kamada data by Tyler Busby, Apr 14 2024

A323139 Integers that are not multiples of 6 and are the sum of two consecutive primes.

Original entry on oeis.org

5, 8, 52, 68, 100, 112, 128, 152, 172, 268, 308, 320, 340, 352, 410, 434, 472, 508, 520, 532, 548, 668, 712, 740, 752, 772, 872, 892, 946, 1012, 1030, 1064, 1088, 1120, 1132, 1148, 1180, 1192, 1208, 1220, 1250, 1300, 1312, 1334, 1360, 1460, 1472, 1508, 1606, 1888, 1900, 1948, 1960, 1988, 2006, 2032, 2072, 2132, 2156

Offset: 1

Pedro Caceres, Jan 05 2019

nonn

All primes, except 2 and 3, are of the form 6k+1 or 6k-1 for k a positive integer. The converse statement is not true for all k, so the sum of two consecutive primes is not always a multiple of 6. This sequence lists the sums of two consecutive primes that are not multiple of 6.

			52 = 23 + 29 is not a multiple of 6.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001043, A047253, A323138.

p:= 2:
count:= 0: Res:= NULL:
while count < 100 do
  q:= nextprime(p);
  if p+q mod 6 <> 0 then
     count:= count+1; Res:= Res, p+q;
  fi;
  p:= q;
od:
Res; # Robert Israel, Jan 09 2019

Select[Total /@ Partition[Prime@ Range@ 180, 2, 1], Mod[#, 6] != 0 &] (* Michael De Vlieger, Jan 07 2019 *)

my(p=2); forprime(q=3, 1e3, if ((p+q) % 6, print1(p+q", ")); p=q); \\ Michel Marcus, Jan 05 2019

A364902 Let x, y be the greatest exponents of 2, 3 respectively such that 2^x, 3^y do not exceed n and let k_2, k_3 be n - 2^x, and n - 3^y respectively. Then for n such that k_2 = 0 or k_3 = 0, a(n) = n, else a(n) is the least novel number Min{pa(k_2), qa(k_3)}, where p, q are primes not equal to either 2 or 3.

Original entry on oeis.org

1, 2, 3, 4, 5, 10, 15, 8, 9, 7, 14, 20, 25, 35, 50, 16, 11, 22, 21, 28, 55, 70, 75, 40, 45, 49, 27, 13, 26, 33, 44, 32, 17, 34, 39, 52, 65, 98, 100, 56, 63, 77, 80, 121, 110, 105, 140, 112, 143, 154, 147, 196, 245, 135, 91, 130, 165, 220, 160, 85, 170, 195, 260, 64, 19, 38, 51, 68

Offset: 1

David James Sycamore and Michael De Vlieger Sep 21 2023

nonn

Motivated by the recursion D(2) known to reproduce A005940, this sequence uses a compound version based on a squarefree semiprime (6) rather than a prime, in which the terms are generated by a greedy algorithm related to the distances between n and the greatest powers of 2, and 3 not exceeding n. After a(9) = 9 each power of 2 or 3 is followed by the smallest prime not yet in the sequence. (e.g. 11 follows 16, 13 follows 27, etc).
There are no multiples of 6 in this sequence.
For k > 2, if a(i) = prime(k) = p and a(j) = p^2 then j-i is a term in A006899 (e.g. a(17) = 11, a(44) = 121 and 44 - 17 = 27 = 3^3).
Conjectures: (i). This is a permutation of A047253 with primes in order; (ii). All terms between consecutive prime terms, prime(k), prime(k+1) are prime(k)-smooth.

			a(n) = n for n <= 4 because all such n are powers of 2 or 3.
a(5) = least novel Min{a(1)*p,a(2)*q} = Min{p,2*q} for o,q prime != 2 or 3, so a(5) = 5.
17=16+1=9+8, so a(17) = least novel Min{a(1)*p,a(8)*q} = Min{p,8*q} = 11.
Data can be shown in tabular form in two distinct ways: First row starts with 1 and then rows start with a prime; alternatively each row starts with 2^i or 3^j:
 1;                       1;
 2;                       2;
 3,4;                     3;
 5,10,15,8,9;             4,5,10,15;
 7,14,20,25,35,50,16;     8;
 11,22,21,28,55...        9,7,14,20,25,35,50

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16.

Cf. A005940, A006899, A047253, A108906, A356867, A364611, A364628.

nn = 120; c[_] = False; s = {1, 2}; w = Length[s]; t = Prime[s]; flag = 0;
Array[Set[{q[#1], p[#1],
      r[#1]}, {#1, #2,
        Prepend[#2^Range[Floor@Log[#2, nn]], 1]} & @@ {#2,
       Prime[#2]}] & @@ {#, s[[#]]} &, w];
Do[If[n == 1,
   Set[{a[n], c[1]}, {1, True}],
   Array[Set[m[#], 1] &, w];
   Array[Set[j[#], n - p[#]^(-1 + LengthWhile[r[#], # < n + 1 &])] &, w];
   Array[
    If[j[#] == 0,
      k[#] = n; flag = #,
      While[Set[k[#], Prime[m[#]] a[j[#]]];
       Or[MemberQ[s, m[#]], c[k[#]]], m[#]++]] &, w];
   If[flag > 0,
    Set[{a[n], c[k[flag]]}, {k[flag], True}]; flag = 0,
    Set[{a[n], c[#]}, {#, True}] &[Min@ Array[k, w]] ]], {n, nn}];
Array[a, nn] (* Michael De Vlieger, Sep 24 2023 *)

For n > 6, a(A006899(n) + 1) = prime(n-2).

A292163 a(n) is the least prime p such that the orderly concatenation of the n successive powers of p yields a prime number; a(n)=0 if n is a multiple of 6.

Original entry on oeis.org

3, 3, 337, 23, 0, 373, 37, 839, 421, 7, 0, 1447, 2113, 29, 43, 17, 0, 1789, 523, 84737, 7669, 397, 0, 3851, 3583, 99149, 146023, 157, 0, 14173, 38329, 1229, 8017, 1021, 0, 18979, 5437, 17207, 6571, 47, 0, 347, 43669, 25847, 257353, 2887, 0, 33889, 71287

Offset: 2

Michel Marcus, Sep 10 2017

See in the Prime Puzzle link the discussion for when n is a multiple of 6.

			For n=2, the concatenation of 3^0 and 3^1 is 13 which is prime (while 12 was not prime); so a(2) = 3.
For n=3, the concatenation of 3^0, 3^1 and 3^2 is 139 which is prime (while 124 was not prime); so a(3) = 3.

Carlos Rivera, Puzzle 892. Primes as a concatenation of a series of powers of a prime, The Prime Puzzles and Problems Connection.

Cf. A047253.

g:= proc(p,n) local i,t;
  t:= p^(n-1):
  for i from n-2 to 0 by -1 do
    t:= t + 10^(1+ilog10(t))*p^i
  od;
  t
end proc:
f:= proc(n)
  local p;
  if n mod 6 = 0 then return 0 fi;
  p:= 3;
  while not isprime(g(p,n)) do
    p:= nextprime(p);
    if n mod 3 = 0 then while p mod 3 = 1 do p:= nextprime(p) od fi:
  od;
  p
end proc:
map(f, [$2..30]); # Robert Israel, Sep 10 2017

pconc(p, n) = {my(s = "1"); for (k=1, n, s = concat(s, Str(p^k));); eval(s);}
a(n) = {if (!(n % 6), return (0)); n --; my(p = 2); while (! isprime(pconc(p, n)), p = nextprime(p+1)); p;}

a(27)-a(50) from Robert Israel, Sep 10 2017

cp's OEIS Frontend

A055678 Integers not congruent to 0 (mod 6) that are divisible by the number of their divisors.

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A108612 Beatty-2 (or nested Beatty) sequence for 1/sin(1).

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A108613 Excess of Beatty-2 function of 1/sin(1) over n^2.

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A215898 a(4n) = 1+4n, a(1+4n) = -2-6n, a(2+4n) = 4+6n, a(3+4n) = -3-4n.

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A274677 Numbers k such that 7*10^k + 19 is prime.

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A323139 Integers that are not multiples of 6 and are the sum of two consecutive primes.

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A292163 a(n) is the least prime p such that the orderly concatenation of the n successive powers of p yields a prime number; a(n)=0 if n is a multiple of 6.

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