Displaying 1-10 of 84 results found.
Filter sequence for all such sequences b, for which b( A006254(k)) = constant for all k >= 3.
+20
6
1, 2, 3, 4, 5, 4, 4, 6, 4, 4, 7, 4, 8, 9, 4, 4, 10, 11, 4, 12, 4, 4, 13, 4, 14, 15, 4, 16, 17, 4, 4, 18, 19, 4, 20, 4, 4, 21, 22, 4, 23, 4, 24, 25, 4, 26, 27, 28, 4, 29, 4, 4, 30, 4, 4, 31, 4, 32, 33, 34, 35, 36, 37, 4, 38, 4, 39, 40, 4, 4, 41, 42, 43, 44, 4, 4, 45, 46, 4, 47, 48, 4, 49, 4, 50, 51, 4, 52, 53, 4, 4, 54, 55, 56, 57, 4, 4, 58, 4, 4, 59, 60, 61
FORMULA
For n <= 3, a(n) = n, and for n >= 4, a(n) = 4 if 2n-1 is a prime (for all n in A006254[3..] = 4, 6, 7, 9, 10, 12, 15, ...), and for all other n (numbers n such that 2n-1 is composite), a(n) = running count from 5 onward.
PROG
(PARI)
up_to = 1000;
partialsums(f, up_to) = { my(v = vector(up_to), s=0); for(i=1, up_to, s += f(i); v[i] = s); (v); }
v_partsums = partialsums(x -> isprime(x+x-1), up_to);
A305901(n) = if(n<=3, n, if(isprime(n+n-1), 4, 3+n-v_partsums[n]));
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 16, 17, 18, 19, 20, 26, 27, 28, 29, 30, 36, 37, 38, 39, 40, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65
COMMENTS
Name was: Numbers just more than half-prime.
REFERENCES
M. J. Halm, More Sequences, Mpossibilities 83, 2003.
EXAMPLE
a(11) = 16 because [(31)/2 + 1/2]=16.
4, 9, 25, 49, 121, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 16129, 17161, 18769, 19321, 22201, 22801, 24649, 26569, 27889, 29929, 32041, 32761, 36481
COMMENTS
Also 4, together with numbers n such that Sum_{d|n}(-1)^d = - A048272(n) = -3. - Benoit Cloitre, Apr 14 2002 Also, all solutions to the equation sigma(x) + phi(x) = 2x + 1. - Farideh Firoozbakht, Feb 02 2005 Unique numbers having 3 divisors (1, their square root, themselves). - Alexandre Wajnberg, Jan 15 2006 Smallest (or first) new number deleted at the n-th step in an Eratosthenes sieve. - Lekraj Beedassy, Aug 17 2006 Integers having only 1 factor other than 1 and the number itself. Every number in the sequence is a multiple of 1 factor other than 1 and the number itself. 4 : 2 is the only factor other than 1 and 4; 9 : 3 is the only factor other than 1 and 9; and so on. - Rachit Agrawal (rachit_agrawal(AT)daiict.ac.in), Oct 23 2007
The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime. - Omar E. Pol, May 06 2008 There are 2 Abelian groups of order p^2 (C_p^2 and C_p x C_p) and no non-Abelian group. - Franz Vrabec, Sep 11 2008 Also numbers n such that phi(n) = n - sqrt(n). - Michel Lagneau, May 25 2012 Let r(n) = (a(n) - 1)/(a(n) + 1); then Product_{n>=1} r(n) = (3/5) * (4/5) * (12/13) * (24/25) * (60/61) * ... = 2/5. - Dimitris Valianatos, Feb 26 2019
FORMULA
For n > 2: a(n) = 1 (mod 24). - Zak Seidov, Dec 07 2011 Product_{n>=1} a(n)/(a(n)-1) = Pi^2/6. - Daniel Suteu, Feb 06 2017 Product_{n>=1} (1 + 1/a(n)) = zeta(2)/zeta(4) = 15/Pi^2 ( A082020). Product_{n>=1} (1 - 1/a(n)) = 1/zeta(2) = 6/Pi^2 ( A059956). (End)
PROG
(Haskell)
a001248 n = a001248_list !! (n-1)
(SageMath) [n^2 for n in prime_range(1, 301)] # G. C. Greubel, May 02 2024 (Python)
from sympy import prime
1, 2, 3, 5, 6, 8, 9, 11, 14, 15, 18, 20, 21, 23, 26, 29, 30, 33, 35, 36, 39, 41, 44, 48, 50, 51, 53, 54, 56, 63, 65, 68, 69, 74, 75, 78, 81, 83, 86, 89, 90, 95, 96, 98, 99, 105, 111, 113, 114, 116, 119, 120, 125, 128, 131, 134, 135, 138, 140, 141, 146, 153, 155, 156
COMMENTS
Or, numbers k such that 2k+1 is prime.
Also numbers not of the form 2xy + x + y. - Jose Brox (tautocrona(AT)terra.es), Dec 29 2005
This sequence arises if you factor the product of a large number of the first odd numbers into the form 3^n(3)5^n(5)7^n(7)11^n(11)... Then n(3)/n(5) = 2, n(3)/n(7) = 3, n(3)/n(11) = 5, ... . - Andrzej Staruszkiewicz (astar(AT)th.if.uj.edu.pl), May 31 2007
Kohen shows: A king invites n couples to sit around a round table with 2n+1 seats. For each couple, the king decides a prescribed distance d between 1 and n which the two spouses have to be seated from each other (distance d means that they are separated by exactly d-1 chairs). We will show that there is a solution for every choice of the distances if and only if 2n+1 is a prime number [i.e., iff n is in A005097], using a theorem known as Combinatorial Nullstellensatz. - Jonathan Vos Post, Jun 14 2010 Starting from 6, positions at which new primes are seen for Goldbach partitions. E.g., 31 is first seen at 34 from 31+3, so position = 1 + (34-6)/2 = 15. - Bill McEachen, Jul 05 2010 Perfect error-correcting Lee codes of word length n over Z: it is conjectured that these always exist when 2n+1 is a prime, as mentioned in Horak. - Jonathan Vos Post, Sep 19 2011 I conjecture that the set of pairwise sums of terms of this sequence ( A005097) is the set of integers greater than 1, i.e.: 1+1=2, 1+2=3, ..., 5+5=10, ... (This is equivalent to Goldbach's conjecture: every even integer greater than or equal to 6 can be expressed as the sum of two odd primes.) - Lear Young, May 20 2014 See conjecture and comments from Richard R. Forberg, in Links section below, on the relationship of this sequence to rules on values of c that allow both p^q+c and p^q-c to be prime, for an infinite number of primes p. - Richard R. Forberg, Jul 13 2016 The sequence represents the minimum number Ng of gears which are needed to draw a complete graph of order p using a Spirograph(R), where p is an odd prime. The resulting graph consists of Ng hypotrochoids whose respective nodes coincide. If the teethed ring has a circumference p then Ng = (p-1)/2. Examples: A complete graph of order three can be drawn with a Spirograph(R) using a ring with 3n teeth and one gear with n teeth. n is an arbitrary number, only related to the geometry of the gears. A complete graph of order 5 can be drawn using a ring with diameter 5 and 2 gears with diameters 1 and 2 respectively. A complete graph of order 7 can be drawn using a ring with diameter 7 and 3 gears with diameters 1, 2 and 3 respectively. - Bob Andriesse, Mar 31 2017
MAPLE
with(numtheory): p:=n-> ithprime(n):seq((p(n+1)^2-1)/(2*sigma(p(n+1))), n= 1..64) # Gary Detlefs, May 02 2012
PROG
(Haskell)
(Python)
from sympy import prime
CROSSREFS
A130290 is an essentially identical sequence.
Cf. A005384 (subsequence of primes), A266400 (their indices in this sequence).
Primes p such that 2p-1 is also prime.
(Formerly M0849)
+10
122
2, 3, 7, 19, 31, 37, 79, 97, 139, 157, 199, 211, 229, 271, 307, 331, 337, 367, 379, 439, 499, 547, 577, 601, 607, 619, 661, 691, 727, 811, 829, 877, 937, 967, 997, 1009, 1069, 1171, 1237, 1279, 1297, 1399, 1429, 1459, 1531, 1609, 1627, 1657, 1759, 1867, 2011
COMMENTS
Sequence gives values of p such Sum_{i=1..p} gcd(p,i) = A018804(p) is prime. - Benoit Cloitre, Jan 25 2002 Let q = 2n-1. For these n (and q), the sum of two cyclotomic polynomials can be written as a product of cyclotomic polynomials and as a cyclotomic polynomial in x^2: Phi(q,x) + Phi(2q,x) = 2 Phi(n,x) Phi(2n,x) = 2 Phi(n,x^2). - T. D. Noe, Nov 04 2003 If a(n) is in A168421 then A005383(n) is a twin prime with a Ramanujan prime, A005383(n) - 2. If this sequence has an infinite number of terms in A168421, then the twin prime conjecture can be proved. - John W. Nicholson, Dec 05 2013
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
MAPLE
f := proc(Q) local t1, i, j; t1 := []; for i from 1 to 500 do j := ithprime(i); if isprime(2*j-Q) then t1 := [op(t1), j]; fi; od: t1; end; f(1);
MATHEMATICA
Select[Prime[Range[300]], PrimeQ[2#-1]&]
PROG
(Magma) [n: n in [0..1000] | IsPrime(n) and IsPrime(2*n-1)] // Vincenzo Librandi, Nov 18 2010 (Haskell)
a005382 n = a005382_list !! (n-1)
a005382_list = filter
((== 1) . a010051 . (subtract 1) . (* 2)) a000040_list
(PARI) forprime(n=2, 10^3, if(ispseudoprime(2*n-1), print1(n, ", "))) \\ Felix Fröhlich, Jun 15 2014
CROSSREFS
Cf. A005383, A005384 (2p+1), A057326, A057327, A057328, A057329, A057330, A005603, A063908 (2p-3), A063909 (2p-5), A023204 (2p+3), A000384, A001358.
Numbers k such that 2*k + 3 is a prime.
+10
65
0, 1, 2, 4, 5, 7, 8, 10, 13, 14, 17, 19, 20, 22, 25, 28, 29, 32, 34, 35, 38, 40, 43, 47, 49, 50, 52, 53, 55, 62, 64, 67, 68, 73, 74, 77, 80, 82, 85, 88, 89, 94, 95, 97, 98, 104, 110, 112, 113, 115, 118, 119, 124, 127, 130, 133, 134, 137, 139, 140, 145, 152, 154, 155
COMMENTS
The following sequences (allowing offset of first term) all appear to have the same parity: A034953, triangular numbers with prime indices; A054269, length of period of continued fraction for sqrt(p), p prime; A082749, difference between the sum of next prime(n) natural numbers and the sum of next n primes; A006254, numbers n such that 2n-1 is prime; A067076, 2n+3 is a prime. - Jeremy Gardiner, Sep 10 2004 n is in the sequence iff none of the numbers (n-3k)/(2k+1), 1 <= k <= (n-1)/5, is positive integer. - Vladimir Shevelev, May 31 2009
MAPLE
select(t -> isprime(2*t+3), [$0..1000]); # Robert Israel, Feb 19 2015
MATHEMATICA
Select[Range[0, 200], PrimeQ[2#+3]&] (* Harvey P. Dale, Jun 10 2014 *)
PROG
(PARI) [k | k<-[0..99], isprime(2*k+3)] \\ for illustration
(Sage) [n for n in (0..200) if is_prime(2*n+3) ] # G. C. Greubel, May 21 2019 (GAP) Filtered([0..200], k-> IsPrime(2*k+3) ) # G. C. Greubel, May 21 2019
CROSSREFS
Numbers n such that 2n+k is prime: A005097 (k=1), this seq(k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), A101448 (k=11), A153081 (k=13), A089559 (k=15), A173059 (k=17), A153143 (k=19). - Jason Kimberley, Sep 07 2012
EXTENSIONS
Offset changed from 0 to 1 in 2008: some formulas here and elsewhere may need to be corrected.
Triangular numbers ( A000217) with prime indices.
+10
50
3, 6, 15, 28, 66, 91, 153, 190, 276, 435, 496, 703, 861, 946, 1128, 1431, 1770, 1891, 2278, 2556, 2701, 3160, 3486, 4005, 4753, 5151, 5356, 5778, 5995, 6441, 8128, 8646, 9453, 9730, 11175, 11476, 12403, 13366, 14028, 15051, 16110, 16471, 18336, 18721, 19503
COMMENTS
The following sequences (allowing offset of first term) all appear to have the same parity: A034953, triangular numbers with prime indices; A054269, length of period of continued fraction for sqrt(p), p prime; A082749, difference between the sum of next prime(n) natural numbers and the sum of next n primes; A006254, numbers n such that 2n-1 is prime; A067076, 2n+3 is a prime. - Jeremy Gardiner, Sep 10 2004 Given a rectangular prism with sides 1, p, p^2 for p = n-th prime (n > 1), the area of the six sides divided by the volume gives a remainder which is 4*a(n). - J. M. Bergot, Sep 12 2011
MAPLE
a:= n-> (p-> p*(p+1)/2)(ithprime(n)):
MATHEMATICA
With[{nn=200}, Pick[Accumulate[Range[nn]], Table[If[PrimeQ[n], 1, 0], {n, nn}], 1]] (* Harvey P. Dale, Mar 05 2023 *)
PROG
(Haskell)
a034953 n = a034953_list !! (n-1)
a034953_list = map a000217 a000040_list
Average of twin prime pairs ( A014574), divided by 2. Equivalently, 2*a(n)-1 and 2*a(n)+1 are primes.
+10
43
2, 3, 6, 9, 15, 21, 30, 36, 51, 54, 69, 75, 90, 96, 99, 114, 120, 135, 141, 156, 174, 210, 216, 231, 261, 285, 300, 309, 321, 330, 405, 411, 414, 429, 441, 510, 516, 525, 531, 546, 576, 615, 639, 645, 651, 660, 714, 726, 741, 744, 804, 810, 834, 849, 861, 894
COMMENTS
The only possible pairs for 2a(n)+-1 are prime/prime (this sequence), not prime/not prime ( A104278), prime/notprime ( A104279) and not prime/prime ( A104280), ... this sequence + A104280 + A104279 + A104278 = the odd numbers. These numbers are never k mod (2k+1) or (k+1) mod (2k+1) with 2k+1 < a(n). - Jon Perry, Sep 04 2012 Excluding the first term, all remaining terms have digital root 3, 6 or 9. - J. W. Helkenberg, Jul 24 2013 Positive numbers x such that the difference between x^2 and adjacent squares are prime (both x^2-(x-1)^2 and (x+1)^2-x^2 are prime). - Doug Bell, Aug 21 2015
FORMULA
For n > 1, a(n) = A167379(n-1) * 3/2. (End)
MAPLE
P := select(isprime, [$1..1789]): map(p->(p+1)/2, select(p->member(p+2, P), P)); # Peter Luschny, Mar 03 2011
MATHEMATICA
Select[Range[900], And @@ PrimeQ[{-1, 1} + 2# ] &] (* Ray Chandler, Oct 12 2005 *)
PROG
(PARI) p=2; forprime(b=3, 1e4, if(b-p==2, print1((p+1)/2", ")); p=b) \\ Altug Alkan, Nov 10 2015 (Haskell)
Nonnegative numbers k such that 2k+5 is prime.
+10
34
0, 1, 3, 4, 6, 7, 9, 12, 13, 16, 18, 19, 21, 24, 27, 28, 31, 33, 34, 37, 39, 42, 46, 48, 49, 51, 52, 54, 61, 63, 66, 67, 72, 73, 76, 79, 81, 84, 87, 88, 93, 94, 96, 97, 103, 109, 111, 112, 114, 117, 118, 123, 126, 129, 132, 133, 136, 138, 139, 144, 151, 153, 154, 156, 163
FORMULA
a(n) = ( A086304(n-1) + 1)/2, n > 1.
PROG
(Sage) [n for n in (0..200) if is_prime(2*n+5) ] # G. C. Greubel, May 21 2019 (GAP) Filtered([0..200], k-> IsPrime(2*k+5) ) # G. C. Greubel, May 21 2019
Numbers k such that 2k-3 is prime.
+10
34
3, 4, 5, 7, 8, 10, 11, 13, 16, 17, 20, 22, 23, 25, 28, 31, 32, 35, 37, 38, 41, 43, 46, 50, 52, 53, 55, 56, 58, 65, 67, 70, 71, 76, 77, 80, 83, 85, 88, 91, 92, 97, 98, 100, 101, 107, 113, 115, 116, 118, 121, 122, 127, 130, 133, 136, 137, 140, 142, 143, 148, 155, 157, 158
FORMULA
Half of p + 3, where p is a prime greater than 2.
PROG
(Sage) [n for n in (1..200) if is_prime(2*n-3) ] # G. C. Greubel, May 21 2019 (GAP) Filtered([1..200], k-> IsPrime(2*k-3) ) # G. C. Greubel, May 21 2019
AUTHOR
Douglas Winston (douglas.winston(AT)srupc.com), Sep 14 2004
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