Displaying 1-10 of 40 results found.
19, 29, 59, 79, 89, 109, 139, 149, 179, 199, 229, 239, 269, 349, 359, 379, 389, 409, 419, 439, 449, 479, 499, 509, 569, 599, 619, 659, 709, 719, 739, 769, 809, 829, 839, 859, 919, 929, 1009, 1019, 1039, 1049, 1069, 1109, 1129, 1229, 1249, 1259, 1279, 1289
COMMENTS
Also primes of form 5*k + 4.
Conjecture: Primes p such that ((x+1)^5-1)/x has 2 distinct irreducible factors of degree 2 over GF(p). - Federico Provvedi, Apr 01 2018 The digital root of a(n) is 1, 2, 4, 5, 7 or 8. - Muniru A Asiru, Apr 28 2018 Primes p such that the ideal (p) factors into two prime ideals in Z[zeta_5], where zeta_5 = exp(2*Pi*i/5). Since Z[zeta_5] is a PID, this is equivalent to saying that this sequence lists primes p that are the product of two non-associate prime elements Z[zeta_5]. In particular, the factorization of p == 4 (mod 5) in Z[zeta_5] coincides with the factorization in Z[(1+sqrt(5))/2] (e.g., 19 = (8+3*sqrt(5))*(8-3*sqrt(5)) is the factorization of 19 in both Z[(1+sqrt(5))/2] and Z[zeta_5]).
Also primes p such that x^4 + x^3 + x^2 + x + 1 factors into two irreducible quadratic polynomials over GF(p) (cf. A327753). (End)
MAPLE
select(isprime, [seq(10*n+9, n=1..500)]); # Muniru A Asiru, Apr 27 2018
MATHEMATICA
Select[Prime@Range[210], Mod[ #, 10] == 9 &] (* Ray Chandler, Nov 07 2006 *) Prime@Flatten@Position[Length@FactorList[((1+d)^5-1)/d, Modulus->#]&/@Prime@Range@200, 3] (* Federico Provvedi, Apr 04 2018 *)
PROG
(PARI) for(n=1, 1e3, if(isprime(p=10*n+9), print1(p, ", "))); \\ Altug Alkan, Apr 19 2018 (GAP) Filtered(List([1..500], n->10*n+9), IsPrime); # Muniru A Asiru, Apr 27 2018
a(n) = numerator of n/(n+20).
+10
19
0, 1, 1, 3, 1, 1, 3, 7, 2, 9, 1, 11, 3, 13, 7, 3, 4, 17, 9, 19, 1, 21, 11, 23, 6, 5, 13, 27, 7, 29, 3, 31, 8, 33, 17, 7, 9, 37, 19, 39, 2, 41, 21, 43, 11, 9, 23, 47, 12, 49, 5, 51, 13, 53, 27, 11, 14, 57, 29, 59, 3, 61, 31, 63, 16, 13, 33, 67, 17, 69, 7, 71, 18, 73, 37, 15, 19, 77, 39, 79
COMMENTS
Multiplicative and also a strong divisibility sequence: gcd(a(n),a(m)) = a(gcd(n,m)) for n, m >= 1. - Peter Bala, Feb 24 2019
LINKS
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1).
FORMULA
a(n) = 9*a(n-20) - 36*a(n-40) + 84*a(n-60) - 126*a(n-80) + 126*a(n-100) - 84*a(n-120) + 36*a(n-140) - 9*a(n-160) + a(n-180).
a(n) = (5*(119*m^9 - 4923*m^8 + 86250*m^7 - 832230*m^6 + 4807887*m^5 - 16882299*m^4 + 34770400*m^3 - 37855620m^2 + 16581744*m + 54432)*floor(n/10) + 72*m*(3*m^8 - 120*m^7 + 2030*m^6 - 18900*m^5 + 105329*m^4 - 356580*m^3 + 706220*m^2 - 733200*m + 300258) + ((19*m^9 - 855*m^8 + 15810*m^7 - 154350*m^6 + 849387*m^5 - 2597175*m^4 + 4037840*m^3 - 2600100*m^2 + 540144*m - 90720)*floor(n/10) - 72*m*(m^7 - 35*m^6 + 490*m^5 - 3500*m^4 + 13489*m^3 - 27335*m^2 + 26340*m - 9450))*(-1)^floor(n/10))/362880 where m = (n mod 10). (End)
a(n) = n/gcd(n,20) is a quasi-polynomial in n since gcd(n,20) is a purely periodic sequence of period 20.
O.g.f.: F(x) - F(x^2) - F(x^4) - 4*F(x^5) + 4*F(x^10) + 4*F(x^20), where F(x) = x/(1 - x)^2.
O.g.f. for reciprocals: Sum_{n >= 1} x^n/a(n) = Sum_{d divides 20} (phi(d)/d) * log(1/(1 - x^d)) = log(1/(1 - x)) + (1/2)*log(1/(1 - x^2)) + (2/4)*log(1/(1 - x^4)) + (4/5)*log(1/(1 - x^5)) + (4/10)*log(1/(1 - x^10)) + (8/20)*log(1/(1 - x^20)), where phi(n) denotes the Euler totient function A000010. (End) Multiplicative with a(2^e) = 2^max(0, e-2), a(5^e) = 5^max(0,e-1), and a(p^e) = p^e otherwise.
Dirichlet g.f.: zeta(s-1)*(1 - 1/2^s - 1/4^s - 4/5^s + 4/10^s + 4/20^s).
Sum_{k=1..n} a(k) ~ (231/800) * n^2. (End)
PROG
(Sage) [lcm(20, n)/20 for n in range(0, 80)] # Zerinvary Lajos, Jun 12 2009 (GAP) List([0..80], n->NumeratorRat(n/(n+20))); # Muniru A Asiru, Feb 19 2019
Decimal representation ends with either 2 or 9.
+10
15
2, 9, 12, 19, 22, 29, 32, 39, 42, 49, 52, 59, 62, 69, 72, 79, 82, 89, 92, 99, 102, 109, 112, 119, 122, 129, 132, 139, 142, 149, 152, 159, 162, 169, 172, 179, 182, 189, 192, 199, 202, 209, 212, 219, 222, 229, 232, 239, 242, 249, 252, 259, 262, 269, 272, 279, 282, 289, 292, 299, 302, 309, 312, 319, 322, 329, 332, 339
FORMULA
a(n) = 10*(((n-2)+ A000035(n))/2) + 2 [when n is odd], or + 9 [when n is even]. For n >= 5, a(n) = 2*a(n-2) - a(n-4).
Other identities. For all n >= 1:
G.f.: x*(x^2+7*x+2)/((x+1)*(x-1)^2).
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt((1+1/sqrt(5))/2)*phi^2*Pi/10 - log(phi)/(2*sqrt(5)) - log(2)/5, where phi is the golden ratio ( A001622). - Amiram Eldar, Apr 15 2023
MATHEMATICA
Select[Range@ 340, MemberQ[{2, 9}, Mod[#, 10]] &] (* or *)
Table[{10 n + 2, 10 n + 9}, {n, 0, 33}] // Flatten (* or *)
CoefficientList[Series[(-5/(1 - x) + (11 - x)/(-1 + x)^2 - 2/(1 + x))/2, {x, 0, 67}], x] (* Michael De Vlieger, Aug 07 2016 *)
PROG
(Scheme)
(define ( A273669 n) (+ (* 10 (/ (+ (- n 2) (if (odd? n) 1 0)) 2)) (if (odd? n) 2 9)))
a(n) = 10*binomial(n,2) + 9*n.
+10
14
0, 9, 28, 57, 96, 145, 204, 273, 352, 441, 540, 649, 768, 897, 1036, 1185, 1344, 1513, 1692, 1881, 2080, 2289, 2508, 2737, 2976, 3225, 3484, 3753, 4032, 4321, 4620, 4929, 5248, 5577, 5916, 6265, 6624, 6993, 7372, 7761, 8160, 8569, 8988, 9417, 9856, 10305, 10764
COMMENTS
Also, second 12-gonal (or dodecagonal) numbers. Identity for the numbers b(n)=n*(h*n+h-2)/2 (see Crossrefs): Sum_{i=0..n} (b(n)+i)^2 = (Sum_{i=n+1..2*n} (b(n)+i)^2) + h*(h-4)* A000217(n)^2 for n>0. - Bruno Berselli, Jan 15 2011 Sequence found by reading the line from 0, in the direction 0, 28, ..., and the line from 9, in the direction 9, 57, ..., in the square spiral whose vertices are the generalized 12-gonal numbers A195162. - Omar E. Pol, Jul 24 2012
FORMULA
O.g.f.: x*(9+x)/(1-x)^3.
a(n) = n*(5*n+4). (End)
Sum_{n>=1} 1/a(n) = 5/16 + sqrt(1 + 2/sqrt(5))*Pi/8 - 5*log(5)/16 - sqrt(5)*log((1 + sqrt(5))/2)/8 = 0.2155517745488486003038... . - Vaclav Kotesovec, Apr 27 2016 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
E.g.f.: x*(9 + 5*x)*exp(x). (End)
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {0, 9, 28}, 50] (* or *) Table[5*n^2 + 4*n, {n, 0, 50}] (* G. C. Greubel, Oct 29 2016 *) Table[10 Binomial[n, 2]+9n, {n, 0, 60}] (* Harvey P. Dale, Jun 14 2023 *)
Positive integers k that are the product of two integers ending with 3.
+10
11
9, 39, 69, 99, 129, 159, 169, 189, 219, 249, 279, 299, 309, 339, 369, 399, 429, 459, 489, 519, 529, 549, 559, 579, 609, 639, 669, 689, 699, 729, 759, 789, 819, 849, 879, 909, 939, 949, 969, 989, 999, 1029, 1059, 1079, 1089, 1119, 1149, 1179, 1209, 1219, 1239, 1269
COMMENTS
All the terms end with 9 ( A017377).
FORMULA
Limit_{n->oo} a(n)/a(n-1) = 1.
EXAMPLE
9 = 3*3, 39 = 3*13, 69 = 3*23, 99 = 3*33, 129 = 3*43, 159 = 3*53, 169 = 13*13, 189 = 3*63, ...
MATHEMATICA
a={}; For[n=0, n<=250, n++, For[k=0, k<=n, k++, If[Mod[10*n+9, 10*k+3]==0 && Mod[(10*n+9)/(10*k+3), 10]==3&& 10*n+9>Max[a], AppendTo[a, 10*n+9]]]]; a
PROG
(Python)
def aupto(lim): return sorted(set(a*b for a in range(3, lim//3+1, 10) for b in range(a, lim//a+1, 10)))
Number of positive integers with n digits that are the product of two integers ending with 3.
+10
8
1, 3, 37, 398, 4303, 45765, 480740, 5005328, 51770770, 532790460, 5461696481, 55814395421, 568944166801, 5787517297675
COMMENTS
a(n) is the number of n-digit numbers in A346950.
FORMULA
Conjecture: Lim_{n->infinity} a(n)/a(n-1) = 10.
MATHEMATICA
Table[{lo, hi}={10^(n-1), 10^n}; Length@Select[Union@Flatten@Table[a*b, {a, 3, Floor[hi/3], 10}, {b, a, Floor[hi/a], 10}], lo<#<hi&], {n, 7}] (* Giorgos Kalogeropoulos, Aug 16 2021 *)
PROG
(Python)
def a(n):
lo, hi = 10**(n-1), 10**n
return len(set(a*b for a in range(3, hi//3+1, 10) for b in range(a, hi//a+1, 10) if lo <= a*b < hi))
Positive integers that are the product of two integers ending with 7.
+10
6
49, 119, 189, 259, 289, 329, 399, 459, 469, 539, 609, 629, 679, 729, 749, 799, 819, 889, 959, 969, 999, 1029, 1099, 1139, 1169, 1239, 1269, 1309, 1369, 1379, 1449, 1479, 1519, 1539, 1589, 1649, 1659, 1729, 1739, 1799, 1809, 1819, 1869, 1939, 1989, 2009, 2079, 2109
FORMULA
Lim_{n->infinity} a(n)/a(n-1) = 1.
EXAMPLE
49 = 7*7, 119 = 7*17, 189 = 7*27, 259 = 7*37, 289 = 17*17, 329 = 7*47, 399 = 7*57, ...
MATHEMATICA
a={}; For[n=0, n<=210, n++, For[k=0, k<=n, k++, If[Mod[10*n+9, 10*k+7]==0 && Mod[(10*n+9)/(10*k+7), 10]==7 && 10*n+9>Max[a], AppendTo[a, 10*n+9]]]]; a
PROG
(Python)
def aupto(lim): return sorted(set(a*b for a in range(7, lim//7+1, 10) for b in range(a, lim//a+1, 10)))
Carmichael numbers ending in 9.
+10
5
1729, 294409, 1033669, 1082809, 1773289, 5444489, 7995169, 8719309, 17098369, 19384289, 23382529, 26921089, 37964809, 43620409, 45890209, 50201089, 69331969, 84311569, 105309289, 114910489, 146843929, 168659569, 172947529, 180115489, 188516329, 194120389, 214852609, 228842209, 230996949, 246446929, 271481329
COMMENTS
The first term is the Hardy-Ramanujan number.
MATHEMATICA
Select[10*Range[0, 3*10^7] + 9, CompositeQ[#] && Divisible[# - 1, CarmichaelLambda[#]] &] (* Amiram Eldar, May 28 2022 *)
PROG
(Python)
from itertools import islice
from sympy import factorint, nextprime
def A352970_gen(): # generator of terms p, q = 3, 5
while True:
for n in range(p+11-((p+2) % 10), q, 10):
f = factorint(n)
if max(f.values()) == 1 and not any((n-1) % (p-1) for p in f):
yield n
p, q = q, nextprime(q)
Replace decimal digits with their binary values and convert back to decimal representation.
+10
4
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 3, 6, 7, 12, 13, 14, 15, 24, 25, 4, 5, 10, 11, 20, 21, 22, 23, 40, 41, 6, 7, 14, 15, 28, 29, 30, 31, 56, 57, 8, 9, 18, 19, 36, 37, 38, 39, 72, 73, 10, 11, 22, 23, 44, 45, 46, 47, 88, 89, 12, 13, 26, 27, 52, 53, 54, 55, 104, 105, 14, 15, 30, 31, 60, 61, 62
COMMENTS
m is a local maximum iff m == 9 modulo 10 (see A017377).
EXAMPLE
n=27 -> '2''7' -> '10''111' -> '10111' -> 23: a(27)=23.
MATHEMATICA
Table[FromDigits[Flatten[IntegerDigits[#, 2]&/@IntegerDigits[n]], 2], {n, 80}] (* Harvey P. Dale, Aug 30 2014 *)
PROG
(Haskell)
import Data.Maybe (mapMaybe)
a080719 = foldr (\b v -> 2 * v + b) 0 .
concat . mapMaybe (flip lookup bin) . a031298_row
where bin = zip [0..9] a030308_tabf
(Python)
....return int(''.join((format(int(d), 'b') for d in str(n))), 2)
Squares ending in digit 9.
+10
4
9, 49, 169, 289, 529, 729, 1089, 1369, 1849, 2209, 2809, 3249, 3969, 4489, 5329, 5929, 6889, 7569, 8649, 9409, 10609, 11449, 12769, 13689, 15129, 16129, 17689, 18769, 20449, 21609, 23409, 24649, 26569, 27889, 29929, 31329, 33489, 34969, 37249, 38809
COMMENTS
A quasipolynomial of order two and degree two: a(n) = 25n^2 - 30n + 9 if n is even and 25n^2 - 20n + 4 if n is odd. - Charles R Greathouse IV, Nov 03 2021
FORMULA
G.f.: x*(9 + 40*x + 102*x^2 + 40*x^3 + 9*x^4)/((1 + x)^2*(1 - x)^3).
a(n) = 6 + (50*(n-1)*n - 5*(2*n-1)*(-1)^n + 1)/2.
Sum_{n>=1} 1/a(n) = Pi^2*(3-sqrt(5))/50. - Amiram Eldar, Feb 16 2023
MATHEMATICA
Table[6 + (50 (n - 1) n - 5 (2 n - 1) (-1)^n + 1)/2, {n, 1, 50}]
PROG
(Magma) /* By definition: */ [n^2: n in [0..200] | Modexp(n, 2, 10) eq 9];
(Magma) [6+(50*(n-1)*n-5*(2*n-1)*(-1)^n+1)/2: n in [1..50]];
CROSSREFS
Cf. similar sequences listed in A273373.
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