A017377 -id:A017377 - cp's OEIS frontend

A030433 Primes of form 10*k + 9.

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

Offset: 1

Warut Roonguthai

nonn

Also primes of form 5*k + 4.
5 is quadratic residue of primes of form 10*k-1. - Vincenzo Librandi, Jun 25 2014
Also, primes p such that 5 divides sigma(p), cf. A274397. - M. F. Hasler, Jul 10 2016
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
From Jianing Song, Sep 13 2022: (Start)
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)

Michael B. Porter, Table of n, a(n) for n = 1..100000
A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
Erika Klarreich, Mathematicians Discover Prime Conspiracy, Quanta Magazine, 2016.
R. J. Lemke Oliver and K. Soundararajan, Unexpected biases in the distribution of consecutive primes, arXiv:1603.03720 [math.NT], 2016.

Cf. A045457, A049510, A102700.

Filtered(List([1..500],n->10*n+9),IsPrime); # Muniru A Asiru, Apr 27 2018

select(isprime,[seq(10*n+9,n=1..500)]); # Muniru A Asiru, Apr 27 2018

Select[Prime@Range[210], Mod[ #, 10] == 9 &] (* Ray Chandler, Nov 07 2006 *)
Select[Range[9, 1300, 10], PrimeQ] (* Harvey P. Dale, Jun 01 2012 *)
Prime@Flatten@Position[Length@FactorList[((1+d)^5-1)/d,Modulus->#]&/@Prime@Range@200,3] (* Federico Provvedi, Apr 04 2018 *)

select(n->n%10==9, primes(100)) \\ Charles R Greathouse IV, Apr 29 2015

for(n=1, 1e3, if(isprime(p=10*n+9), print1(p, ", "))); \\ Altug Alkan, Apr 19 2018

a(n) = 10*A102700(n) + 9.
Union of A132234 and A132236. - Ray Chandler, Apr 07 2009
Intersection of A000040 and A017377. - Iain Fox, Dec 30 2017

Extended by Ray Chandler, Nov 07 2006

A106621 a(n) = numerator of n/(n+20).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, May 15 2005

Contains as subsequences A026741, A017281, A017305, A005408, A017353, and A017377. - Luce ETIENNE, Nov 04 2018

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

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Peter Bala, A note on the sequence of numerators of a rational function, 2019.
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).

Cf. A000010, A010879.

Cf. Sequences given by the formula numerator(n/(n + k)): A026741 (k = 2), A051176 (k = 3), A060819 (k = 4), A060791 (k = 5), A060789 (k = 6), A106608 thru A106612 (k = 7 thru 11), A051724 (k = 12), A106614 thru A106620 (k = 13 thru 19).

List([0..80],n->NumeratorRat(n/(n+20))); # Muniru A Asiru, Feb 19 2019

[Numerator(n/(n+20)): n in [0..100]]; // Vincenzo Librandi, Mar 06 2018

seq(numer(n/(n+20)),n=0..80); # Muniru A Asiru, Feb 19 2019

f[n_]:=Numerator[n/(n+20)];Array[f,100,0] (* Vladimir Joseph Stephan Orlovsky, Feb 17 2011 *)

a(n) = numerator(n/(n+20)); \\ Michel Marcus, Mar 07 2018

[lcm(20,n)/20 for n in range(0, 80)] # Zerinvary Lajos, Jun 12 2009

a(n) = lcm(20, n)/20. - Zerinvary Lajos, Jun 12 2009
a(n) = n/gcd(n, 20). - Andrew Howroyd, Jul 25 2018
From Luce ETIENNE, Nov 04 2018: (Start)
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)
From Peter Bala, Feb 24 2019: (Start)
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)
From Amiram Eldar, Nov 25 2022: (Start)
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)

Keyword:mult added by Andrew Howroyd, Jul 25 2018

A273669 Decimal representation ends with either 2 or 9.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 06 2016

Natural numbers not in A273664.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature(1,1,-1).

Sequences A017293 and A017377 interleaved.
Cf. A000035, A001622, A007310, A084967, A126760, A249745, A249746.
Cf. also A273664, A249824, A275716.

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 *)

(define (A273669 n) (+ (* 10 (/ (+ (- n 2) (if (odd? n) 1 0)) 2)) (if (odd? n) 2 9)))

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).
a(n) = A126760(A084967(n)).
a(n) = A249746((3*A249745(n))-1).
Other identities. For all n >= 1:
A084967(n) = 5*A007310(n) = A007310(a(n)).
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

A135705 a(n) = 10binomial(n,2) + 9n.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Mar 04 2008

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
Bisection of A195162. - Omar E. Pol, Aug 04 2012

Ivan Panchenko, Table of n, a(n) for n = 0..1000
L. Hogben, Choice and Chance by Cardpack and Chessboard, Vol. 1, Max Parrish and Co, London, 1950, p. 36.
Leo Tavares, Illustration: Diamond Clipped Stars
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Second n-gonal numbers: A005449, A014105, A147875, A045944, A179986, A033954, A062728, this sequence.
Cf. A003154, A000290.
Cf. A195162.

List([0..50], n-> n*(5*n+4)); # G. C. Greubel, Jul 04 2019

[n*(5*n+4): n in [0..50]]; // G. C. Greubel, Jul 04 2019

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 *)

a(n) = 10*binomial(n,2) + 9*n \\ Charles R Greathouse IV, Jun 11 2015

[n*(5*n+4) for n in (0..50)] # G. C. Greubel, Jul 04 2019

From R. J. Mathar, Mar 06 2008: (Start)
O.g.f.: x*(9+x)/(1-x)^3.
a(n) = n*(5*n+4). (End)
a(n) = a(n-1) + 10*n - 1 (with a(0)=0). - Vincenzo Librandi, Nov 24 2009
a(n) = Sum_{i=0..n-1} A017377(i) for n>0. - Bruno Berselli, Jan 15 2011
a(n) = A131242(10n+8). - Philippe Deléham, Mar 27 2013
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
From G. C. Greubel, Oct 29 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
E.g.f.: x*(9 + 5*x)*exp(x). (End)
a(n) = A003154(n+1) - A000290(n+1). - Leo Tavares, Mar 29 2022

A346950 Positive integers k that are the product of two integers ending with 3.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Aug 08 2021

All the terms end with 9 (A017377).

			9 = 3*3, 39 = 3*13, 69 = 3*23, 99 = 3*33, 129 = 3*43, 159 = 3*53, 169 = 13*13, 189 = 3*63, ...

Stefano Spezia, Table of n, a(n) for n = 1..10000

Cf. A017377 (supersequence), A053742 (ending with 5), A139245 (ending with 2), A324297 (ending with 6), A346951, A346952, A346953.

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

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)))
print(aupto(1270)) # Michael S. Branicky, Aug 08 2021

Limit_{n->oo} a(n)/a(n-1) = 1.

A346952 Number of positive integers with n digits that are the product of two integers ending with 3.

Original entry on oeis.org

1, 3, 37, 398, 4303, 45765, 480740, 5005328, 51770770, 532790460, 5461696481, 55814395421, 568944166801, 5787517297675

Offset: 1

Stefano Spezia, Aug 08 2021

a(n) is the number of n-digit numbers in A346950.

Cf. A017377, A052268, A346509 (ending with 1), A337855 (ending with 5), A337856 (ending with 6), A346950.

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<#Giorgos Kalogeropoulos, Aug 16 2021 *)

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))
print([a(n) for n in range(1, 9)]) # Michael S. Branicky, Aug 09 2021

a(n) < A052268(n).

Conjecture: Lim_{n->infinity} a(n)/a(n-1) = 10.

a(6)-a(11) from Michael S. Branicky, Aug 09 2021

a(12)-a(14) from Martin Ehrenstein, Aug 22 2021

A348054 Positive integers that are the product of two integers ending with 7.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Sep 26 2021

			49 = 7*7, 119 = 7*17, 189 = 7*27, 259 = 7*37, 289 = 17*17, 329 = 7*47, 399 = 7*57, ...

Cf. A017377 (supersequence), A053742 (ending with 5), A139245 (ending with 2), A324297 (ending with 6), A346950 (ending with 3), A347253 (ending with 4), A348055.

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

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)))
print(aupto(2110)) # Michael S. Branicky, Sep 26 2021

Lim_{n->infinity} a(n)/a(n-1) = 1.

A352970 Carmichael numbers ending in 9.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Apr 12 2022

The first term is the Hardy-Ramanujan number.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..109 from Chai Wah Wu)
Index entries for sequences related to Carmichael numbers.

Intersection of A002997 and A017377.
Subsequence of A053181.
Cf. A001235, A354609, A355305, A355307, A355309.

Select[10*Range[0, 3*10^7] + 9, CompositeQ[#] && Divisible[# - 1, CarmichaelLambda[#]] &] (* Amiram Eldar, May 28 2022 *)

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)
A352970_list = list(islice(A352970_gen(),5)) # Chai Wah Wu, May 11 2022

A080719 Replace decimal digits with their binary values and convert back to decimal representation.

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, Mar 06 2003

m is a local maximum iff m == 9 modulo 10 (see A017377).

A257831 seen as binary numbers: A007088(a(n)) = A257831(n). - Reinhard Zumkeller, May 10 2015

			n=27 -> '2''7' -> '10''111' -> '10111' -> 23: a(27)=23.
See also A257831.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A007088.

Cf. A257831, A030308, A031298.

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
-- Reinhard Zumkeller, May 10 2015

Table[FromDigits[Flatten[IntegerDigits[#,2]&/@IntegerDigits[n]],2],{n,80}] (* Harvey P. Dale, Aug 30 2014 *)

def A080719(n):
    return int(''.join((format(int(d),'b') for d in str(n))),2)
# Chai Wah Wu, May 10 2015

a(0)=0 prepended and offset changed by Reinhard Zumkeller, May 10 2015

A273374 Squares ending in digit 9.

Original entry on oeis.org

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

Offset: 1

Vincenzo Librandi, May 21 2016

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

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000290, A016754, A063226.
Cf. A017377 (numbers ending in 9), A017379 (cubes ending in 9).
Cf. similar sequences listed in A273373.

/* By definition: */ [n^2: n in [0..200] | Modexp(n,2,10) eq 9];

[6+(50*(n-1)*n-5*(2*n-1)*(-1)^n+1)/2: n in [1..50]];

Table[6 + (50 (n - 1) n - 5 (2 n - 1) (-1)^n + 1)/2, {n, 1, 50}]

a(n)=(5*n-3+n%2)^2 \\ Charles R Greathouse IV, Nov 03 2021

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.
a(n) = A063226(n)^2. - Seiichi Manyama, May 25 2016
Sum_{n>=1} 1/a(n) = Pi^2*(3-sqrt(5))/50. - Amiram Eldar, Feb 16 2023

Corrected and extended by Bruno Berselli, May 21 2016

cp's OEIS Frontend

A030433 Primes of form 10*k + 9.

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A106621 a(n) = numerator of n/(n+20).

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A273669 Decimal representation ends with either 2 or 9.

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A135705 a(n) = 10*binomial(n,2) + 9*n.

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A346950 Positive integers k that are the product of two integers ending with 3.

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A346952 Number of positive integers with n digits that are the product of two integers ending with 3.

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A135705 a(n) = 10binomial(n,2) + 9n.