A017377 -id:A017377 - cp's OEIS frontend

A154419 Primes of the form 20k^2 + 36k + 17.

17, 73, 953, 1249, 2377, 2833, 3329, 4441, 8737, 12401, 13417, 15569, 17881, 20353, 21649, 28729, 33457, 36809, 49801, 51817, 62497, 67049, 71761, 74177, 86857, 89513, 100537, 103393, 118273, 121369, 127681, 134153, 144161, 161641, 168913

Offset: 1

Vincenzo Librandi, Jan 09 2009

Also primes of the form 5*j^2 + 18*j + 17. (Proof: this format implies that j=2*k, even, because otherwise 5*j^2 + 18*j + 17 is even and cannot be prime. So 5*j^2 + 18*j + 17 = 20*k^2 + 36*k + 17.) - R. J. Mathar, Jan 12 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A017377, A154418.

[a: n in [0..100] | IsPrime(a) where a is 20*n^2+36*n+17]; // Vincenzo Librandi, Jul 23 2012

Select[Table[20n^2+36n+17,{n,0,6001}],PrimeQ] (* Vincenzo Librandi, Jul 23 2012 *)

select(isprime, vector(100, n, 20*(n-1)^2 + 36*(n-1) + 17)) \\ Robert C. Lyons,  Feb 27 2025

A161365 a(n) = 3/2 + 5n - 5(-1)^n/2.

Original entry on oeis.org

9, 9, 19, 19, 29, 29, 39, 39, 49, 49, 59, 59, 69, 69, 79, 79, 89, 89, 99, 99, 109, 109, 119, 119, 129, 129, 139, 139, 149, 149, 159, 159, 169, 169, 179, 179, 189, 189, 199, 199, 209, 209, 219, 219, 229, 229, 239, 239, 249, 249, 259, 259, 269, 269, 279, 279, 289

Offset: 1

Vincenzo Librandi, Nov 25 2009

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

I:=[9, 9, 19]; [n le 3 select I[n] else Self(n-1)+Self(n-2)-Self(n-3): n in [1..60]]; // Vincenzo Librandi, Mar 02 2012

LinearRecurrence[{1, 1, -1}, {9, 9, 19}, 60] (* Vincenzo Librandi, Mar 02 2012 *)
Table[3/2+5n-(5(-1)^n)/2,{n,60}] (* or *) nxt[{n_,a_}]:={n+1,10(n+1)-a-2}; NestList[nxt,{1,9},60][[;;,2]] (* Harvey P. Dale, Nov 04 2024 *)

for(n=1, 60, print1(3/2+5*n-5*(-1)^n/2", ")); \\ Vincenzo Librandi, Mar 02 2012

a(n) = 10*n - a(n-1) - 2, n > 1.
a(n+1) = A017377(floor(n/2)). - R. J. Mathar, Jan 05 2011
G.f.: x*(9+x^2) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Jan 05 2011

Definition rewritten by R. J. Mathar, Jan 05 2011

A168419 a(n) = 9*floor(n/2).

Original entry on oeis.org

0, 9, 9, 18, 18, 27, 27, 36, 36, 45, 45, 54, 54, 63, 63, 72, 72, 81, 81, 90, 90, 99, 99, 108, 108, 117, 117, 126, 126, 135, 135, 144, 144, 153, 153, 162, 162, 171, 171, 180, 180, 189, 189, 198, 198, 207, 207, 216, 216, 225, 225, 234, 234, 243, 243, 252, 252, 261

Offset: 1

Vincenzo Librandi, Nov 25 2009

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

Cf. A004526, A017377.

[9*Floor(n/2): n in [1..70]]; // Vincenzo Librandi, Sep 19 2013

Table[9 Floor[n/2], {n, 70}] (* or *) CoefficientList[Series[9 x/((1 + x) (x - 1)^2), {x, 0, 70}], x] (* Vincenzo Librandi, Sep 19 2013 *)
LinearRecurrence[{1,1,-1},{0,9,9},60] (* Harvey P. Dale, Apr 21 2019 *)

a(n) = 9*n - a(n-1) - 9, with n>1, a(1)=0.
G.f.: 9*x^2/((1+x)*(x-1)^2). - Vincenzo Librandi, Sep 19 2013
a(n) = a(n-1) +a(n-2) -a(n-3). - Vincenzo Librandi, Sep 19 2013
E.g.f.: (9/4)*(1 + (2*x - 1)*exp(2*x))*exp(-x). - G. C. Greubel, Jul 22 2016

New definition by Vincenzo Librandi, Sep 19 2013

A239794 5n^2 + 4n - 15.

Original entry on oeis.org

-6, 13, 42, 81, 130, 189, 258, 337, 426, 525, 634, 753, 882, 1021, 1170, 1329, 1498, 1677, 1866, 2065, 2274, 2493, 2722, 2961, 3210, 3469, 3738, 4017, 4306, 4605, 4914, 5233, 5562, 5901, 6250, 6609, 6978, 7357, 7746, 8145, 8554, 8973, 9402, 9841, 10290

Offset: 1

Katherine Guo, Mar 26 2014

Follows the integer values from 1 on the quadratic equation 5*x^2 + 4*n - 15, this is the case x=n.

			For n=3, a(3) = 5*3^2 + 4*3 - 15 = 42; for n=6, a(6) = 5*6^2 + 4*6 - 15 = 189.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
WolframAlpha, Table of 5n^2+4n-15
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A008590, A017377.

Magma
```
[5*n^2+4*n-15: n in [1..50]];
```

Table[5 n^2 + 4 n - 15, {n, 50}]
CoefficientList[Series[(6 - 31 x + 15 x^2)/(x - 1)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Mar 29 2014 *)
LinearRecurrence[{3,-3,1},{-6,13,42},50] (* Harvey P. Dale, May 30 2025 *)

a(n)=5*n^2+4*n-15 \\ Charles R Greathouse IV, Jun 17 2017

From Bruno Berselli, Mar 27 2014: (Start)
G.f.: -x*(6 - 31*x + 15*x^2)/(1 - x)^3.
a(n+1) - a(n) = A017377(n).
a(n) - a(-n) = A008590(n). (End)

A250024 a(n) = 40*n - 21.

Original entry on oeis.org

19, 59, 99, 139, 179, 219, 259, 299, 339, 379, 419, 459, 499, 539, 579, 619, 659, 699, 739, 779, 819, 859, 899, 939, 979, 1019, 1059, 1099, 1139, 1179, 1219, 1259, 1299, 1339, 1379, 1419, 1459, 1499, 1539, 1579, 1619, 1659, 1699, 1739, 1779, 1819, 1859, 1899

Offset: 1

Karl V. Keller, Jr., Nov 10 2014

This is the sequence of numbers congruent to 19 mod 40.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..10000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A017377 (10*n+9), A142190 (prime terms).

[40*n-21: n in [1..60]]; // Vincenzo Librandi, Nov 11 2014

40 Range[40] - 21 (* Alonso del Arte, Nov 10 2014 *)
CoefficientList[Series[(19 + 21 x) / (1 - x)^2, {x, 0, 50}], x] (* Vincenzo Librandi, Nov 11 2014 *)

a(n)=40*n-21 \\ Charles R Greathouse IV, May 23 2019

for n in range(1,101): print(40*n-21,end=",")

From Vincenzo Librandi, Nov 11 2014: (Start)
G.f.: x*(19+21*x)/(1-x)^2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Elmo R. Oliveira, Apr 04 2025: (Start)
E.g.f.: 21 + exp(x)*(40*x - 21).
a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)

A266297 Numbers whose last digit is a square.

Original entry on oeis.org

0, 1, 4, 9, 10, 11, 14, 19, 20, 21, 24, 29, 30, 31, 34, 39, 40, 41, 44, 49, 50, 51, 54, 59, 60, 61, 64, 69, 70, 71, 74, 79, 80, 81, 84, 89, 90, 91, 94, 99, 100, 101, 104, 109, 110, 111, 114, 119, 120, 121, 124, 129, 130, 131, 134, 139, 140, 141, 144, 149

Offset: 1

Wesley Ivan Hurt, Dec 26 2015

Numbers ending in 0, 1, 4 and 9.
Union of A008592, A017281, A017317 and A017377. - Hurt
None of these numbers are prime in Z[phi] (where phi = 1/2 + sqrt(5)/2 is the golden ratio), since the numbers in this sequence that are prime in Z can be expressed in the form (a - b sqrt(5))(a + b sqrt(5)). - Alonso del Arte, Dec 30 2015
Union of A197652 and A016897. - Wesley Ivan Hurt, Dec 31 2015
Union of A146763 and A090771. - Wesley Ivan Hurt, Jan 01 2016

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A008592, A016897, A017281, A017317, A017377, A090771, A091084, A141158, A146763, A197652.

[(10*n-11+(-1)^n+(4+2*(-1)^n)*(-1)^((2*n-1+(-1)^n) div 4))/4: n in [1..60]]; // Vincenzo Librandi, Dec 27 2015

A266297:=n->(10*n-11+(-1)^n+(4+2*(-1)^n)*(-1)^((2*n-1+(-1)^n)/4))/4: seq(A266297(n), n=1..100);

Table[(10 n - 11 + (-1)^n + (4 + 2 (-1)^n)*(-1)^((2 n - 1 + (-1)^n)/4))/4, {n, 50}] (* G. C. Greubel, Dec 27 2015 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {0, 1, 4, 9, 10}, 60] (* Vincenzo Librandi, Dec 27 2015 *)
CoefficientList[Series[x*(1 + 3*x + 5*x^2 + x^3)/((x - 1)^2*(1 + x + x^2 + x^3)), {x, 0, 100}], x] (* Wesley Ivan Hurt, Dec 30 2015 *)
Flatten[Table[10n + {0, 1, 4, 9}, {n, 0, 19}]] (* Alonso del Arte, Dec 30 2015 *)
Select[Range[0,150],MemberQ[{0,1,4,9},Mod[#,10]]&] (* Harvey P. Dale, Jul 30 2019 *)

is(n) = issquare(n%10); \\ Altug Alkan, Dec 29 2015

G.f.: x^2*(1 + 3*x + 5*x^2 + x^3)/((x - 1)^2*(1 + x + x^2 + x^3)).
a(n) = a(n - 1) + (n - 4) - a(n - 5) for n > 5.
a(n) = (10n - 11 + (-1)^n + (4 + 2(-1)^n) * (-1)^((2n - 1 + (-1)^n)/4))/4.
a(n+1) - a(n) = A091084(n+1) for n>0.
Sum_{n>=2} (-1)^n/a(n) = (14*sqrt(5)*arccoth(sqrt(5)) - 2*Pi*sqrt(1-2/sqrt(5)) + 16*log(2) + 5*log(5))/40. - Amiram Eldar, Jul 30 2024

A348544 Positive integers that are equal both to the product of two integers ending with 3 and to that of two integers ending with 7.

Original entry on oeis.org

189, 399, 459, 609, 729, 819, 969, 999, 1029, 1239, 1269, 1449, 1479, 1539, 1659, 1729, 1809, 1869, 1989, 2079, 2109, 2289, 2349, 2499, 2619, 2639, 2679, 2709, 2889, 2919, 3009, 3059, 3129, 3159, 3219, 3249, 3339, 3429, 3519, 3549, 3699, 3759, 3819, 3969, 4029

Offset: 1

Stefano Spezia, Oct 22 2021

Intersection of A346950 and A348054.

			189 = 7*27 = 3*63, 399 = 3*133 = 7*57, 459 = 3*153 = 17*27, 609 = 3*203 = 7*87, ...

Cf. A017377 (supersequence), A346950, A348054, A348546.

max=4050; Select[Intersection[Union@Flatten@Table[a*b, {a, 3, Floor[max/3], 10}, {b, a, Floor[max/a], 10}], Union@Flatten@Table[a*b, {a, 7, Floor[max/7], 10}, {b, a, Floor[max/a], 10}]], 0<#

isok(m) = my(ok3=0, ok7=0); fordiv(m, d, if (((d % 10) == 3) && ((m/d % 10) == 3), ok3++); if (((d % 10) == 7) && ((m/d % 10) == 7), ok7++); if (ok3 && ok7, return(1))); \\ Michel Marcus, Oct 22 2021

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)) & set(a*b for a in range(7, lim//7+1, 10) for b in range(a, lim//a+1, 10)))
print(aupto(4029)) # Michael S. Branicky, Oct 22 2021

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

A348545 Positive integers with final digit 9 that are equal to the product of two integers ending with the same digit.

Original entry on oeis.org

9, 39, 49, 69, 99, 119, 129, 159, 169, 189, 219, 249, 259, 279, 289, 299, 309, 329, 339, 369, 399, 429, 459, 469, 489, 519, 529, 539, 549, 559, 579, 609, 629, 639, 669, 679, 689, 699, 729, 749, 759, 789, 799, 819, 849, 879, 889, 909, 939, 949, 959, 969, 989, 999

Offset: 1

Stefano Spezia, Oct 22 2021

Union of A346950 and A348054.

			9 = 3*3, 39 = 3*13, 49 = 7*7, 69 = 3*23, 99 = 3*33, 119 = 7*17, 129 = 3*43, 159 = 3*53, 169 = 13*13, 189 = 3*63 = 7*27, ...

Cf. A017377 (supersequence), A346950, A348054, A348547.

a={}; For[n=0, n<=100, 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] || 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

isok(m) = ((m%10) == 9) && sumdiv(m, d, (d % 10) == (m/d % 10)); \\ Michel Marcus, Oct 22 2021

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)) | set(a*b for a in range(7, lim//7+1, 10) for b in range(a, lim//a+1, 10)))
print(aupto(999)) # Michael S. Branicky, Oct 22 2021

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

A157917 A decimation of the EKG sequence: A064413(10n+9).

Original entry on oeis.org

10, 22, 39, 45, 56, 66, 75, 88, 47, 106, 117, 133, 201, 219, 162, 166, 180, 189, 101, 216, 225, 234, 242, 258, 275, 137, 286, 149, 306, 315, 341, 343, 519, 360, 368, 361, 416, 402, 391, 434, 425, 446, 454, 687, 476, 717, 495, 518, 513, 524, 538, 549, 554, 562, 283, 588

Offset: 0

Paul Curtz, Mar 09 2009

nonn

The average growth is roughly 10.5*n, and two streaks of subsequences growing roughly proportional to 15.6*n and 5.2*n appear also in the graph.

a(n)=A064413(A017377(n)).

Edited and extended by R. J. Mathar, Mar 15 2009

A270968 Reduced 5x+1 function R applied to the odd integers: a(n) = R(2n-1), where R(k) = (5k+1)/2^r, with r as large as possible.

Original entry on oeis.org

3, 1, 13, 9, 23, 7, 33, 19, 43, 3, 53, 29, 63, 17, 73, 39, 83, 11, 93, 49, 103, 27, 113, 59, 123, 1, 133, 69, 143, 37, 153, 79, 163, 21, 173, 89, 183, 47, 193, 99, 203, 13, 213, 109, 223, 57, 233, 119, 243, 31, 253, 129, 263, 67, 273, 139, 283, 9, 293, 149, 303

Offset: 1

Michel Lagneau, Mar 27 2016

The odd-indexed terms a(2i+1) = 10i+3 = A017305(i), i>=0;
a(4i+4) = 10i+9 = A017377(i), i>=0;
a(8i+6) = 10i+7 = A017353(i), i>=0;
a(16i+2) = 10i+1 = A017281(i), i>=0.
Note that a(n) = a(16n-6) = a(6n-2)/3. No multiple of 5 is in this sequence.
a(n) = R(2n-1) < 2n-1 for n = 2, 6, 10, ..., 2+4i,...

			a(4)=9 because (2*4-1) = 7  -> (5*7+1)/2^2 = 9.

Cf. A075677, A017281, A017305, A017353, A017377, A133423, A133424, A174795, A133419, A133420, A232711, A259207, A267969.

nextOddK[n_] := Module[{m=5n+1}, While[EvenQ[m], m=m/2]; m]; (* assumes odd n *) Table[nextOddK[n], {n, 1, 200, 2}]

a(n) = my(m = 2*n-1, c = 5*m+1); c/2^valuation(c, 2); \\ Michel Marcus, Mar 27 2016

a(n) = A000265(A017341(n-1)). - Michel Marcus, Mar 27 2016

cp's OEIS Frontend

A154419 Primes of the form 20*k^2 + 36*k + 17.

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A161365 a(n) = 3/2 + 5*n - 5*(-1)^n/2.

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A168419 a(n) = 9*floor(n/2).

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A239794 5*n^2 + 4*n - 15.

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A250024 a(n) = 40*n - 21.

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A266297 Numbers whose last digit is a square.

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A348544 Positive integers that are equal both to the product of two integers ending with 3 and to that of two integers ending with 7.

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A154419 Primes of the form 20k^2 + 36k + 17.

A161365 a(n) = 3/2 + 5n - 5(-1)^n/2.

A239794 5n^2 + 4n - 15.