A017329 -id:A017329 - cp's OEIS frontend

A017339 a(n) = (10*n + 5)^11.

48828125, 8649755859375, 2384185791015625, 96549157373046875, 1532278301220703125, 13931233916552734375, 87507831740087890625, 422351360321044921875, 1673432436896142578125, 5688000922764599609375, 17103393581163134765625, 46523913960640966796875, 116415321826934814453125

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A013669, A016763, A017329.

[(10*n+5)^11: n in [0..10]]; // Vincenzo Librandi, Aug 02 2011

Table[(10*n + 5)^11, {n, 0, 15}] (* Amiram Eldar, Apr 18 2023 *)

G.f.: 48828125*(x+1)*(x^10 + 177134*x^9 + 46525293*x^8 + 1356555432*x^7 + 9480267666*x^6 + 19107752148*x^5 + 9480267666*x^4 + 1356555432*x^3 + 46525293*x^2 + 177134*x + 1)/(x-1)^12. - Colin Barker, Nov 14 2012
From Amiram Eldar, Apr 18 2023: (Start)
a(n) = A017329(n)^11.
a(n) = 5^11 * A016763(n).
Sum_{n>=0} 1/a(n) = 2047*zeta(11)/100000000000.
Sum_{n>=0} (-1)^n/a(n) = 50521*Pi^11/725760000000000000. (End)

A017340 a(n) = (10*n + 5)^12.

Original entry on oeis.org

244140625, 129746337890625, 59604644775390625, 3379220508056640625, 68952523554931640625, 766217865410400390625, 5688009063105712890625, 31676352024078369140625, 142241757136172119140625, 540360087662636962890625, 1795856326022129150390625, 5350250105473711181640625

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A016764, A017329.

[(10*n+5)^12: n in [0..10]]; // Vincenzo Librandi, Aug 02 2011

(10*Range[0,20]+5)^12 (* Harvey P. Dale, May 09 2022 *)

G.f.: -244140625*(x^12 + 531428*x^11 + 237231970*x^10 + 10708911188*x^9 + 121383780207*x^8 + 477020564424*x^7 + 743288515164*x^6 + 477020564424*x^5 + 121383780207*x^4 + 10708911188*x^3 + 237231970*x^2 + 531428*x + 1)/(x-1)^13. - Colin Barker, Nov 14 2012
From Amiram Eldar, Apr 18 2023: (Start)
a(n) = A017329(n)^12.
a(n) = 5^12 * A016764(n).
Sum_{n>=0} 1/a(n) = 691*Pi^12/155925000000000000. (End)

A032587 Lucky numbers ending with digit 5.

Original entry on oeis.org

15, 25, 75, 105, 115, 135, 195, 205, 235, 285, 385, 415, 475, 495, 535, 615, 645, 655, 685, 735, 745, 805, 855, 885, 895, 925, 975, 1095, 1105, 1155, 1245, 1275, 1285, 1365, 1395, 1435, 1455, 1485, 1495, 1545, 1575, 1585, 1645, 1675, 1705, 1765, 1915

Offset: 1

Patrick De Geest, Apr 15 1998

Also, lucky numbers (A000959) which are congruent to 0 mod 5. - R. J. Mathar, Apr 29 2008

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

Intersection of A000959 and A017329.

A038860 Numbers ending with '5' that are the difference of two positive cubes.

Original entry on oeis.org

215, 335, 485, 665, 875, 1115, 1385, 1685, 2015, 2375, 2765, 3185, 3635, 4095, 4115, 4625, 4905, 5165, 5735, 5805, 6335, 6795, 6965, 7625, 7875, 8315, 9035, 9045, 9785, 10305, 10565, 11375, 11655, 12215, 13085, 13095, 13985, 14625, 14915, 15875

Offset: 1

Jeff Burch

Contains (k+5(2j+1))^3-k^3 for any integers k,j>=0. - M. F. Hasler, May 31 2007

Intersection of A017329 and A181123.

A038860(Nmax=20000,a=[]) = { local(t, j5); forstep( j=1,Nmax^(1/3)/5,2, j5=5*j; for(k=1, sqrt((Nmax/j5-j5^2-3*j5)/3), if(NmaxM. F. Hasler, Jun 07 2007

A number is in this sequence iff it is of the form (k+10j-5)^3-k^3, where k,j are any positive integers, since (k+d)^3 - k^3 = d(3(k+d/2)^2+d^2/4) == 5 (mod 10) iff d is odd and d == 0 (mod 5) (cf. A038853) - M. F. Hasler, Jun 07 2007

Corrected by M. F. Hasler, Jun 07 2007

A070330 Nontrivial numbers in which suffixing or inserting a 0 anywhere yields only composite numbers.

Original entry on oeis.org

1, 7, 23, 29, 31, 43, 47, 73, 77, 83, 91, 101, 103, 107, 127, 143, 157, 173, 199, 209, 221, 233, 241, 247, 251, 257, 259, 271, 307, 313, 317, 343, 353, 359, 373, 377, 391, 409, 431, 433, 437, 439, 443, 461, 467, 469, 487, 497, 527, 541, 563, 569, 583, 589, 601

Offset: 1

Amarnath Murthy, May 11 2002

"Nontrivial" excludes multiples of 2, 3, or 5.

Cf. A002808, A070331.

Cf. A008585, A005843, A017329.

More terms from Sascha Kurz, Feb 01 2003

a(1)=1 inserted by Sean A. Irvine, Jun 08 2024

A172292 Triangle read by rows: T(n, k) = (2n+1)(2*k+1), n>=1, 1<=k<=n.

Original entry on oeis.org

9, 15, 25, 21, 35, 49, 27, 45, 63, 81, 33, 55, 77, 99, 121, 39, 65, 91, 117, 143, 169, 45, 75, 105, 135, 165, 195, 225, 51, 85, 119, 153, 187, 221, 255, 289, 57, 95, 133, 171, 209, 247, 285, 323, 361, 63, 105, 147, 189, 231, 273, 315, 357, 399, 441, 69, 115, 161

Offset: 1

Vincenzo Librandi, Nov 24 2010

A number m belongs to this sequence if and only if it is odd and composite.
First column: A016945(n, n>=1), second column: A017329(n, n>=2), third column: A147587(n, n>=3). - Vincenzo Librandi, Nov 20 2012
The number of occurrences of m corresponds to the number of nontrivial factorizations of m, i.e., A072670(m-1). - Daniel Forgues, Apr 22 2014

			Triangle begins:
9;
15, 25;
21, 35,  49;
27, 45,  63,  81;
33, 55,  77,  99,  121;
39, 65,  91,  117, 143, 169;
45, 75,  105, 135, 165, 195, 225;
51, 85,  119, 153, 187, 221, 255, 289;
57, 95,  133, 171, 209, 247, 285, 323, 361;
63, 105, 147, 189, 231, 273, 315, 357, 399, 441; etc.
Number of occurrences:
  63 = 9*7 = 21*3 has two nontrivial factorizations, thus occurs twice.

Vincenzo Librandi, Rows n = 1..100, flattened
OEIS Wiki, Odd composites

Cf. A144562, A083487, A016945, A017329, A147587.

[4*n*k + 2*n + 2*k + 1: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 20 2012

t[n_,k_]:= 4 n*k + 2n + 2k + 1; Table[t[n, k], {n,15}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 20 2012 *)

T(n, k) = A144562(n,k)*2+3 read by rows. (Was old name.)

T(n, k) = 2*A083487(n, k)+1. - Daniel Forgues, Sep 20 2011

A187715 a(n) = 5*n - (9 + (-1)^n)/2.

Original entry on oeis.org

1, 5, 11, 15, 21, 25, 31, 35, 41, 45, 51, 55, 61, 65, 71, 75, 81, 85, 91, 95, 101, 105, 111, 115, 121, 125, 131, 135, 141, 145, 151, 155, 161, 165, 171, 175, 181, 185, 191, 195, 201, 205, 211, 215, 221, 225

Offset: 1

Vincenzo Librandi, Mar 13 2011

Numbers congruent to {1,5} mod 10. - Bruno Berselli, Mar 31 2012

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

Cf. A001622, A010711 (first differences), A017281, A017329.

Filtered([1..250],n-> n mod 10 =1 or n mod 10 = 5); # Muniru A Asiru, Nov 25 2018

Magma
```
[5*n -(9+(-1)^n)/2: n in [1..60]];
```

[5*n-(9+(-1)^n)/2$n=1..50]; # Muniru A Asiru, Nov 25 2018

nxt[{n_,a_}]:={n+1,If[EvenQ[n+1],a+4,a+6]}; Transpose[NestList[nxt,{1,1},50]][[2]] (* Harvey P. Dale, Feb 16 2013 *)
Table[BitOr[5*n, 1], {n, 0, 50}] (* Jon Maiga, Nov 24 2018 *)

vector(50, n, (10*n -9-(-1)^n)/2) \\ G. C. Greubel, Dec 04 2018

for n in range(1,60): print(int(5*n - (9 + (-1)**n)/2), end=', ') # Stefano Spezia, Nov 30 2018

[(10*n -9-(-1)^n)/2 for n in (1..50)] # G. C. Greubel, Dec 04 2018

a(n) = a(n-1) + 4 if n is even, a(n) = a(n-1) + 6 if n is odd.
a(n) = 2*a(n-1) - a(n-2) - 2*(-1)^n.
From R. J. Mathar, Mar 15 2011: (Start)
G.f.: x*(1 + 4*x + 5*x^2)/( (1+x)*(1-x)^2 ).
Bisections: a(2*n+1) = A017281(n), a(2*n) = A017329(n-1). (End)
a(n) = 5*(n-1) bitwise-OR 1. - Jon Maiga, Nov 24 2018
E.g.f.: ((10*x-9)*exp(x) - exp(-x) + 10)/2. - G. C. Greubel, Dec 04 2018
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(5+2*sqrt(5))*Pi/20 + 3*log(phi)/(4*sqrt(5)) + log(5)/8, where phi is the golden ratio (A001622). - Amiram Eldar, Apr 15 2023

Definition rewritten by R. J. Mathar, Mar 15 2011

A290781 a(n) = 20*n + 15.

Original entry on oeis.org

15, 35, 55, 75, 95, 115, 135, 155, 175, 195, 215, 235, 255, 275, 295, 315, 335, 355, 375, 395, 415, 435, 455, 475, 495, 515, 535, 555, 575, 595, 615, 635, 655, 675, 695, 715, 735, 755, 775, 795, 815, 835, 855, 875, 895, 915, 935, 955, 975, 995, 1015, 1035

Offset: 0

Arkadiusz Wesolowski, Aug 10 2017

Bisection of A017329.

None of the numbers in this sequence is a Fermat pseudoprime to base 2.

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A001567, A004767, A008587, A016885, A017329.

Magma
```
[n: n in [15..1035 by 20]];
```
Mathematica
```
Range[15, 1035, 20]
```

G.f.: 5*(3 + x)/(1 - x)^2.
a(n) = A004767(A016885(n)) = A004767(A004767(n) + n). - Torlach Rush, Oct 10 2019
E.g.f.: 5*exp(x)*(3 + 4*x). - Stefano Spezia, Oct 12 2019
From Elmo R. Oliveira, Apr 12 2025: (Start)
a(n) = 5*A004767(n) = A017329(2*n+1) = A008587(4*n+3).
a(n) = 2*a(n-1) - a(n-2). (End)

A063284 Dimension of the space of weight n cuspidal newforms for Gamma_1( 11 ).

Original entry on oeis.org

-1, 1, 5, 10, 15, 20, 25, 30, 35, 40, 45, 48, 55, 60, 65, 68, 75, 78, 85, 88, 95, 98, 105, 106, 115, 118, 125, 126, 135, 136, 145, 146, 155, 156, 165, 164, 175, 176, 185, 184, 195, 194, 205, 204, 215, 214, 225, 222, 235, 234, 245, 242, 255, 252

Offset: 2

N. J. A. Sloane, Jul 14 2001

sign

William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_1(N))
William A. Stein, The modular forms database
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 1, 0, 1, 0, 0, 0, -1).

Cf. A017329 (bisection),

From Colin Barker, Feb 24 2016: (Start)
a(n) = a(n-4) + a(n-6) - a(n-10) for n>13.
G.f.: -x^2*(1 -x -5*x^2 -10*x^3 -16*x^4 -19*x^5 -21*x^6 -19*x^7 -15*x^8 -10*x^9 -4*x^10 +x^11) / ((1 -x)^2*(1 +x)^2*(1 -x +x^2)*(1 +x^2)*(1 +x +x^2)).
(End)

A122124 Numbers n such that 25 divides Sum[ Prime[k]^n, {k,1,n}].

Original entry on oeis.org

3, 5, 7, 11, 15, 19, 23, 25, 27, 31, 35, 39, 43, 45, 47, 51, 55, 59, 63, 65, 67, 71, 75, 79, 83, 85, 87, 91, 95, 99, 103, 105, 107, 111, 115, 119, 123, 125, 127, 131, 135, 139, 143, 145, 147, 151, 155, 159, 163, 165, 167, 171, 175, 179, 183, 185, 187, 191, 195, 199

Offset: 1

Alexander Adamchuk, Aug 21 2006, Sep 18 2006, Sep 21 2006

nonn

a(n) up to a(7) = 23 coincides with A007665[n+1] = Tower of Hanoi with 5 pegs. It appears that a(n) includes all A007665[n] = {1, 3, 5, 7, 11, 15, 19, 23, 27, 31, 39, 47, 55, 63, 71, 79, 87, 95, 103, 111, 127, 143, 159, 175, 191, 207, 223, 239, 255, 271, 287, 303, 319, 335, 351, 383, 415, 447, 479, 511, 543, 575, 607, 639, 671, 703, 735, 767, 799, ...} except A007665[1] = 1.

Primes in this sequence include 5 and all primes of the form 4k+3, A002145[n]. Terms include all numbers of the form 10k+5 (with nonnegative k), A017329[n].

			There are 25 primes p < 100, p(n) = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97}.
a(1) = because 25 divides Sum[p(n)^3,{n,1,25}] = 2^3 + 3^3 + ... + 89^3 + 97^3 = A098999[25] and does not divide Sum[p(n)^1,{n,1,25}] = A007504[25] and Sum[p(n)^2,{n,1,25}] = A024450[25].
The next a(2) = 5 because 25 divides Sum[p(n)^5,{n,1,25}] = A122103[25] and does not divide Sum[p(n)^4,{n,1,25}] = A122102[25].

Cf. A007504, A024450, A098999, A122102, A122103, A007665.

Cf. A002144, A002145, A017329.

Select[Range[300],IntegerQ[Sum[ Prime[k]^#1, {k,1,25}]/25]&]

for(n=1,100,if(sum(k=1,25,prime(k)^n)%25==0,print1(n,",")));
print;print("Alternative method not using primes:");
for(n=1,100,m=(n-1)%6;print1((n-m)*3+(n-m+if(m>1,(m-1)*12-1,m*6-1))/3,",")) \\ K. Spage, Oct 23 2009

cp's OEIS Frontend

A017339 a(n) = (10*n + 5)^11.

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A017340 a(n) = (10*n + 5)^12.

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A032587 Lucky numbers ending with digit 5.

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A038860 Numbers ending with '5' that are the difference of two positive cubes.

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A070330 Nontrivial numbers in which suffixing or inserting a 0 anywhere yields only composite numbers.

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A172292 Triangle read by rows: T(n, k) = (2*n+1)*(2*k+1), n>=1, 1<=k<=n.

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A187715 a(n) = 5*n - (9 + (-1)^n)/2.

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A290781 a(n) = 20*n + 15.

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A172292 Triangle read by rows: T(n, k) = (2n+1)(2*k+1), n>=1, 1<=k<=n.