Toshitaka Suzuki - cp's OEIS frontend

A384507 Initial term of first run of exactly n consecutive numbers with exactly 5 distinct prime factors.

2310, 254540, 1042404, 21871365, 129963314, 830692265, 4617927894, 18297409143, 41268813542, 287980277114, 1182325618032, 6455097761454, 14207465691240, 54049709480208, 90987640183352, 1164838922284960, 546525829796442

Offset: 1

Toshitaka Suzuki, Jun 23 2025

See A087978 for further details and an explanation of why this sequence is finite.
First differs from A087978 at n=16.
a(18) > 2 * 10^15.

			a(16) > a(17) because the first run of 16 consecutive integers i with A001221(i)=5 is not a maximal run.

Cf. A001221, A046387, A080569, A087977, A087978, A185032, A185042.

A378928 Numbers k such that pi(k) = rad(k), where pi = A000720, rad = A007947.

Original entry on oeis.org

4, 120, 350, 1116, 64700, 1304511, 3523884, 9559575, 189961956, 189964332, 514275185, 514275546, 1394194200, 3779851131, 75370121712, 204475052375, 204475052625, 204475053625, 204475054125, 204475054750

Offset: 1

Toshitaka Suzuki, Dec 11 2024

a(21) > 10^12.

			a(1) = 4 because the first solution of pi(x) = rad(x) is 4 since pi(4) = rad(4) = 2.

Cf. A000720 (pi), A007947 (rad).

A373859 Smallest prime obtained by appending one or more 9's to n, -1 if no such prime exists.

Original entry on oeis.org

19, 29, -1, 499, 59, -1, 79, 89, -1, 109, 1199999, -1, 139, 149, -1, 1699, 179, -1, 199, 2099, -1, 229, 239, -1, 25999, 269, -1, 289999, 2999, -1, 319999999999999999999999999999, 3299, -1, 349, 359, -1, 379, 389, -1, 409, 419, -1, 439, 449, -1

Offset: 1

Toshitaka Suzuki, Jun 19 2024

Toshitaka Suzuki, Table of n, a(n) for n = 1..448

Cf. A030433, A373201, A112386, A372056, A113076.

from itertools import count
from sympy import isprime
def A373859(n): return next(p for p in ((n+1)*10**m-1 for m in count(1)) if isprime(p)) if n%3 else -1 # Chai Wah Wu, Jul 08 2024

a(11) = 1199999 because 119, 1199, 11999 and 119999 are not primes.

A373201 a(n) = smallest number m > 0 such that n followed by m 9's yields a prime, or -1 if no such m exists.

Original entry on oeis.org

1, 1, -1, 2, 1, -1, 1, 1, -1, 1, 5, -1, 1, 1, -1, 2, 1, -1, 1, 2, -1, 1, 1, -1, 3, 1, -1, 4, 2, -1, 28, 2, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 16, 1, -1, 1, 1, -1, 3, 2, -1, 5, 1, -1, 15, 1, -1, 1, 2, -1, 12, 1, -1, 4, 2, -1, 1, 1, -1, 1, 2, -1, 1, 3

Offset: 1

Toshitaka Suzuki, May 27 2024

a(n) = -1 when n = 3*k because no matter how many 9's are appended to n, the resulting number is always divisible by 3 and therefore cannot be prime.
a(n) = -1 when n = 37037*k + 8424, 10175, 10998, 12210, 17600, 19360, 25826, 27146, 27685, 27861, 35915 or 36047, because n followed by any positive number, m say, of 9's is divisible by at least one of the primes {7,11,13,37}.
a(4420) > 300000 or a(4420) = -1.

			a(25)=3 because 259 and 2599 are composite but 25999 is prime.

Toshitaka Suzuki, Table of n, a(n) for n = 1..4419

Cf. A090465, A069568, A372262, A363922.

A372262 a(n) = smallest number m > 0 such that n followed by m 3's yields a prime, or -1 if no such m exists.

Original entry on oeis.org

1, 1, -1, 1, 1, -1, 1, 1, -1, 1, 1, -1, 14, 2, -1, 1, 1, -1, 1, 3, -1, 1, 1, -1, 8, 1, -1, 1, 1, -1, 1, 4, -1, 2, 1, -1, 1, 1, -1, 483, 2, -1, 1, 1, -1, 1, 2, -1, 2, 1, -1, 1, 2, -1, 3, 1, -1, 6, 1, -1, 1, 5, -1, 1, 1, -1, 1, 1, -1, 5, 3, -1, 1, 1, -1, 3, 1, -1, 2, 4

Offset: 1

Toshitaka Suzuki, Apr 24 2024

a(n) = -1 when n = 3*k because no matter how many 3's are appended to n, the resulting number is always divisible by 3 and therefore cannot be prime.
a(n) = -1 when n = 37037*k + 2808, 3666, 4070, 9287, 18799, 21574, 28083, 30558, 33300, 33740, 36663 or 36707, because n followed by any positive number, m say, of 3's is divisible by at least one of the primes {7,11,13,37}.
a(817) > 300000 or a(817) = -1.

			a(20)=3 because 203 and 2033 are composite but 20333 is prime.

Toshitaka Suzuki, Table of n, a(n) for n = 1..816

Cf. A372056, A090584, A069568, A363922.

snm[n_]:=Module[{k=1},If[Mod[n,3]==0,-1,While[CompositeQ[FromDigits[ PadRight[ IntegerDigits[ n],k+ IntegerLength[ n],3]]],k++];k]]; Array[snm,80] (* Harvey P. Dale, Aug 06 2024 *)

A372056 Smallest prime obtained by appending one or more 3's to n, or -1 if no such prime exists.

Original entry on oeis.org

13, 23, -1, 43, 53, -1, 73, 83, -1, 103, 113, -1, 1333333333333333, 1433, -1, 163, 173, -1, 193, 20333, -1, 223, 233, -1, 2533333333, 263, -1, 283, 293, -1, 313, 323333, -1, 3433, 353, -1, 373, 383, -1

Offset: 1

Toshitaka Suzuki, Mar 30 2024

Next term is 40 followed by 483 3's and is too large to display here (see the b-file).

			For n = 13, a(13) = 1333333333333333 is a prime (but 133,1333,13333 etc. are not primes).

Toshitaka Suzuki, Table of n, a(n) for n = 1..409

See A112394 for another version.

Cf. A030431, A090584, A112386, A113076.

Edited by N. J. A. Sloane, Apr 24 2024

A363922 a(n) = smallest number m > 0 such that n followed by m 7's yields a prime, or -1 if no such m exists.

Original entry on oeis.org

1, 2, 1, 1, 2, 1, -1, 2, 1, 1, 3, 1, 1, -1, 1, 1, 2, 2, 1, 6, -1, 1, 2, 2, 1, 2, 1, -1, 48, 1, 1, 5, 1, 1, -1, 1, 10, 2, 1, 12, 2, -1, 3, 3, 1, 1, 3, 1, -1, 2, 8, 7, 3, 1, 1, -1, 1, 1, 9, 1, 1, 2, -1, 1, 2, 5, 1, 3, 2, -1, 2, 1, 66, 2, 1, 3, -1, 1, 1, 3

Offset: 1

Toshitaka Suzuki, Jul 12 2023

a(n) = -1 when n = 7*k because no matter how many 7's are appended to n, the resulting number is always divisible by 7 and therefore cannot be prime.
a(n) = -1 when n = 15873*k + 891, 1261, 2889, 3263, 3300, 7810, 8917, 9812, 12617, 13024, 14615 or 15066, because n followed by any positive number, m say, of 7's is divisible by at least one of the primes {3,11,13,37}.
Similarly,
a(n) = -1 when n = 11111111*k + 964146, 1207525, 2342974, 3567630, 7525789, 8134540, 8591231 or 9641467 by primes {11,73,101,137};
a(n) = -1 when n = 429000429*k + 23928593, 27079312, 36492115, 41207969, 52285750, 80569929, 89920882, 93857078, 133928703, 217208145, 223492302, 236849444, 239285937, 247857232, 259793116, 270793127, 323985244, 332698824, 333570182, 334985255, 346849554, 364921157, 376698868 or 412079697 by primes {3,11,13,101,9901};
a(n) = -1 when n = 1221001221*k + 14569863, 28792885, 145698637, 167698659, 225079510, 235985156, 247079532, 287928857, 331921124, 399492478, 415286113, 421492500, 437286135, 455985376, 489857474, 529929099, 551921344, 635208563, 709857694, 877208805, 896850104, 993570842, 1029793886 or 1138850346 by primes {3,11,37,101,9901};
a(n) = -1 when n = 1443001443*k + 85928655, 167698659, 176928746, 218921011, 233985154, 247079532, 310492389, 326286024, 376857361, 585793442, 655208583, 700699192, 746208674, 780080065, 791570640, 805850013, 843492922, 859286557, 882570731, 896850104, 1027793884, 1219922012, 1234986155 or 1377858362 by primes {3,13,37,101,9901}.
a(4444) > 300000 or a(4444) = -1.

			a(11)=3 because 117 and 1177 are composite but 11777 is prime.

Toshitaka Suzuki, Table of n, a(n) for n = 1..4443

Cf. A008589, A069568, A090464, A113076.

a(n) = if ((n%7), my(m=1); while (!isprime(eval(concat(Str(n), Str(7*(10^m-1)/9)))), m++); m, -1); \\ Michel Marcus, Jul 17 2023

A104519 Sufficient number of monominoes to exclude X-pentominoes from an n X n board.

Original entry on oeis.org

1, 2, 3, 4, 7, 10, 12, 16, 20, 24, 29, 35, 40, 47, 53, 60, 68, 76, 84, 92, 101, 111, 121, 131, 141, 152, 164, 176, 188, 200, 213, 227, 241, 255, 269, 284, 300, 316, 332, 348, 365, 383, 401, 419, 437, 456, 476, 496, 516, 536, 557, 579, 601, 623, 645, 668, 692

Offset: 3

Toshitaka Suzuki, Apr 19 2005

a(n+2) is also the domination number (size of minimum dominating set) in an n X n grid graph (Alanko et al.).

Apparently also the minimal number of X-polyominoes needed to cover an n X n board. - Rob Pratt, Jan 03 2008

Harvey P. Dale, Table of n, a(n) for n = 3..1000
Samu Alanko, Simon Crevals, Anton Isopoussu, Patric R. J. Östergård, and Ville Pettersson, Computing the Domination Number of Grid Graphs, The Electronic Journal of Combinatorics, 18 (2011), #P141.
Daniel Gonçalves, Alexandre Pinlou, Michaël Rao, and Stéphan Thomassé, The Domination Number of Grids, SIAM Journal on Discrete Mathematics, 25 (2011), 1443.
Stephan Mertens, Domination Polynomials of the Grid, the Cylinder, the Torus, and the King Graph, arXiv:2408.08053 [math.CO], 2024. See p. 15.
Eric Weisstein's World of Mathematics, Domination Number
Eric Weisstein's World of Mathematics, Grid Graph
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).

Cf. A193764, A269706 (size of a minimum dominating set in an n X n X n grid).

Table[Piecewise[{{n - 2, n <= 6}, {7, n == 7}, {10, n == 8}, {40, n == 15}}, Floor[n^2/5] - 4], {n, 3, 51}] (* Eric W. Weisstein, Apr 12 2016 *)
LinearRecurrence[{2,-1,0,0,1,-2,1},{1,2,3,4,7,10,12,16,20,24,29,35,40,47,53,60,68,76,84,92},60] (* Harvey P. Dale, Aug 30 2024 *)

a(n) = n^2 - A193764(n). - Colin Barker, Oct 05 2014
Empirical g.f.: -x^3*(x^19 -2*x^18 +x^17 -x^14 +2*x^13 -3*x^12 +2*x^11 +x^10 -2*x^9 +2*x^7 -x^6 -x^5 +2*x^4 +1) / ((x -1)^3*(x^4 +x^3 +x^2 +x +1)) - Colin Barker, Oct 05 2014
Empirical recurrence a(n) = 2*a(n-1)-a(n-2)+a(n-5)-2*a(n-6)+a(n-7) with a(3)=-3, a(4)=-1, a(5)=1, a(6)=3, a(7)=5, a(8)=8, a(9)=12 matches the sequence for 9 <= n <= 14 and 16 <= n <= 51. - Eric W. Weisstein, Jun 27 2017
a(n) = floor(n^2/5) - 4 for n > 15. (Conçalves et al.) - Stephan Mertens, Jan 24 2024
Empirical g.f. and recurrence confirmed by above formula. - Ray Chandler, Jan 25 2024

Extended to a(29) by Alanko et al.

More terms from Colin Barker, Oct 05 2014

A103535 Number of nets in a regular right prism.

Original entry on oeis.org

9, 29, 99, 354, 1290, 4762, 17663, 65733, 244923, 913383, 3407329, 12713796, 47443092, 177050612, 660741597, 2465886087, 9202736493, 34344949105, 128176812671, 478361888166, 1785269817246, 6662715837966, 24865590090907, 92799638767689, 346332952127775

Offset: 3

Toshitaka Suzuki, Mar 22 2005

nonn

The second term is the number of nets in a general regular right 4-prism, not in a cube.

Takashi Horiyama and Wataru Shoji, The Number of Different Unfoldings of Polyhedra. In: L. Cai, S.-W. Cheng, and T.-W. Lam (Eds.): ISAAC2013, LNCS 8283, pp. 623-633, Springer-Verlag, 2013.

a(n) = 1/(8*sqrt(3))*( 2*sqrt(3)*n + sqrt(3)*(2 + sqrt(3))^n + (2 + sqrt(3))^floor(n/2)*(4+2*sqrt(3)) + (2 - sqrt(3))^floor(n/2)*(2*sqrt(3) - 4) + sqrt(3)*((2 - sqrt(3))^n - 2)) for n >= 3 odd; a(n) = 1/24*(6*n + 3*(2 + sqrt(3))^n + 4*sqrt(3)*(2 + sqrt(3))^(n/2) - 4*sqrt(3)*(2 - sqrt(3))^(n/2) + 3*(2 - sqrt(3))^n - 6) for n >= 4 even. - Humberto Bortolossi, Mar 31 2017

Empirical g.f.: x^3*(9 - 25*x - 21*x^2 + 96*x^3 - 60*x^4 - 12*x^5 + 17*x^6 - 3*x^7) / ((1 - x)^2*(1 - 4*x + x^2)*(1 - 4*x^2 + x^4)). - Colin Barker, Mar 31 2017

More terms from Alois P. Heinz, Mar 31 2017

A103536 Number of geometrically distinct edge-unfoldings of a regular n-gonal pyramid.

Original entry on oeis.org

4, 8, 15, 33, 67, 152, 340, 791, 1845, 4411, 10557, 25600, 62332, 152780, 375859, 928841, 2302191, 5724425, 14269196, 35655157, 89277769, 223982893, 562912585, 1417014038, 3572323492, 9018370892, 22796073015, 57691327693, 146165207035, 370706641856, 941111617892, 2391394225355, 6081869637093

Offset: 3

Toshitaka Suzuki, Mar 22 2005

nonn

The first term is the number of nets of a general regular triangular pyramid, not of a tetrahedron.

Z. Lengvárszky and Rick Mabry, Enumerating nets of prism-like polyhedra, Acta Sci. Math. (Szeged) 83:3-4 (2017), 377-392.

-1 + (1/2) Fibonacci[n] + (1/(2 n)) Sum[Fibonacci[2 GCD[j, n] - 1] + Fibonacci[2 GCD[j, n] + 1], {j, 1, n}] (* Rick Mabry, Apr 10 2023 *)

a(n) = {sum(j=1, n, fibonacci(2*gcd(j,n) - 1) + fibonacci(2*gcd(j,n) + 1))/(2*n) + fibonacci(n)/2 - 1} \\ Andrew Howroyd, Apr 10 2023

a(n) = -1 + (1/2)*F(n) + (1/(2*n))*Sum_{j=1..n} (F(2*gcd(j,n)-1) + F(2*gcd(j,n)+1)), where F(n) is the usual n-th Fibonacci number. Simplified from link. - Rick Mabry, Apr 10 2023

More terms from Rick Mabry, Apr 10 2023

Definition changed by Rick Mabry, Apr 17 2023

cp's OEIS Frontend

User: Toshitaka Suzuki

A384507 Initial term of first run of exactly n consecutive numbers with exactly 5 distinct prime factors.

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A378928 Numbers k such that pi(k) = rad(k), where pi = A000720, rad = A007947.

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A373859 Smallest prime obtained by appending one or more 9's to n, -1 if no such prime exists.

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A373201 a(n) = smallest number m > 0 such that n followed by m 9's yields a prime, or -1 if no such m exists.

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A372262 a(n) = smallest number m > 0 such that n followed by m 3's yields a prime, or -1 if no such m exists.

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A372056 Smallest prime obtained by appending one or more 3's to n, or -1 if no such prime exists.

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A363922 a(n) = smallest number m > 0 such that n followed by m 7's yields a prime, or -1 if no such m exists.

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A104519 Sufficient number of monominoes to exclude X-pentominoes from an n X n board.

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A103535 Number of nets in a regular right prism.

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