Don N. Page has authored 10 sequences.
A370531
The smallest number in base n such that two digits (and no fewer) need to be changed to get a prime.
Original entry on oeis.org
8, 24, 24, 90, 90, 119, 200, 117, 200, 319, 528, 1131, 1134, 525, 1328, 1343, 1332, 1330, 1340, 2478, 7260, 1334, 5352, 4300, 5954, 4833, 13188, 8468, 10800, 15686, 11744, 19338, 19618, 22575, 19620, 15688, 28234, 19617, 25480, 31406, 19614, 40291, 25476, 31410
Offset: 2
a(2) = 8 = 1000_2 can be changed to the prime 1011_2 (11 in decimal) by changing the last two digits. Although 4 = 100_2 can be changed to the prime 111_2 by changing two digits, it can also be changed to the prime 101_2 by only one base-2 digit, so 4 is not a(2).
a(3) = 24 = 220_3 can be changed to 212_3 = 23. 24 is not prime and no single base-3 digit change works.
a(4) = 24 = 120_4 can be changed to 113_4 = 23.
a(5) = 90 = 330_5 -> 324_5 = 89.
a(6) = 90 = 230_6 -> 225_6 = 89.
a(7) = 119 = 230_7 -> 221_7 = 113.
a(8) = 200 = 310_8 -> 307_8 = 199.
a(9) = 117 = 140_9 -> 135_9 = 113.
Often, there are alternative ways to change two digits to get alternative primes, but for each a(n), there is not any way to get a prime by changing 0 or 1 digits in base n.
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from sympy import isprime
from sympy.ntheory import digits
from itertools import combinations, count, product
def fromdigits(d, b): return sum(di*b**i for i, di in enumerate(d[::-1]))
def PEN(base, k):
if isprime(k): return 0
d = digits(k, base)[1:]
for j in range(1, len(d)+1):
for c in combinations(range(len(d)), j):
for p in product(*[[i for i in range(base) if i!=d[c[m]]] for m in range(j)]):
dd = d[:]
for i in range(j): dd[c[i]] = p[i]
if isprime(fromdigits(dd, base)): return j
def a(n): return next(k for k in count(n) if PEN(n, k) == 2)
print([a(n) for n in range(2, 32)]) # Michael S. Branicky, Feb 21 2024
A367992
Smallest nonprime that is the n-th prime plus a multiple of the (n-1)-st primorial.
Original entry on oeis.org
4, 9, 35, 187, 221, 2323, 120137, 1021039, 19399403, 223092899, 6469693261, 200560490167, 7420738134851, 304250263527253, 39248283995010137, 614889782588491463, 65178316954380089519, 3845520700308425278201, 117288381359406970983337, 7858321551080267055879161
Offset: 1
For n = 1, the n-th prime (2) plus a multiple m of the (n-1) primorial (1) is 2+m, giving the smallest nonprime, a(1)=4, when m=2.
For n = 4, the n-th prime (7) plus a multiple m of the (n-1) primorial (30) is 7+30m, giving the smallest nonprime, a(4)=187, when m=6.
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a[n_] := Module[{p = Prime[n], r = Product[Prime[i], {i, 1, n - 1}]}, While[p += r; PrimeQ[p]]; p]; Array[a, 20] (* Amiram Eldar, Dec 07 2023 *)
A370572
The smallest number which in base n requires 3 digit changes to convert k into a prime.
Original entry on oeis.org
84, 1953, 34560, 7000485, 354748446, 77478704205, 1878528135128, 48398467146642
Offset: 2
A250302
Positive integers k whose square is, for some prime p, the largest integer m such that both m and m-1 factor into primes less than or equal to p.
Original entry on oeis.org
3, 9, 99, 351, 13311, 1205645, 138982582999
Offset: 1
Here are the largest pairs of consecutive integers with prime factors p or smaller:
p : pair
--------------------------
3 : 3^2-1 and 3^2;
5 : 9^2-1 and 9^2;
11 : 99^2-1 and 99^2;
13 : 351^2-1 and 351^2;
29 : 13311^2-1 and 13311^2;
53 : 1205645^2-1 and 1205645^2;
103 : 138982582999^2-1 and 138982582999^2.
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lista(v_002072) = {v = v_002072; for (i=1, #v, vi = v[i]; if (issquare(vi+1), print1(sqrtint(vi+1), ", ")););} \\ Michel Marcus, Feb 28 2015
A250298
Primes p such that the largest integer m such that both m and m-1 factor into primes less than or equal to p is a perfect square, m = k^2.
Original entry on oeis.org
3, 5, 11, 13, 29, 53, 103
Offset: 1
p = 3 gives m = 3^2;
p = 5 gives m = 9^2;
p = 11 gives m = 99^2;
p = 13 gives m = 351^2;
p = 29 gives m = 13311^2;
p = 53 gives m = 1205645^2;
p = 103 gives m = 138982582999^2.
A228611
Primes p such that the largest consecutive pair of p-smooth integers is the same as the largest consecutive pair of (p-1)-smooth integers.
Original entry on oeis.org
23, 67, 83, 89, 97, 101
Offset: 1
For n = 1, a(1) = 23 is a prime such that the largest consecutive pair of 23-smooth integers, (11859210,11859211), is the same as the largest consecutive pair of 22-smooth integers (or of 19-smooth integers, 19 being the next smaller prime).
A228610
Numbers k such that the largest consecutive pair of prime(k)-smooth integers is the same as the largest consecutive pair of prime(k-1)-smooth integers.
Original entry on oeis.org
9, 19, 23, 24, 25, 26
Offset: 1
For n = 1, k = a(1) = 9 gives prime(k) = 23 such that the largest consecutive pair of 23-smooth integers, (11859210,11859211), is the same as the largest consecutive pair of prime(k-1)-smooth integers (19-smooth integers).
A111203
x such that pi(x)/li(x) is greater than it is for all smaller x > 1.5.
Original entry on oeis.org
2, 1051, 1063, 1069, 1097, 1103, 1109, 1123, 1129, 1303, 1307, 1321, 1327, 1619, 1621, 1627, 2399, 2447, 2477, 2719, 2731, 2753, 2801, 2803, 3929, 3931, 3947, 4273, 4289, 4297, 5851, 5857, 5861, 5867, 5869, 5881, 6367, 6373, 6379, 9433, 9437, 9439
Offset: 1
For 1.5<x<2, li(x)>0 and pi(x)=0, so pi(x)/li(x)=0. a(1)=2 because at x=2, pi(x)/li(x) has its increase, to 1/li(2)=0.9567878442. a(2)=1051 because x=1051 gives the next time pi(x)/li(x) gives a higher value, 177/Li(1051)=0.956932676.
A104767
a(n)=n for n <= 3, a(n) = 2a(n-1) - 2a(n-2) + 2a(n-3) for n >= 4.
Original entry on oeis.org
0, 1, 2, 3, 4, 6, 10, 16, 24, 36, 56, 88, 136, 208, 320, 496, 768, 1184, 1824, 2816, 4352, 6720, 10368, 16000, 24704, 38144, 58880, 90880, 140288, 216576, 334336, 516096, 796672, 1229824, 1898496, 2930688, 4524032, 6983680, 10780672, 16642048, 25690112, 39657472
Offset: 0
From _Michael Somos_, May 17 2018: (Start)
For n=3, (x - y) * (x - y) = x^2 - 2*x*y + y^2 has a(3) = 3 terms.
For n=4, (x - y) * (x - y) * (x^2 - y^2) = x^4 - 2*x^3*y + 2*x*y^3 - y^4 has a(4) = 4 terms.
for n=2, (x - y) * (x^2 - y^2) = x^3 - x^2*y - x*y^2 + y^3 has a(2) = 2 terms with + sign and also with - sign.
For n=3, (x - y) * (x^2 - y^2) * (x^3 - y^3) = x^6 - x^5*y - x^4*y^2 + x^2*y^4 + x*y^5 - y^6 has a(3) = 3 terms with + sign and also with - sign. (End)
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a:=[0,1,2,3,4];; for n in [5..50] do a[n]:=2*a[n-1]-2*a[n-2]+2*a[n-3]; od; a; # Muniru A Asiru, May 17 2018
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f:=proc(n) option remember; if n <= 4 then RETURN(n); fi; 2*f(n-4)+f(n-1); end;
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a[n_] := a[n] = If[n < 4, n, 2a[n - 1] - 2a[n - 2] + 2a[n - 3]]; Table[ a[n], {n, 0, 39}] (* Robert G. Wilson v *)
Join[{0}, LinearRecurrence[{2, -2, 2}, {1, 2, 3}, 41]] (* Robert G. Wilson v, May 12 2013 *)
Join[{0}, LinearRecurrence[{1, 0, 0, 2}, {1, 2, 3, 4}, 41]] (* Robert G. Wilson v, May 12 2013 *)
a[n_] := Length@ ExpandAll@ Product[1 - x^Fibonacci[k], {k, n-1}]; a[1] = 1; (* Robert G. Wilson v, May 12 2013 *)
nxt[{a_,b_,c_}]:={b,c,2c-2b+2a}; Join[{0},NestList[nxt,{1,2,3},40][[All,1]]] (* Harvey P. Dale, Nov 30 2021 *)
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a=vector(100); a[1]=1;a[2]=2;a[3]=3; for(n=4, #a, a[n] = 2*a[n-1]-2*a[n-2]+2*a[n-3]); concat(0,a) \\ Altug Alkan, May 18 2018
A029549
a(n + 3) = 35*a(n + 2) - 35*a(n + 1) + a(n), with a(0) = 0, a(1) = 6, a(2) = 210.
Original entry on oeis.org
0, 6, 210, 7140, 242556, 8239770, 279909630, 9508687656, 323015470680, 10973017315470, 372759573255306, 12662852473364940, 430164224521152660, 14612920781245825506, 496409142337836914550, 16863297918705209269200, 572855720093639278238256
Offset: 0
- Reinhard Zumkeller, Table of n, a(n) for n = 0..100
- H. J. Hindin, Stars, hexes, triangular numbers and Pythagorean triples, J. Rec. Math., 16 (1983/1984), 191-193. (Annotated scanned copy)
- Shyam Sunder Gupta Fascinating Triangular Numbers
- Roger B. Nelson, Multi-Polygonal Numbers, Mathematics Magazine, Vol. 89, No. 3 (June 2016), pp. 159-164.
- Vladimir Pletser, Recurrent Relations for Multiple of Triangular Numbers being Triangular Numbers, arXiv:2101.00998 [math.NT], 2021.
- Vladimir Pletser, Closed Form Equations for Triangular Numbers Multiple of Other Triangular Numbers, arXiv:2102.12392 [math.GM], 2021.
- Vladimir Pletser, Triangular Numbers Multiple of Triangular Numbers and Solutions of Pell Equations, arXiv:2102.13494 [math.NT], 2021.
- Vladimir Pletser, Congruence Properties of Indices of Triangular Numbers Multiple of Other Triangular Numbers, arXiv:2103.03019 [math.GM], 2021.
- Vladimir Pletser, Searching for multiple of triangular numbers being triangular numbers, 2021.
- Vladimir Pletser, Using Pell equation solutions to find all triangular numbers multiple of other triangular numbers, 2021.
- Index entries for linear recurrences with constant coefficients, signature (35,-35,1).
Cf.
A000217,
A001109,
A001652,
A002315,
A002378,
A011900,
A029546,
A046090,
A046729,
A053141,
A084159,
A096979,
A157259,
A165518.
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List([0..20], n-> (Lucas(2,-1, 4*n+2)[2] -6)/32 ); # G. C. Greubel, Jan 13 2020
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a029549 n = a029549_list !! n
a029549_list = [0,6,210] ++
zipWith (+) a029549_list
(map (* 35) $ tail delta)
where delta = zipWith (-) (tail a029549_list) a029549_list
-- Reinhard Zumkeller, Sep 19 2011
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(makelist(binom(n,2),n,1,999999),intersection(%%,2*%%)) /* Bill Gosper, Feb 07 2010 */
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R:=PowerSeriesRing(Integers(), 25); [0] cat Coefficients(R!(6/(1-35*x+35*x^2-x^3))); // G. C. Greubel, Jul 15 2018
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A029549 := proc(n)
option remember;
if n <= 1 then
op(n+1,[0,6]) ;
else
34*procname(n-1)-procname(n-2)+6 ;
end if;
end proc: # R. J. Mathar, Feb 05 2016
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Table[Floor[(Sqrt[2] + 1)^(4n + 2)/32], {n, 0, 20} ] (* Original program from author, corrected by Ray Chandler, Jul 09 2015 *)
CoefficientList[Series[6/(1 - 35x + 35x^2 - x^3), {x, 0, 14}], x]
Intersection[#, 2#] &@ Table[Binomial[n, 2], {n, 999999}] (* Bill Gosper, Feb 07 2010 *)
LinearRecurrence[{35, -35, 1}, {0, 6, 210}, 20] (* Harvey P. Dale, Jun 06 2011 *)
(LucasL[4Range[20] - 2, 2] -6)/32 (* G. C. Greubel, Jan 13 2020 *)
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concat(0,Vec(6/(1-35*x+35*x^2-x^3)+O(x^25))) \\ Charles R Greathouse IV, Jun 13 2013
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[(lucas_number2(4*n+2, 2, -1) -6)/32 for n in (0..20)] # G. C. Greubel, Jan 13 2020
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val triNums = (0 to 39999).map(n => (n * n + n)/2)
triNums.filter( % 2 == 0).filter(n => (triNums.contains(n/2))) // _Alonso del Arte, Jan 12 2020
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