Brooke Logan - cp's OEIS frontend

A274516 Place n equally-spaced points around a circle, labeled 0,1,2,...,n-1. For each i = 0..n-1 such that 5i != i mod n, draw an (undirected) chord from i to (5i mod n). Then a(n) is the total number of distinct chords.

0, 0, 0, 1, 0, 4, 2, 6, 2, 7, 8, 10, 4, 12, 12, 13, 10, 16, 14, 18, 16, 19, 20, 22, 10, 24, 24, 25, 24, 28, 26, 30, 26, 31, 32, 34, 28, 36, 36, 37, 34, 40, 38, 42, 40, 43, 44, 46, 34, 48, 48, 49, 48, 52, 50, 54, 50, 55, 56, 58, 52, 60, 60

Offset: 0

Brooke Logan, Jun 25 2016

nonn

Brooke Logan, Table of n, a(n) for n = 0..10000

If 5i in the definition is replaced by 2i we get A117571, if 5i is replaced by 3i we get A273724, and if 5i is replaced by 4i we get A274462.

We argue as in A273724. There are n-1 choices for i.
For nontrivial chords we need i != 5i mod n, which means 4i != 0 mod n, and so when n == 0 mod 4 we must subtract 3 from n-1 and when n == 2 mod 4 we must subtract 1 from n-1.
A chord occurs twice (but must be counted only once) when j==5i mod n and i==5j mod n, thus when 24i == 0 mod n. If n == +/- 3, +/- 9 mod 24 then subtract another 1, if n == +/- 6, +/- 8 mod 24 then subtract another 2, if n==12 mod 24 subtract 4, and if n == 0 mod 24 then subtract another 10.
Putting the pieces together, we obtain the g.f.
x^2/(1-x)^2-(3+x^2)/(1-x^4)-(x^3+x^9+x^15+x^21)/(1-x^24)-2(x^6+x^8+x^16+x^18)/(1-x^24)-(4*x^12+10)/(1-x^24)+13.
The g.f. can also be written as
(14*x^25 - 12*x^24 + 2*x^23 + x^22 + 3*x^21 - 2*x^20 + 2*x^19 + 4*x^17 - x^16 + x^15 + 8*x^13 - 6*x^12 + 2*x^11 + x^10 + 3*x^9 - 2*x^8 + 2*x^7 + 4*x^5 - x^4 + x^3 + 2*x - 2) / ((1-x)*(1-x^24)).

A274462 Place n equally-spaced points around a circle, labeled 0,1,2,...,n-1. For each i = 0..n-1 such that 4i != i mod n, draw an (undirected) chord from i to (4i mod n). Then a(n) is the total number of distinct chords.

Original entry on oeis.org

0, 0, 1, 0, 3, 2, 3, 6, 7, 6, 7, 10, 9, 12, 13, 6, 15, 16, 15, 18, 17, 18, 21, 22, 21, 22, 25, 24, 27, 28, 21, 30, 31, 30, 33, 32, 33, 36, 37, 36, 37, 40, 39, 42, 43, 36, 45, 46, 45, 48, 47, 48, 51, 52, 51, 52, 55, 54, 57, 58, 51

Offset: 0

Brooke Logan, Jun 24 2016

nonn

Kival Ngaokrajang, Illustration of initial terms

If 4i in the definition is replaced by 2i we get A117571, and if 4i is replaced by 3i we get A273724.

M:=4; # M is the multiplier (2 for A117571, 3 for A273724, 4 for the present sequence)
ans:=[0,0];
for n from 2 to 100 do
h:=Array(0..n-1,0..n-1,0); ct:=0;
for i from 1 to n-1 do j := (M*i mod n);
if ij then if h[j,i]=0 then ct:=ct+1; h[j,i]:=1; fi;
fi;
od:
ans:=[op(ans),ct];
od:
ans;  # N. J. A. Sloane, Jun 24 2016

We argue as in A273724. There are n-1 choices for i.
For nontrivial chords we need i != 4i mod n, which means 3i != 0 mod n, and so when n == 0 mod 3 we must subtract 2 from n-1.
A chord occurs twice (but must be counted only once) when j==4i mod n and i==4j mod n, thus when 15i==0 mod n. If n==+/- 5 mod 15 then subtract another 2, if n==0 mod 15 subtract 6.
Putting the pieces together, we obtain the g.f.
8 + x^2/(1-x)^2 - 2/(1-x^3) - 2(x^5+x^10)/(1-x^15) - 6/(1-x^15),
which can be rewritten as
x^2*(9*x^14-7*x^13+x^12+3*x^11-x^10+3*x^9+x^8-x^7+x^6+3*x^5+x^4-x^3+3*x^2-x+1)/((1-x)*(1-x^15)).

A272685 a(n) = smallest m such that A265432(m) = A272684(n), or -1 if no such m exists.

Original entry on oeis.org

1, 15, 14, 13, 12, 217, 0, 215, 45, 213, 44, 43, 209, 42, 207, 2, 573, 1327, 572, 130, 185, 570, 1492, 569, 78, 568, 128, 567, 1318, 1498, 565, 188, 564, 10, 563, 1312, 562, 1504, 1309, 560, 1507, 693, 74, 558, 1510, 557, 192, 2360, 6085, 350, 1480, 708, 6083, 3643

Offset: 1

Nathan Fox, Brooke Logan, and N. J. A. Sloane, May 24 2016

It is conjectured that all terms of A272684 eventually appear in A265432.

See A273369 for a version that ignores the fact that terms in A272671 ending in 0 cannot appear in A265432.

			A272684(1) = 6 first appears in A265432 at index 1, so a(1) = 1.
A272684(7) = 225 first appears in A265432 at index 0, so a(7) = 0.

Cf. A272684, A272671, A265432.

See A273369 for another version.

A272684 Terms of A272671 which are not a power of 100 times an earlier term of A272671.

Original entry on oeis.org

6, 21, 44, 69, 96, 156, 225, 296, 369, 444, 521, 681, 764, 849, 936, 1025, 1236, 1449, 1664, 1881, 2321, 2544, 2769, 2996, 3225, 3456, 3689, 3924, 4161, 4641, 4884, 5129, 5376, 5625, 5876, 6129, 6384, 6641, 7161, 7424, 7689, 7956, 8225, 8496, 8769, 9044, 9321, 9881

Offset: 1

Nathan Fox, Brooke Logan, and N. J. A. Sloane, May 24 2016

These are the only terms in A272671 that are candidates for terms of A265432.

Cf. A272671, A265432, A272685.

A272683 Square root of concatenation of 1 and A265432(n).

Original entry on oeis.org

35, 4, 105, 4, 35, 4465115625, 425, 35, 345, 10714375, 125, 435259453125, 14, 13, 12, 11, 109515625, 109556109375, 35, 4, 136875, 14133578125, 125, 1365, 425, 4, 375, 35, 1375, 34040625, 10642755859375, 3333202626953125, 351966828673004150390625, 11076674002613037109375, 35, 351172546875, 33875, 11, 12, 13

Offset: 0

Nathan Fox, Brooke Logan, and N. J. A. Sloane, May 23 2016

The square of a(n) is one of the two squares that are required by the definition of A265432(n). The other is in A272682.

Of course if it ever happens that A265432(n) = -1, the definition will need to be changed.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Table of n, A265432(n), sqrt(concat(n,A265432(n))), sqrt(concat(1,A265432(n))) (based on _David W. Wilson_ 10000-term b-file for A265432)

Cf. A265432, A272682.

A272682 Square root of concatenation of n and A265432(n).

Original entry on oeis.org

15, 4, 145, 6, 65, 7741915625, 825, 85, 905, 30245625, 325, 1090619453125, 36, 37, 38, 39, 402484375, 414731890625, 135, 14, 456875, 46901578125, 475, 4885, 1575, 16, 1625, 165, 5375, 170759375, 54893244140625, 17638317373046875, 1795516819326995849609375, 57642802733386962890625, 185

Offset: 0

Nathan Fox, Brooke Logan, and N. J. A. Sloane, May 23 2016

The square of a(n) is one of the two squares that are required by the definition of A265432(n). The other is in A272683.

Of course if it ever happens that A265432(n) = -1, the definition will need to be changed.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Table of n, A265432(n), sqrt(concat(n,A265432(n))), sqrt(concat(1,A265432(n))) (based on _David W. Wilson_ 10000-term b-file for A265432)

Cf. A265432, A272683.

A272678 Smallest number m such that A272677(m) = n.

Original entry on oeis.org

0, 2, 5, 35, 296, 2600, 25317, 251416, 2504474, 25010000, 250044723, 2500100000, 25000316228, 250002000003, 2500004472137, 25000010000000, 250000044721361, 2500000141421358, 25000000316227767

Offset: 0

Nathan Fox, Brooke Logan, and N. J. A. Sloane, May 21 2016

Given n, this is the smallest number m with the property that the smallest square beginning with m has n more digits than n.
a(n) >= 25*10^(n-3). Conjecture: a(n)/(25*10^(n-3)) -> 1 as n -> oo. - Chai Wah Wu, May 21 2016
For odd n > 2, it seems that a(n) is about 25 * 10^(n-3) + 10^(floor((n-1)/2)), although a(13) breaks that pattern. - David A. Corneth, May 22 2016
Except for n = 1 and 13, a(n) appears to be approximately equal to either 25*10^(n-3)+sqrt(10^(n-1)) (for n = 0, 2, 3, 5, 6, 9, 11, 12, 15, 18, ... ) or 25*10^(n-3)+sqrt(2*10^(n-1)) (for n = 4, 7, 8, 14, 16, 17, ...). For n = 1, a(n) is approximately 25*10^(n-3)+sqrt(3*10^(n-1)) and for n = 13, a(n) is about equal to 25*10^(n-3)+sqrt(4*10^(n-1)). Conjecture: a(n) is always  approximately to 25*10^(n-3)+sqrt(k*10^(n-1)) for some small integer k > 0. - Chai Wah Wu, May 22 2016
Using the above conjecture as a guide, upper bounds for a(n) can be computed (see file in links) which coincide with a(n) for n <= 19. - Chai Wah Wu, May 23 2016

			The smallest square beginning with 5 is 529, which has two more digits than 5, and corresponds to a(2) = 5.

Chai Wah Wu, Upper bounds for a(n), n = 0..1000

Cf. A018851, A018796, A272677.

a(6)-a(8) from Chai Wah Wu, May 21 2016
a(9)-a(10), a(15)-a(18) and corrected a(12) from Chai Wah Wu, May 22 2016
a(11)-a(14) from David A. Corneth, May 22 2016

A272677 Number of digits in A018796 - number of digits in n.

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 1, 2, 1, 0, 1, 2, 1, 2, 1, 2, 0, 2, 2, 1, 2, 2, 1, 2, 2, 0, 2, 2, 1, 2, 2, 2, 1, 2, 2, 3, 0, 2, 2, 2, 1, 3, 2, 2, 1, 3, 2, 2, 1, 0, 2, 2, 1, 2, 2, 3, 2, 1, 3, 2, 2, 3, 1, 3, 0, 2, 3, 1, 2, 3, 2, 3, 1, 2, 3, 2, 3, 2, 1, 2, 3, 0, 2, 3, 1, 3, 2, 3, 2, 3

Offset: 0

Nathan Fox, Brooke Logan, and N. J. A. Sloane, May 21 2016

nonn

These are the numbers d mentioned in Robert Israel's comment in A018851.

N. J. A. Sloane, Table of n, a(n) for n = 0..20000

Cf. A018851, A018796.

A272676 Smallest number greater than previous term such that a(n)^2 begins with n.

Original entry on oeis.org

0, 1, 5, 6, 7, 23, 25, 27, 29, 30, 32, 34, 35, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 188, 190, 193, 195, 198, 200, 203, 205, 208

Offset: 0

Nathan Fox, Brooke Logan, and N. J. A. Sloane, May 21 2016

			a(4) = 7 because 7^2 = 49 begins with 4. (Without the monotonic condition, we could have used 4 - this would give A018851.)

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A077502 (the squares), A018851, A018796.

A272675 Indices where records occur in A265432.

Original entry on oeis.org

0, 2, 5, 11, 30, 31, 32, 98, 101, 1092, 1116, 1143, 1250, 1251, 1601, 1715, 1757, 2712, 2714, 2771, 2778, 2781, 2844, 2981, 4398, 5298, 5421, 5423, 9189, 9191, 9195, 9200, 9204, 9210, 9407, 9411, 9414, 9420, 9422, 9432, 9434, 9438, 9440, 9630, 9632, 9644, 9650, 9656, 9662, 9666, 9671

Offset: 1

Nathan Fox, Brooke Logan, and N. J. A. Sloane, May 21 2016

nonn

N. J. A. Sloane and Chai Wah Wu, Table of n, a(n) for n = 1..112 a(n) for n = 1..96 from N. J. A. Sloane

Cf. A265432, A272674.

cp's OEIS Frontend

User: Brooke Logan

A274516 Place n equally-spaced points around a circle, labeled 0,1,2,...,n-1. For each i = 0..n-1 such that 5i != i mod n, draw an (undirected) chord from i to (5i mod n). Then a(n) is the total number of distinct chords.

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A274462 Place n equally-spaced points around a circle, labeled 0,1,2,...,n-1. For each i = 0..n-1 such that 4i != i mod n, draw an (undirected) chord from i to (4i mod n). Then a(n) is the total number of distinct chords.

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A272685 a(n) = smallest m such that A265432(m) = A272684(n), or -1 if no such m exists.

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A272684 Terms of A272671 which are not a power of 100 times an earlier term of A272671.

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A272683 Square root of concatenation of 1 and A265432(n).

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A272682 Square root of concatenation of n and A265432(n).

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A272678 Smallest number m such that A272677(m) = n.

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A272677 Number of digits in A018796 - number of digits in n.

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A272676 Smallest number greater than previous term such that a(n)^2 begins with n.

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A272675 Indices where records occur in A265432.

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