Roger Hui - cp's OEIS frontend

A197487 Number of nonsingular n X n matrices with elements from {0,1,2}.

1, 2, 50, 12792, 30844560, 671869521960, 129553882116606720

Offset: 0

Roger Hui, Nov 29 2011

Minfeng Wang, C++ program to calculate A197487

(* 2x2 case *) cnt = 0; Do[d = Det[{{a, b}, {c, d}}]; If[d != 0, cnt++], {a, 0, 2}, {b, 0, 2}, {c, 0, 2}, {d, 0, 2}]; cnt (* T. D. Noe, Nov 29 2011 *)

a(5)-a(6) from Minfeng Wang, May 29 2024

a(0)=1 prepended by Alois P. Heinz, May 29 2024

A112660 a(n) = (p-1)! mod p^2 where p = n-th prime.

Original entry on oeis.org

1, 2, 24, 34, 10, 168, 84, 37, 183, 521, 588, 258, 655, 558, 281, 1801, 1592, 3415, 803, 4898, 802, 5766, 1659, 6229, 6789, 7271, 5870, 106, 3269, 10734, 9016, 15588, 7671, 9312, 14005, 12985, 23706, 17603, 3506, 18337, 8591, 13031, 30368, 6754, 28958, 23481, 36502, 40139

Offset: 1

Roger Hui, Dec 28 2005

nonn

Related to the Wilson primes A007540, which are primes p such that (p-1)! = -1 mod p^2.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Claire Levaillant, Wilson's theorem modulo p^2 derived from Faulhaber polynomials, arXiv:1912.06652 [math.CO], 2019.

Cf. A001248, A007540, A177771.

[Factorial(NthPrime(n)-1) mod NthPrime(n)^2 : n in [1..50]]; // G. C. Greubel, Dec 17 2019

seq(`mod`(factorial(ithprime(n)-1), ithprime(n)^2), n = 1..50); # G. C. Greubel, Dec 17 2019

Table[Mod[(Prime[n]-1)!, Prime[n]^2], {n, 50}] (* G. C. Greubel, Dec 17 2019 *)

a(n) = my(p=prime(n)); (p-1)! % p^2; \\ Michel Marcus, Dec 17 2019

[mod(factorial(nth_prime(n)-1), nth_prime(n)^2) for n in (1..50)] # G. C. Greubel, Dec 17 2019

a(n) = A177771(n) mod A001248(n). - Michel Marcus, Dec 17 2019

Offset 1 and more terms from Michel Marcus, Dec 17 2019

A112632 Excess of 3k - 1 primes over 3k + 1 primes, beginning with 2.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 4, 3, 2, 3, 2, 1, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 4, 3, 4, 3, 2, 1, 2, 3, 4, 3, 4, 3, 4, 3, 2, 1, 2, 1, 2, 3, 2, 3, 4, 5, 6, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 3, 4, 3, 4, 5, 4, 3, 2, 3, 4, 3, 4, 3, 4, 3, 4, 3, 2, 3, 4, 3, 4, 3, 4, 5, 4, 5, 4, 5, 6, 7, 6, 5

Offset: 1

Roger Hui, Dec 22 2005

Cumulative sums of A134323, negated. The first negative term is a(23338590792) = -1 for the prime 608981813029. See page 4 of the paper by Granville and Martin. - T. D. Noe, Jan 23 2008 [Corrected by Jianing Song, Nov 24 2018]

See the comment about "Chebyshev's bias" in A321856. - Jianing Song, Nov 24 2018

			a(1) = 1 because 2 == -1 (mod 3).
a(2) = 1 because 3 == 0 (mod 3) and does not change the counting.
a(3) = 2 because 5 == -1 (mod 3).
a(4) = 1 because 7 == 1 (mod 3).

T. D. Noe, Table of n, a(n) for n = 1..10000
A. Granville and G. Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), pp. 1-33.
Wikipedia, Chebyshev's bias

Cf. A007352, A098044, A102283, A134323.
Let d be a fundamental discriminant.
Sequences of the form "a(n) = -Sum_{primes p<=n} Kronecker(d,p)" with |d| <= 12: A321860 (d=-11), A320857 (d=-8), A321859 (d=-7), A066520 (d=-4), A321856 (d=-3), A321857 (d=5), A071838 (d=8), A321858 (d=12).
Sequences of the form "a(n) = -Sum_{i=1..n} Kronecker(d,prime(i))" with |d| <= 12: A321865 (d=-11), A320858 (d=-8), A321864 (d=-7), A038698 (d=-4), this sequence (d=-3), A321862 (d=5), A321861 (d=8), A321863 (d=12).

a112632 n = a112632_list !! (n-1)
a112632_list = scanl1 (+) $ map negate a134323_list
-- Reinhard Zumkeller, Sep 16 2014

a[n_] := a[n] = a[n-1] + If[Mod[Prime[n], 6] == 1, -1, 1]; a[1] = a[2] = 1; Table[a[n], {n, 1, 100}]  (* Jean-François Alcover, Jul 24 2012 *)
Accumulate[Which[IntegerQ[(#+1)/3],1,IntegerQ[(#-1)/3],-1,True,0]& /@ Prime[ Range[100]]] (* Harvey P. Dale, Jun 06 2013 *)

a(n) = -sum(i=1, n, kronecker(-3, prime(i))) \\ Jianing Song, Nov 24 2018

a(n) = -Sum_{primes p<=n} Legendre(prime(i),3) = -Sum_{primes p<=n} Kronecker(-3,prime(i)) = -Sum_{i=1..n} A102283(prime(i)). - Jianing Song, Nov 24 2018

A112516 Numbers k such that the first 9 decimal digits of the k-th Fibonacci number is 1-9 pandigital.

Original entry on oeis.org

2749, 4589, 7102, 7727, 8198, 9383, 12633, 15708, 19014, 21206, 21303, 21434, 21566, 22706, 22890, 25790, 28244, 29877, 32174, 32717, 34433, 34883, 37965, 44691, 47422, 48635, 54473, 60438, 60536, 63902, 68340, 72424, 73147, 75873

Offset: 1

Roger Hui, Dec 22 2005

			The 2749th Fibonacci number is:
14372 68955 33879 17661 82964 56715 64334 14434 76345 06448 91772 ...
which is 1-9 pandigital in its first 9 digits.

Cf. A000045, A112371.

NB. (www.jsoftware.com):
plus=: 4 : 0
'x xe'=. +. x.
'y ye'=. +. y.
e=. xe>.ye
z=. (x*10^xe-e)+y*10^ye-e
(z%10^b) j. e+b=. 10<:z
)
g =: 3 : '{."1 ({:,plus/)^:(

filter:= n -> convert(convert(combinat:-fibonacci(n),base,10)[-9..-1],set) = {$1..9}:
select(filter, [$40.. 5 * 10^4]); # Robert Israel, May 31 2015

fQ[n_] := Sort@Take[IntegerDigits@Fibonacci@n, 9] == {1, 2, 3, 4, 5, 6, 7, 8, 9}; Select[ Range[40, 77705], fQ[ # ] &] (* Robert G. Wilson v, Dec 27 2005 *)

a(31)-a(34) from Robert G. Wilson v, Dec 27 2005

A111194 Permanent of the inverse Hilbert matrix.

Original entry on oeis.org

1, 1, 84, 1397520, 5314794912000, 4855173934730716800000, 1090093558153665322315192780800000, 60907190511553979457004412118419080463155200000

Offset: 0

Roger Hui, Oct 22 2005

nonn

Cf. A005249 = determinant of inverse Hilbert matrix; and A092326 = (permanent/determinant) of inverse Hilbert matrix.

NB. www.jsoftware.com
H =: % @: >: @: (+/~) @: i. @ x:
perm=: +/ .*
perm@%.@H n

Permanent[m_List] := With[{v = Array[x, Length[m]]}, Coefficient[Times @@ (m . v), Times @@ v]]; f[n_] := Block[{i = Inverse[Table[1/(i + j - 1), {i, n}, {j, n}]]}, Permanent[i]]; Table[ f[n], {n, 7}] (* Robert G. Wilson v, Oct 24 2005 *)

A111237 LCM of the absolute values of the inverse Hilbert matrix.

Original entry on oeis.org

1, 12, 2880, 226800, 101606400, 6985440000, 35961045120000, 1431699108840000, 692306057963520000, 181128033944995737600, 344143264495491901440000, 6651046563131624734080000, 22028266217091941119272960000

Offset: 1

Roger Hui, Oct 28 2005

nonn

Cf. A005249 (determinant), A111194 (permanent), A092326 (permanent/determinant).

NB. http://www.jsoftware.com
H=: % @: >: @: (+/~) @: i. @: x:
*./ @: | @:, @: %. @: H n

A112371 Numbers n such that the last 9 decimal digits of the n-th Fibonacci number is pandigital 1-9.

Original entry on oeis.org

541, 919, 1788, 6355, 16257, 17799, 20411, 24347, 28837, 36485, 40784, 43450, 45136, 45196, 51973, 54453, 54833, 57128, 57969, 63692, 67188, 67952, 69931, 74765, 76259, 78102, 78196, 78826, 81070, 81726, 87123, 87362, 91636, 91932

Offset: 1

Roger Hui, Dec 22 2005

Since the Fibonacci sequence mod 10^9 is periodic with period 1500000000, there is some positive M such that this sequence satisfies a(n+M) = a(n) + 1500000000. - Robert Israel, Jan 18 2015

			The 541st Fibonacci number is:
51621 23292 73937 94428 28328 17223 02417 68441 62155 65352
08137 22196 49050 89439 99028 11978 84249 30258 98332 77779
69788 39725 641
which is pandigital 1-9 in its last 9 digits.

Clifford A. Pickover, "Wonders of Numbers".

Norman Morton and Michael Satteson, Table of n, a(n) for n = 1..10000, (first 150 terms from Norman Morton)

Cf. A000045, A112516.

NB. In J (www.jsoftware.com).
f=: 3 : '{."(1) 1e9&|@(+/\)@|.^:( ":&.> f n

f:= proc(n) option remember; f(n-1)+f(n-2) mod 10^9 end proc:
f(0):= 0: f(1):= 1:
filter:= n -> convert(convert(f(n),base,10),set)={$1..9};
select(filter, [$1..10^5]); # Robert Israel, Jan 18 2015

cp's OEIS Frontend

User: Roger Hui

A197487 Number of nonsingular n X n matrices with elements from {0,1,2}.

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A112660 a(n) = (p-1)! mod p^2 where p = n-th prime.

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A112632 Excess of 3k - 1 primes over 3k + 1 primes, beginning with 2.

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A112516 Numbers k such that the first 9 decimal digits of the k-th Fibonacci number is 1-9 pandigital.

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J

Maple

Mathematica

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A111194 Permanent of the inverse Hilbert matrix.

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J

Mathematica

A111237 LCM of the absolute values of the inverse Hilbert matrix.

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J

A112371 Numbers n such that the last 9 decimal digits of the n-th Fibonacci number is pandigital 1-9.

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Keywords

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Links

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J

Maple