A329126 -id:A329126 - cp's OEIS frontend

A329338 a(n) = {1{0}^(A268336(n)-1)}^(n-1){1}{0}^A051903(n): upper bound for A329126(n).

1, 110, 101010, 111100, 100010001000100010, 1111110, 10000010000010000010000010000010000010, 11111111000, 1010101010101010100, 10101010101010101010, 100000000010000000001000000000100000000010000000001000000000100000000010000000001000000000100000000010

Offset: 1

M. F. Hasler, Nov 13 2019

nonn

This is the upper bound for A329126 as explained in the "FORMULA" there.

It is sharp for all n except 10, 14, 15, ...

Cf. A329126, A329000, A329339 (this converted from binary to decimal), A268336, A051903.

apply( {A329338(n)=my(k=lcm(lcm([p-1|p<-factor(n)[,1]]), n)/n); fromdigits(concat(vector(n, i, Vec(1, if(i1, vecmax(factor(n)[,2])+1)))))}, [1..16])

a(n) = A007088(A329339(n)), where A007088 = binary numbers and A329339(n) = 2^A051903(n)*(m^n-1)/(m-1) with m = 2^A268336(n).

A329440 Triangle read by rows: n-th row gives positions of ones in A329126(n) in decreasing order.

Original entry on oeis.org

0, 2, 1, 5, 3, 1, 5, 4, 3, 2, 17, 13, 9, 5, 1, 6, 5, 4, 3, 2, 1, 37, 31, 25, 19, 13, 7, 1, 10, 9, 8, 7, 6, 5, 4, 3, 18, 16, 14, 12, 10, 8, 6, 4, 2, 18, 17, 14, 13, 10, 9, 6, 5, 2, 1, 101, 91, 81, 71, 61, 51, 41, 31, 21, 11, 1, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4

Offset: 1

Peter Kagey, Nov 13 2019

Does the n-th row always contain n entries?
Do the rows always form an n term arithmetic progression?
Conjecture: the last value in each row is A051903(n).

			Table begins:
   0
   2,  1
   5,  3,  1
   5,  4,  3,  2
  17, 13,  9,  5,  1
   6,  5,  4,  3,  2, 1
  37, 31, 25, 19, 13, 7, 1
For example, when n = 5, x^17 + x^13 + x^9 + x^5 + x is a multiple of 5 for all integers x > 1.

Cf. A051903, A329126.

A329000 a(n) is the least positive number which yields a multiple of n when its binary digit string, S(n), is read in any numeric base; a(n) is displayed in base 10.

Original entry on oeis.org

1, 6, 42, 60, 139810, 126, 139620524162, 2040, 349524, 419430, 2537779500750160131246576896002, 16380, 44612382091907903486070965589630128805126146, 418861572486, 146602109610, 1048560

Offset: 1

N. J. A. Sloane, Nov 12 2019

If we consider sequence terms to be character strings, a(n) is A329126(n) read in base 2 and converted to base 10. Based on b-file for A329126.

Least k >= 1 such that n divides A329443(k). - Peter Munn, Dec 02 2021

			The strings S(1), S(2), S(3), ... are 1, 110, 101010, 111100, 100010001000100010, 1111110, ... (A329126); converted from binary to decimal these give the current sequence.

Cf. A329126, A329339, A329338, A067029, A268336, A329443.

\\ See A329339. - M. F. Hasler, Nov 14 2019

a(n) <= A329339(n) = 2^A051903(n)*(m^n-1)/(m-1) with m = 2^A268336(n), equality except for n = 10, 14, 15, ... - M. F. Hasler, Nov 14 2019

Definition corrected: not lexicographically earliest string, but smallest binary number. - M. F. Hasler, Nov 09 2021

Name aligned with new A329126 name by Peter Munn, Dec 02 2021

A329339 a(n) = 2^A051903(n)Sum_{k=0..n-1} 2^(A268336(n)k): upper bound for A329000(n).

Original entry on oeis.org

1, 6, 42, 60, 139810, 126, 139620524162, 2040, 349524, 699050, 2537779500750160131246576896002, 16380, 44612382091907903486070965589630128805126146, 1256584717458, 153722867280912930, 1048560, 231587712222682663714935471840371426842813815977643091627066215779128553111554, 1048572

Offset: 1

M. F. Hasler, Nov 13 2019

nonn

This corresponds to the upper bound for A329000 as explained in the "FORMULA" for A329126.

Differs from A329000(n) for n = 10, 14, 15, ...

Cf. A329000, A329126, A329338 (a(n) written in binary), A067029, A051903.

apply( A329339(n)={my(m=2^(lcm(lcm(znstar(n)[2]),n)/n)); (m^n-1)\(m-1)<1,vecmax(factor(n)[,2]))}, [1..20])

a(n) = 2^A051903(n)*(m^n-1)/(m-1) with m = 2^A268336(n).

A329443 a(n) is the GCD of the binary representation of n interpreted in any numeric base.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 2

Offset: 0

Rémy Sigrist, Nov 13 2019

			For n = 42:
- the binary representation of 42 is "101010",
- the corresponding interpretations in the first bases b, alongside their GCD, are:
  b   b+b^3+b^5  GCD
  --  ---------  ---
   2         42   42
   3        273   21
   4       1092   21
   5       3255   21
   6       7998    3
- as b + b^3 + b^5 is always divisible by 3, we have a(42) = 3.

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Cf. A329000, A329126.

a(n) = my (g=n, d=binary(n)); for (b=3, oo, g = gcd(g, fromdigits(d,b)); if (g < b, return (g)))

k divides a(A329000(k)) for any k > 0.

A329479 Number of degree n polynomials f with all nonzero coefficients equal to 1 such that f(k) is divisible by 3 for all integers k.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 6, 15, 30, 66, 121, 242, 462, 903, 1806, 3570, 7225, 14450, 29070, 58311, 116622, 233586, 466489, 932978, 1864590, 3727815, 7455630, 14908530, 29822521, 59645042, 119301006, 238612935, 477225870, 954473586, 1908903481, 3817806962, 7635526542

Offset: 1

Peter Kagey, Nov 13 2019

nonn

Equivalently, this counts strings of numbers of length n that start with a 1 and which yield a multiple of 3 when read in any base.

			For n = 7, the a(7) = 6 (0,1)-polynomials of degree seven such that f(0) = f(1) = f(2) = 0 (mod 3) are
x^7 + x^5 + x^3,
x^7 + x^6 + x^5 + x^4 + x^3 + x^2,
x^7 + x^5 + x,
x^7 + x^3 + x,
x^7 + x^6 + x^5 + x^4 + x^2 + x, and
x^7 + x^6 + x^4 + x^3 + x^2 + x.

Peter Kagey, Table of n, a(n) for n = 1..1000

Cf. A024493, A024495, A329126.

A008776(n) gives the number of polynomials of degree n+3 without the coefficient restriction.

a(2n) = A024495(n-1) * A024493(n).
a(2n+1) = A024495(n) * A024493(n).
Conjectured recurrence: a(n) = 2a(n-1) + 2a(n-2) - 5a(n-3) - 2a(n-4) + 10a(n-5) - 4a(n-6) - 4a(n-7) + 8a(n-8).

A309846 Number of degree n polynomials f with all nonzero coefficients equal to 1 such that f(k) is divisible by 4 for all integers k.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 6, 10, 20, 32, 64, 120, 256, 512, 1056, 2080, 4160, 8192, 16384, 32640, 65536, 131072, 262656, 524800, 1049600, 2097152, 4194304, 8386560, 16777216, 33554432, 67117056, 134225920, 268451840, 536870912, 1073741824, 2147450880, 4294967296, 8589934592, 17180000256, 34359869440

Offset: 1

Peter Kagey, Nov 18 2019

nonn

Equivalently, this counts strings of numbers of length n that start with a 1 and which yield a multiple of 4 when read in any base.

Conjecture: All terms are of the form 2^(n-5), 2^k*(2^(n-k-5) + 1), or 2^k*(2^(n-k-5) - 1) for some value of k.

			For n = 7, the a(7) = 6 (0,1)-polynomials of degree seven such that f(0) == f(1) == f(2) == f(3) == 0 (mod 3) are
x^7 + x^6 + x^5 + x^4,
x^7 + x^6 + x^4 + x^3,
x^7 + x^6 + x^5 + x^2,
x^7 + x^5 + x^4 + x^2,
x^7 + x^6 + x^3 + x^2, and
x^7 + x^4 + x^3 + x^2.

Robert Israel, Table of n, a(n) for n = 1..1000
Robert Israel, Proofs of formulas

Cf. A329126, A329479.

f:= proc(n) local k, r;
     if n <= 4 then return 0 fi;
     r:= n mod 8;
     k:= (n-r)/8;
     if r = 0 then 16^k/8 + 256^k/32
     elif r = 1 then 16^k/4 + 256^k/16
     elif r = 2 then 256^k/8
     elif r = 3 then 256^k/4
     elif r = 4 then -16^k/2 + 256^k/2
     elif r = 5 then 256^k
     elif r = 6 then 2 * 256^k
     else 2 * 16^k + 4 * 256^k
     fi
end proc:
map(f, [$1..50]); # Robert Israel, Oct 29 2023

From Robert Israel, Oct 29 2023: (Start)
a(8 k) = 16^k/8 + 256^k/32 for k >= 1.
a(8 k + 1) = 16^k/4 + 256^k/16 for k >= 1.
a(8 k + 2) = 256^k/8 for k >= 1.
a(8 k + 3) = 256^k/4 for k >= 1.
a(8 k + 4) = -16^k/2 + 256^k/2.
a(8 k + 5) = 256^k.
a(8 k + 6) = 2 * 256^k.
a(8 k + 7) = 2 * 16^k + 4 * 256^k.
G.f.: x^5 * (1 - 2*x + 2*x^2 - 2*x^3)/((1 - 2*x) * (1 - 2*x^2) * (1 - 2*x + 2*x^2)). (End)

More terms from Robert Israel, Oct 29 2023

A328994 a(n) = n^2(1+n)(1+n^2)/4.

Original entry on oeis.org

1, 15, 90, 340, 975, 2331, 4900, 9360, 16605, 27775, 44286, 67860, 100555, 144795, 203400, 279616, 377145, 500175, 653410, 842100, 1072071, 1349755, 1682220, 2077200, 2543125, 3089151, 3725190, 4461940, 5310915, 6284475, 7395856, 8659200, 10089585

Offset: 1

N. J. A. Sloane, Nov 12 2019

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A329126, A329000.

[(n^2+n^3+n^4+n^5)/4: n in [1..40]] // Vincenzo Librandi, Nov 13 2019

CoefficientList[Series[(1+9x+15x^2+5x^3)/(1-x)^6,{x,0,33}],x] (* Vincenzo Librandi, Nov 13 2019 *)

From Vincenzo Librandi, Nov 13 2019: (Start)
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
G.f.: x*(1+9*x+15*x^2+5*x^3)/(1-x)^6. (End)

cp's OEIS Frontend

A329338 a(n) = {1{0}^(A268336(n)-1)}^(n-1){1}{0}^A051903(n): upper bound for A329126(n).

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A329440 Triangle read by rows: n-th row gives positions of ones in A329126(n) in decreasing order.

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A329000 a(n) is the least positive number which yields a multiple of n when its binary digit string, S(n), is read in any numeric base; a(n) is displayed in base 10.

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A329339 a(n) = 2^A051903(n)*Sum_{k=0..n-1} 2^(A268336(n)*k): upper bound for A329000(n).

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A329443 a(n) is the GCD of the binary representation of n interpreted in any numeric base.

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A329479 Number of degree n polynomials f with all nonzero coefficients equal to 1 such that f(k) is divisible by 3 for all integers k.

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A309846 Number of degree n polynomials f with all nonzero coefficients equal to 1 such that f(k) is divisible by 4 for all integers k.

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Maple

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A328994 a(n) = n^2*(1+n)*(1+n^2)/4.

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A329339 a(n) = 2^A051903(n)Sum_{k=0..n-1} 2^(A268336(n)k): upper bound for A329000(n).

A328994 a(n) = n^2(1+n)(1+n^2)/4.