A004275 -id:A004275 - cp's OEIS frontend

A110854 A155750(n)-A155067(n) = prime(2n+2)-prime(2n+1)-prime(2n)+prime(2n-1).

1, 0, 0, 4, 0, -4, 4, -4, 2, 2, 0, -2, 0, 0, 0, -2, 0, 4, 0, -4, 0, 0, 10, -10, 4, 4, -4, -4, 2, 6, -6, 0, 2, -4, 4, 0, -2, 4, 0, -6, 0, 2, 4, -6, 10, -8, 0, 8, 6, -8, -4, 0, 0, -4, 4, -4, 8, -6, 2, 6, -6, 4, 0, -4, -2, 2, 2, 6, -2, -2, -6, 6, -6, 0, 0, 0, 0, 6, -6, 2, -2, 2, 0, -2, -2, 0, 8, 0

Offset: 1

Paul Curtz, Aug 25 2008

Do the absolute values cover A004275?

Harvey P. Dale, Table of n, a(n) for n = 1..1000

#[[4]]-#[[3]]-#[[2]]+#[[1]]&/@Partition[Prime[Range[200]],4,2] (* Harvey P. Dale, Oct 11 2020 *)

Edited by R. J. Mathar, Feb 27 2009

A204504 A204512(n)^2 = floor[A055872(n)/8]: Squares such that appending some digit in base 8 yields another square.

Original entry on oeis.org

0, 0, 0, 1, 4, 36, 144, 1225, 4900, 41616, 166464, 1413721, 5654884, 48024900, 192099600, 1631432881, 6525731524, 55420693056, 221682772224, 1882672131025, 7530688524100, 63955431761796, 255821727047184, 2172602007770041, 8690408031080164, 73804512832419600

Offset: 1

M. F. Hasler, Jan 15 2012

Base-8 analog of A202303.

See also A031149=sqrt(A023110) (base 10), A204502=sqrt(A204503) (base 9),
A204514=sqrt(A055872) (base 8), A204516=sqrt(A055859) (base 7),
A204518=sqrt(A055851) (base 6), A204520=sqrt(A055812) (base 5),
A004275=sqrt(A055808) (base 4), A001075=sqrt(A055793) (base 3),
A001541=sqrt(A055792) (base 2).

b=8;for(n=1,2e9,issquare(n^2\b) & print1((n^2\b)","))

a(n)=polcoeff(x^4*(1 + 4*x + x^2 + 4*x^3)/(1 - 35*x^2 + 35*x^4 - x^6+O(x^n)), n)

a(n)=A204512(n)^2.

G.f. = x^4*(1 + 4*x + x^2 + 4*x^3)/(1 - 35*x^2 + 35*x^4 - x^6)

A251862 Numbers m such that m + 3 divides m^m - 3.

Original entry on oeis.org

3, 7, 10, 27, 727, 1587, 9838, 758206, 789223, 1018846, 1588126, 1595287, 2387206, 4263586, 9494746, 12697378, 17379860, 21480726, 25439767, 38541526, 44219926, 55561536, 62072326, 64335356, 70032586, 83142466, 85409276

Offset: 1

Juri-Stepan Gerasimov, Dec 10 2014

nonn

m such that m+3 divides (-3)^m - 3. - Robert Israel, Dec 14 2014

			3 is in this sequence because 3 + 3 = 6 divides 3^3 - 3 = 24.

Chai Wah Wu, Table of n, a(n) for n = 1..96

Cf. ...............Numbers n such that x divides y, where:
...x.....y......k=0.......k=1.......k=2........k=3........
..n-k..n^n-k..A000027...A087156...A242787....A242788......
..n-k..n^n+k..A000027..see below..A249751....A252041......
..n+k..n^n-k..A000027...A004275...A251603..this sequence..
..n+k..n^n+k..A000027...A004273...A213382....A242800......
(For x=n-1 and y=n^n+1, the only terms are 0, 2 and 3. - David L. Harden, Jan 14 2015)

[n: n in [2..10000] | Denominator((n^n-3)/(n+3)) eq 1];

select(t ->((-3) &^ (t) - 3) mod (t+3) = 0, [$1..10^6]); # Robert Israel, Dec 14 2014

a251862[n_] := Select[Range[n], Mod[PowerMod[#, #, # + 3] - 3, # + 3] == 0 &]; a251862[10^6] (* Michael De Vlieger, Dec 14 2014, after Robert G. Wilson v at A252041 *)

isok(n) = Mod(n, n+3)^n == 3; \\ Michel Marcus, Dec 10 2014

A251862_list = [n for n in range(10**6) if pow(-3, n, n+3) == 3] # Chai Wah Wu, Jan 19 2015

[n for n in range(10^4) if (n + 3).divides((-3)^n - 3)] # Peter Luschny, Jan 17 2015

More terms from Michel Marcus, Dec 10 2014

A007457 Number of (j,k): j+k=n, (j,n)=(k,n)=1, j,k squarefree.

Original entry on oeis.org

0, 1, 2, 2, 2, 2, 4, 4, 2, 2, 4, 4, 6, 4, 4, 6, 8, 6, 6, 6, 4, 8, 8, 8, 8, 8, 8, 6, 10, 8, 10, 10, 8, 12, 8, 10, 14, 12, 10, 12, 16, 10, 18, 14, 12, 14, 16, 14, 16, 14, 10, 16, 20, 14, 12, 16, 14, 20, 18, 14, 22, 20, 16, 20

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

Terms are even or 1: range = A004275. [Reinhard Zumkeller, Sep 26 2011]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Ernesto Bruno Cossi, Joachim Herzog, Paul R. Smith and Richard Stong, Problem 6623, Amer. Math. Monthly, 99 (1992), 573-575.
R. G. Wilson v, Letter to N. J. A. Sloane, Oct. 1993

Cf. A004275, A073311.

a007457 n = length [k | k <- [1..n-1], gcd k n == 1, a008966 k == 1,
                        let j = n - k, gcd j n == 1, a008966 j == 1]
-- Reinhard Zumkeller, Sep 26 2011

f:=func; [0] cat [#[i:i in [1..n-1]| f(i,n) and f(n-i,n) ]:n in [2..70]]; // Marius A. Burtea, Nov 19 2019

[0] cat [&+[MoebiusMu(i*(n-i))^2:i in [1..n-1]]:n in [2..70]]; // Marius A. Burtea, Nov 19 2019

with(numtheory): seq(add(mobius(i*(n-i))^2, i=1..n-1), n=1..80); # Ridouane Oudra, Nov 18 2019

a[n_] := Count[ Table[ If[ SquareFreeQ[j] && GCD[j, n] == 1, If[k = n-j; SquareFreeQ[k] && GCD[k, n] == 1, 1]], {j, 1, n-1}], 1]; Table[a[n], {n, 1, 64}](* Jean-François Alcover, Nov 28 2011 *)

a(n) = Sum_{i=1..n-1} mu(i*(n-i))^2. - Ridouane Oudra, Nov 18 2019

A204575 Squares such that [a(n)/2] is again a square (A055792), written in binary.

Original entry on oeis.org

0, 1, 1001, 100100001, 10011001001001, 1010001010010000001, 101011001001001011001001, 10110111001100110101000100001, 1100001001111011011000010110001001, 110011100111010101001010101001000000001

Offset: 1

M. F. Hasler, Jan 16 2012

See also A031149=sqrt(A023110) (base 10), A204502=sqrt(A204503) (base 9), A204514=sqrt(A055872) (base 8), A204516=sqrt(A055859) (base 7), A204518=sqrt(A055851) (base 6), A204520=sqrt(A055812) (base 5), A004275=sqrt(A055808) (base 4), A001075=sqrt(A055793) (base 3), A001541=sqrt(A055792) (base 2).

A204576 Floor[A055792(n-1)/2]=A084703(n-2) (truncated squares), written in binary.

Original entry on oeis.org

0, 0, 100, 10010000, 1001100100100, 101000101001000000, 10101100100100101100100, 1011011100110011010100010000, 110000100111101101100001011000100, 11001110011101010100101010100100000000, 1101101100101100000000000111010111101000100

Offset: 1

M. F. Hasler, Jan 16 2012

A204575 with the last (binary) digit (necessarily = 1, except for a(1)=0) deleted.

Also: Squares, written in binary, such that appending a (binary) digit (necessarily = 1) yields another square (except for a(1)=0 which corresponds to A204575(1)=00, the only square which remains square when multiplied by 2).

David W. Wilson, Table of n, a(n) for n = 1..100

See also A031149=sqrt(A023110) (base 10), A204502=sqrt(A204503) (base 9), A204514=sqrt(A055872) (base 8), A204516=sqrt(A055859) (base 7), A204518=sqrt(A055851) (base 6), A204520=sqrt(A055812) (base 5), A004275=sqrt(A055808) (base 4), A001075=sqrt(A055793) (base 3), A001541=sqrt(A055792) (base 2).

A252606 Numbers j such that j + 2 divides 2^j + 2.

Original entry on oeis.org

3, 4, 16, 196, 2836, 4551, 5956, 25936, 46775, 65536, 82503, 540736, 598816, 797476, 1151536, 3704416, 4290771, 4492203, 4976427, 8095984, 11272276, 13362420, 21235696, 21537831, 21549347, 29640832, 31084096, 42913396, 49960912, 51127259, 55137316, 56786087, 60296571, 70254724, 70836676

Offset: 1

Juri-Stepan Gerasimov, Mar 03 2015

Numbers j such that (2^j + 2)/(j + 2) is an integer. Numbers j such that (2^j - j)/(j + 2) is an integer.
From Robert Israel, Apr 09 2015: (Start)
The even members of this sequence (4, 16, 196, 2836, ...) are the numbers 2*k-2 where k>=3 is odd and 4^k == -8 (mod k).
The odd members of this sequence (3, 4551, 46775, 82503, ...) are the numbers k-2 where k>=3 is odd and 2^k == -8 (mod k). (End)
2^m is in this sequence for m = (2, 4, 16, 36, 120, 256, 456, 1296, 2556, ...), with the subsequence m = 2^k, k = (1, 2, 4, 8, 16, ...). - M. F. Hasler, Apr 09 2015

			3 is in this sequence because (2^3 + 2)/(3 + 2) = 2.

Chai Wah Wu, Table of n, a(n) for n = 1..78

Cf. A001477, A004273, A004275, A081765, A213382, A251603.

[n: n in [0..1200000] | Denominator((2^n+2)/(n+2)) eq 1];

select(t -> 2 &^t + 2 mod (t + 2) = 0, [$1..10^6]); # Robert Israel, Apr 09 2015

Select[Range[10^6],IntegerQ[(2^#+2)/(#+2)]&] (* Ivan N. Ianakiev, Apr 17 2015 *)

for(n=1,10^5,if((2^n+2)%(n+2)==0,print1(n,", "))) \\ Derek Orr, Apr 05 2015

is(n)=Mod(2,n+2)^n==-2 \\ M. F. Hasler, Apr 09 2015

A252606_list = [n for n in range(10**4) if pow(2, n, n+2) == n] # Chai Wah Wu, Apr 16 2015

a(17)-a(22) from Tom Edgar, Mar 03 2015

More terms from Chai Wah Wu, Apr 16 2015

A069470 a(n) = Sum_{k>=1} floor(n/(k*(k+1)/2)).

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 9, 10, 11, 13, 15, 16, 19, 20, 21, 24, 25, 26, 29, 30, 32, 35, 36, 37, 40, 41, 42, 44, 46, 47, 52, 53, 54, 56, 57, 58, 62, 63, 64, 66, 68, 69, 73, 74, 75, 79, 80, 81, 84, 85, 87, 89, 90, 91, 94, 96, 98, 100, 101, 102, 107, 108, 109, 112, 113, 114, 118

Offset: 0

Henry Bottomley, Mar 25 2002

nonn

The summation has floor(1/2 + sqrt(2*n)) = A002024(n) nonzero terms. - Enrique Pérez Herrero, Apr 05 2010

			a(11) = floor(11/1) + floor(11/3) + floor(11/6) + floor(11/10) + floor(11/15) + ... = 11 + 3 + 1 + 1 + 0 + ... = 16.

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A000217, A006218, A007862, A013936, A013937, A013938, A013939.

Cf. A002024, A004275. - Enrique Pérez Herrero, Apr 05 2010

[(&+[Floor(n/(k*(k+1)/2)): k in [1..100]]): n in [0..30]]; // G. C. Greubel, May 23 2018

A069470[n_]:=Sum[Floor[(2*n)/(k*(1 + k))], {k, 1, Floor[1/2 + Sqrt[2*n]]}] (* Enrique Pérez Herrero, Apr 05 2010 *)

for(n=0, 30, print1(sum(k=1, 100, floor(n/(k*(k+1)/2))), ", ")) \\ G. C. Greubel, May 23 2018

a(n) = a(n-1) + A007862(n).
It appears that limit((sum(floor((1/2)*n/(k*(k+1))), k=1..n))/n, n=infinity) = 1/2. - Stephen Crowley, Aug 12 2009
From Enrique Pérez Herrero, Apr 05 2010: (Start)
a(n) <= floor((2*n^2)/(1 + n)) = A004275(n).
a(n) <= floor((2*n*floor((1 + 2*sqrt(2*n))/2))/(1+floor((1+2*sqrt(2*n))/2))). (End)
G.f.: (1/(1 - x)) * Sum_{k>=1} x^(k*(k+1)/2)/(1 - x^(k*(k+1)/2)). - Ilya Gutkovskiy, Jul 11 2019

A204513 A204517(n)^2 = floor[A055859(n)/7]: Squares which written in base 7, with some digit appended, yield another square.

Original entry on oeis.org

0, 0, 0, 1, 9, 36, 289, 2304, 9216, 73441, 585225, 2340900, 18653761, 148644864, 594579456, 4737981889, 37755210249, 151020840996, 1203428746081, 9589674758400, 38358699033600, 305666163522721, 2435739633423369, 9742958533693476, 77638002106025089, 618668277214777344, 2474673108859109376, 19719746868766849921, 157139306672920022025, 628557226691680088100, 5008738066664673854881, 39912765226644470817024, 159651060906577883268096

Offset: 1

M. F. Hasler, Jan 15 2012

Base-7 analog of A202303.

See also A031149=sqrt(A023110) (base 10), A204502=sqrt(A204503) (base 9), A204514=sqrt(A055872) (base 8), A204516=sqrt(A055859) (base 7), A204518=sqrt(A055851) (base 6), A204520=sqrt(A055812) (base 5), A004275=sqrt(A055808) (base 4), A001075=sqrt(A055793) (base 3), A001541=sqrt(A055792) (base 2).

b=7;for(n=0,200,issquare(n^2\b) & print1(n^2\b,","))

A204513(n)=polcoeff((x^4 + 9*x^5 + 36*x^6 + 34*x^7 + 9*x^8 + 36*x^9 + x^10)/(1 - 255*x^3 + 255*x^6 - x^9+O(x^n)),n)

G.f. = (x^4 + 9*x^5 + 36*x^6 + 34*x^7 + 9*x^8 + 36*x^9 + x^10)/(1 - 255*x^3 + 255*x^6 - x^9)

A204577 Sqrt(floor[A204575(n)/2]), written in binary.

Original entry on oeis.org

0, 0, 10, 1100, 1000110, 110011000, 100101001010, 11011000100100, 10011101110001110, 1110010111100110000, 1010011101111110010010, 111101000000111000111100, 101100011100111010111010110

Offset: 1

M. F. Hasler, Jan 16 2012

D. W. Wilson, Table of n, a(n) for n = 1..100

See also A031149=sqrt(A023110) (base 10), A204502=sqrt(A204503) (base 9), A204514=sqrt(A055872) (base 8), A204516=sqrt(A055859) (base 7), A204518=sqrt(A055851) (base 6), A204520=sqrt(A055812) (base 5), A004275=sqrt(A055808) (base 4), A001075=sqrt(A055793) (base 3), A001541=sqrt(A055792) (base 2).

cp's OEIS Frontend

A110854 A155750(n)-A155067(n) = prime(2n+2)-prime(2n+1)-prime(2n)+prime(2n-1).

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A204504 A204512(n)^2 = floor[A055872(n)/8]: Squares such that appending some digit in base 8 yields another square.

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A251862 Numbers m such that m + 3 divides m^m - 3.

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A007457 Number of (j,k): j+k=n, (j,n)=(k,n)=1, j,k squarefree.

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A204575 Squares such that [a(n)/2] is again a square (A055792), written in binary.

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A204576 Floor[A055792(n-1)/2]=A084703(n-2) (truncated squares), written in binary.

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A252606 Numbers j such that j + 2 divides 2^j + 2.

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A069470 a(n) = Sum_{k>=1} floor(n/(k*(k+1)/2)).