A024352 -id:A024352 - cp's OEIS frontend

A139544 Numbers which are not the difference of two squares of positive integers.

1, 2, 4, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 98, 102, 106, 110, 114, 118, 122, 126, 130, 134, 138, 142, 146, 150, 154, 158, 162, 166, 170, 174, 178, 182, 186, 190, 194, 198, 202, 206, 210, 214, 218, 222, 226, 230

Offset: 1

Artur Jasinski, Apr 25 2008

Conjecture: these numbers do not occur in A139491.
Complement sequence to A024352.
All odd numbers 2k+1 for k>0 can be represented by (k+1)^2-k^2. All multiples 4k for k>1 can be represented by (k+1)^2-(k-1)^2. No number of the form 4k+2 is the difference of two squares because, modulo 4, the differences of two squares are 0, 1, or 3. [T. D. Noe, Apr 27 2009]
A024359(a(n)) = 0. - Reinhard Zumkeller, Nov 09 2012

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A024352, A016825.

a139544 n = a139544_list !! (n-1)
a139544_list = 1 : 2 : 4 : tail a016825_list
-- Reinhard Zumkeller, Nov 09 2012

a[1] = 1; a[2] = 2; a[3] = 4; a[n_] := 4*n-10; Array[a, 60] (* Jean-François Alcover, May 27 2015 *)

is(n)=n%4==2||n==1||n==4 \\ Charles R Greathouse IV, May 31 2013

def A139544(n): return 1<Chai Wah Wu, Feb 11 2025

Corrected by T. D. Noe, Apr 27 2009

A097271 Numbers that are neither the sum of two nonzero squares nor the difference of two nonzero squares.

Original entry on oeis.org

1, 4, 6, 14, 22, 30, 38, 42, 46, 54, 62, 66, 70, 78, 86, 94, 102, 110, 114, 118, 126, 134, 138, 142, 150, 154, 158, 166, 174, 182, 186, 190, 198, 206, 210, 214, 222, 230, 238, 246, 254, 258, 262, 266, 270, 278, 282, 286, 294, 302, 310, 318, 322, 326, 330, 334

Offset: 1

Ray Chandler, Aug 19 2004

nonn

Complement of union of A000404 (sum of squares) and A024352 (difference of squares).

Differs from A062316 only in having an initial 1,4.

Eric Weisstein's World of Mathematics, Sum of Squares Function.

Cf. A000404, A024352, A097268-A097270.

Module[{nn=70,sq},sq=Flatten[{Total[#],Abs[#[[1]]-#[[2]]]}&/@Tuples[ Range[ 3nn]^2,2]]//Union;Take[Complement[Range[Max[sq]],sq],nn]] (* Harvey P. Dale, Nov 04 2016 *)

A181125 Difference of two positive 6th powers.

Original entry on oeis.org

0, 63, 665, 728, 3367, 4032, 4095, 11529, 14896, 15561, 15624, 31031, 42560, 45927, 46592, 46655, 70993, 102024, 113553, 116920, 117585, 117648, 144495, 215488, 246519, 258048, 261415, 262080, 262143, 269297, 413792, 468559, 484785, 515816

Offset: 1

T. D. Noe, Oct 06 2010

nonn

No term is a prime number.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A024352 (squares), A181123 (cubes), A147857 (4th powers), A181124-A181128 (5th to 9th powers).

Cf. A022522 (a subsequence, except its first term). - Mathew Englander, Jun 01 2014

nn=10^10; p=6; Union[Reap[Do[n=i^p-j^p; If[n<=nn, Sow[n]], {i,Ceiling[(nn/p)^(1/(p-1))]}, {j,i}]][[2,1]]]

A379815 a(n) is the smallest integer k > n such that sqrt(1/n + 1/k) is a rational number; or 0 if no such k exists.

Original entry on oeis.org

0, 16, 9, 0, 20, 12, 441, 64, 16, 90, 1089, 36, 4212, 98, 225, 0, 272, 144, 549081, 25, 567, 2156, 13225, 48, 144, 650, 81, 98, 142100, 150, 71622369, 256, 363, 578, 1225, 64, 1332, 684, 468, 360, 41984, 252, 521345889, 198, 180, 559682, 108241, 144, 63, 400, 127449, 117, 1755572, 108, 2420, 392, 4275, 568458

Offset: 1

Felix Huber, Feb 07 2025

nonn

a(1) = a(4) = a(16) = 0. Proof: See Huber link.

k > n exists for n > 16.

			a(3) = 9 because sqrt(1/3 + 1/4) = sqrt(7/12) is irrational, sqrt(1/3 + 1/5) = sqrt(8/15) is irrational, sqrt(1/3 + 1/6) = sqrt(1/2) is irrational, sqrt(1/3 + 1/7) = sqrt(10/21) is irrational, sqrt(1/3 + 1/8) = sqrt(11/24) is irrational, but sqrt(1/3 + 1/9) = sqrt(4/9) = 2/3 is rational.

Felix Huber, Proof that a(1) = a(4) = a(16) = 0

Cf. A002350, A024352, A076600, A378501, A379816.

Cf. A357372, A257522.

A379815:=proc(n)
    local k;
    if n=1 or n=4 or n=16 then
        return 0
    else
        for k from n+1 do
            if type(sqrt(1/n+1/k),rational) then
                return k
            fi
        od
    fi;
end proc;
seq(A379815(n),n=1..58);

a(n) = if ((n==1) || (n==4) || (n==16), return(0)); my(k=n+1); while (!issquare(1/n + 1/k), k++); k; \\ Michel Marcus, Feb 08 2025

a(n) <= n*A002350(n)^2 - n if n is not a square; a(m^2) <= A076600(m)^2. - Jinyuan Wang, Feb 11 2025

A379816 a(n) is the smallest integer k > n such that sqrt(1/n - 1/k) is a rational number; or 0 if no such k exists.

Original entry on oeis.org

0, 4, 12, 0, 25, 18, 448, 16, 12, 100, 1100, 18, 4225, 112, 240, 18, 289, 36, 549100, 25, 588, 2178, 13248, 72, 45, 676, 108, 126, 142129, 180, 71622400, 64, 396, 612, 1260, 48, 1369, 722, 507, 400, 42025, 294, 521345932, 242, 225, 559728, 108288, 72, 112, 100, 127500, 169, 1755625, 162, 2475, 448, 4332, 568516, 16573100, 150

Offset: 1

Felix Huber, Feb 07 2025

nonn

a(1) = a(4) = 0. Proof: See Huber link.
k > n exists for n > 4.
Continues a(61) <= 53872731025, a(62) = 1984, a(63) = 112, a(64) = 72, a(65) = 4225, a(66) = 2178, a(67) <= 159831244588, a(68) = 1156, a(69) = 268272, a(70) = 8820, a(71) <= 859838400, a(72) = 144, a(73) <= 83265625, a(74) = 136900, a(75) = 300, a(76) = 6498, a(77) = 13552, a(78) = 2106, a(79) = 505600, a(80) = 100, a(81) = 108, a(82) = 6724, a(83) = 558092, a(84) = 147, a(85) = 12145225, a(86) = 447458, a(87) = 68208, a(88) = 8712, a(89) = 22250089, a(90) = 100, a(91) = 225450316, a(92)=1150, a(93) = 565068, a(97) <= 3046267249, a(101) = 10201, a(103) <= 5332206050752, a(107) = 99022508. - R. J. Mathar, Feb 21 2025
For nonsquare n, solutions k (not necessarily the smallest ones) are given by k= n*A002350(n)^2 such that 1/n-1/k= (A002349(n)/A002350(n))^2. - R. J. Mathar, Feb 25 2025
For n=p^2, squared odd primes, solutions k (not necessarily the smallest) are constructed by k=p*(p+1)^2/4 such that 1/n -1/k = 1/p^2-1/k =  [(p-1)/(p*(p+1))]^2 is a rational square. For other perfect squares n, start from such a solution for a prime factor p|n, n=(p*f)^2, and multiply the 3 terms of both sides of that solution with 1/f^2 to find a solution for n. - R. J. Mathar, Feb 25 2025

			a(16) = 18 because sqrt(1/16 - 1/17) = sqrt(1/272) is irrational but sqrt(1/16 - 1/18) = sqrt(1/144) = 1/12 is rational.

Felix Huber, Proof that a(1) = a(4) = 0

Cf. A024352, A378501, A379815, A002349, A002350.

A379816:=proc(n)
    local k;
    if n=1 or n=4 then
        return 0
    else
        for k from n+1 do
            if type(sqrt(1/n-1/k),rational) then
                return k
            fi
        od
    fi;
end proc;
seq(A379816(n),n=1..58);

a(n) = if ((n==1) || (n==4), return(0)); my(k=n+1); while (!issquare(1/n - 1/k), k++); k; \\ Michel Marcus, Feb 08 2025

Terms a(59-60) from R. J. Mathar, Feb 12 2025

A024411 Short leg of more than one primitive Pythagorean triangle.

Original entry on oeis.org

20, 28, 33, 36, 39, 44, 48, 51, 52, 57, 60, 65, 68, 69, 75, 76, 84, 85, 87, 88, 92, 93, 95, 96, 100, 104, 105, 108, 111, 115, 116, 119, 120, 123, 124, 129, 132, 133, 135, 136, 140, 141, 145, 147, 148, 152, 155, 156, 159, 160, 161, 164, 165, 168, 172, 175, 177, 180, 183, 184

Offset: 1

David W. Wilson

nonn

Every term is composite. - Clark Kimberling, Feb 04 2024

Proof by contradiction: let p prime be the short leg. Then p^2 + b^2 = c^2 i.e., p^2 = (c - b) * (c + b). Then (c - b, c + b) in {(1, p^2), (p, p)}. If (c - b, c + b) = (p, p) then c = p and b = 0 which is impossible. Hence there is at most one solution for (c - b, c + b). A contradiction. - David A. Corneth, Feb 04 2024

Ray Chandler, Table of n, a(n) for n = 1..10000
Ron Knott, Pythagorean Triples and Online Calculators

Cf. A020884, A024352.

aa=1;s="";For[a=1,a<=10^2,For[b=a+1,((b+1)^2-b^2)<=a^2,c=(a^2+b^2)^0.5;If[c==Round[c]&&GCD[a,b]==1,If[a==aa,s=s<>ToString[a]<>","];If[a!=aa,aa=a,aa=1]];b++ ];a++ ];s (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)

is(n) = {
	my(d = divisors(n^2), q = 0, b, c);
	for(i = 1, #d\2,
		if(!bitand(d[#d + 1 - i] - d[i], 1),
			c = (d[i] + d[#d + 1 - i])/2;
			b = d[#d + 1 - i] - c;
			if(gcd(n, b) == 1 && n < b,
				q++;
				if(q >= 2,
					return(1)
				)
			)
		)
	); 0
} \\ David A. Corneth, Feb 04 2024

A097270 Numbers that are the difference of two nonzero squares but not the sum of two nonzero squares.

Original entry on oeis.org

3, 7, 9, 11, 12, 15, 16, 19, 21, 23, 24, 27, 28, 31, 33, 35, 36, 39, 43, 44, 47, 48, 49, 51, 55, 56, 57, 59, 60, 63, 64, 67, 69, 71, 75, 76, 77, 79, 81, 83, 84, 87, 88, 91, 92, 93, 95, 96, 99, 103, 105, 107, 108, 111, 112, 115, 119, 120, 121, 123, 124, 127, 129, 131, 132

Offset: 1

Ray Chandler, Aug 19 2004

nonn

Intersection of A024352 (difference of squares) and complement of A000404 (sum of squares).

Eric Weisstein's World of Mathematics, Sum of Squares Function.

Cf. A000404, A024352, A097268-A097271.

A181126 Difference of two positive 7th powers.

Original entry on oeis.org

0, 127, 2059, 2186, 14197, 16256, 16383, 61741, 75938, 77997, 78124, 201811, 263552, 277749, 279808, 279935, 543607, 745418, 807159, 821356, 823415, 823542, 1273609, 1817216, 2019027, 2080768, 2094965, 2097024, 2097151, 2685817

Offset: 1

T. D. Noe, Oct 06 2010

nonn

Because x^7-y^7 = (x-y)(x^6+x^5*y+x^4*y^2+x^3*y^3+x^2*y^4+x*y^5+y^6), the difference of two 7th powers is a prime number only if x=y+1, in which case all the primes are in A121618.

The number 67675234241018881 = 127^8 is the first of an infinite number of squares of the form (b^(7k)-1)^8 in this sequence. Are any other squares possible?

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A024352 (squares), A181123 (cubes), A147857 (4th powers), A181124-A181128 (5th to 9th powers)

nn=10^12; p=7; Union[Reap[Do[n=i^p-j^p; If[n<=nn, Sow[n]], {i,Ceiling[(nn/p)^(1/(p-1))]}, {j,i}]][[2,1]]]
Join[{0},#[[2]]-#[[1]]&/@Subsets[Range[10]^7,{2}]//Union] (* Harvey P. Dale, Oct 23 2024 *)

A181127 Difference of two positive 8th powers.

Original entry on oeis.org

0, 255, 6305, 6560, 58975, 65280, 65535, 325089, 384064, 390369, 390624, 1288991, 1614080, 1673055, 1679360, 1679615, 4085185, 5374176, 5699265, 5758240, 5764545, 5764800, 11012415, 15097600, 16386591, 16711680, 16770655, 16776960

Offset: 1

T. D. Noe, Oct 06 2010

nonn

No term is a prime number.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A024352 (squares), A181123 (cubes), A147857 (4th powers), A181124-A181128 (5th to 9th powers)

nn=10^14; p=8; Union[Reap[Do[n=i^p-j^p; If[n<=nn, Sow[n]], {i,Ceiling[(nn/p)^(1/(p-1))]}, {j,i}]][[2,1]]]

A303744 Numbers that are not a difference between same powers (greater than 1) of positive numbers.

Original entry on oeis.org

1, 2, 4, 6, 10, 14, 18, 22, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 102, 106, 110, 114, 118, 122, 126, 130, 134, 138, 142, 146, 150, 154, 158, 162, 166, 170, 174, 178, 182, 186, 190, 194, 198, 202, 206, 210, 214, 222, 226, 230, 234, 238, 246, 250, 254, 258, 262, 266, 270, 274, 278, 282, 286, 290

Offset: 1

Adam Kertesz, Apr 29 2018

nonn

Apart from 1 and 4, all terms == 2 (mod 4). - Robert Israel, Jun 25 2018

			Odd numbers greater than 1 are differences of squares, so they are not here.
8 is not a term, 9 - 1: difference of two squares;
26 is not a term, 27 - 1: difference of two cubes.

Robert Israel, Table of n, a(n) for n = 1..10000

Sequences of numbers that are difference of powers: A024352 (squares), A181123 (cubes).

And of further n-th powers: A147857 (4th), A181124 (5th), A181125 (6th), A181126 (7th), A181127 (8th), A181128 (9th).

N:= 1000: # to get all terms <= N
S:= {1,2,4,seq(i,i=6..N,4)}:
for p from 3 to ilog2(N+1) do
  for n from 1 while n^p - (n-1)^p <= N do
    if n^p > N then m0:= ceil((n^p - N)^(1/p)) else m0:= 1 fi;
    for m from m0 to n-1 do
      v:= n^p-m^p;
      S:= S minus {v};
    od
od od:
sort(convert(S,list)); # Robert Israel, Jun 25 2018

cp's OEIS Frontend

A139544 Numbers which are not the difference of two squares of positive integers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Python

Extensions

A097271 Numbers that are neither the sum of two nonzero squares nor the difference of two nonzero squares.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A181125 Difference of two positive 6th powers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A379815 a(n) is the smallest integer k > n such that sqrt(1/n + 1/k) is a rational number; or 0 if no such k exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Formula

A379816 a(n) is the smallest integer k > n such that sqrt(1/n - 1/k) is a rational number; or 0 if no such k exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Extensions

A024411 Short leg of more than one primitive Pythagorean triangle.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A097270 Numbers that are the difference of two nonzero squares but not the sum of two nonzero squares.

Views

Author

Keywords

Comments

Links

Crossrefs

A181126 Difference of two positive 7th powers.

Views

Author

Keywords

Comments

Links

Crossrefs