A018825 -id:A018825 - cp's OEIS frontend

A084349 Squarefree numbers that are not the sum of two squares.

1, 3, 6, 7, 11, 14, 15, 19, 21, 22, 23, 30, 31, 33, 35, 38, 39, 42, 43, 46, 47, 51, 55, 57, 59, 62, 66, 67, 69, 70, 71, 77, 78, 79, 83, 86, 87, 91, 93, 94, 95, 102, 103, 105, 107, 110, 111, 114, 115, 118, 119, 123, 127, 129, 131, 133, 134, 138, 139, 141, 142, 143

Offset: 1

Reinhard Zumkeller, Jun 21 2003

nonn

Intersection of A005117 (squarefree) and A018825. - Michel Marcus, Dec 02 2015

			A005117(10) = 14: 14 - 1^2 = 13, 14 - 2^2 = 10 and 14 - 3^3 = 5 are nonsquares, therefore 14 is a term;
A005117(9) = 13 is not a term: A020893(4) = 13 = 2^2 + 3^2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of squares

Cf. A000290, A005117, A010052, A018825, A020893.

a084349 n = a084349_list !! (n-1)
a084349_list = 1 : filter (\x -> all (== 0) $ map (a010052 . (x -)) $
                           takeWhile (<= x) a000290_list) a005117_list
-- Reinhard Zumkeller, Dec 11 2011

Prepend[Select[Range@ 144, SquareFreeQ@ # && SquaresR[2, #] == 0 &], 1] (* Michael De Vlieger, Dec 02 2015 *)

A214329 Complement of A214328.

Original entry on oeis.org

2, 3, 5, 6, 8, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 29, 30, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 88, 89, 90, 91, 93, 94, 96, 97, 98, 99, 100, 101, 102, 104, 105, 106, 107, 108, 109

Offset: 1

N. J. A. Sloane, Jul 26 2012, following a suggestion from Hans Isdahl, Apr 19 2012

nonn

Numbers that are the sum of 2 or 3 nonzero squares. - Altug Alkan, Jan 13 2016

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000404, A000408, A004214, A018825, A214328.

is2(n) = {for( i=1, #n=factor(n)~%4, n[1, i]==3 && n[2, i]%2 && return); n && ( vecmin(n[1, ])==1 || (n[1, 1]==2 && n[2, 1]%2))}
is3(n) = {my(a, b) ; a=1; while(a^2+1Altug Alkan, Jan 13 2016

A331802 Integers having no representation as sum of two nonsquarefree numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 14, 15, 19, 23

Offset: 1

Bernard Schott, Feb 23 2020

This sequence is finite with 14 terms and 23 is the largest term (see Prime Curios link); a proof can be found in comments of A331801.

			With the two smallest nonsquarefree numbers 4 and 8, it is not possible to get 1, 2, 3, 4, 5, 6, 7, 9, 10 and 11 as sum of two nonsquarefree numbers.

G. L. Honaker, Jr. and Chris K. Caldwell, Prime Curios! 23

Cf. A005117 (squarefree), A013929 (nonsquarefree), A331801 (complement).
Cf. A000404 (sum of 2 nonzero squares), A018825 (not the sum of 2 nonzero squares).
Cf. A001694 (squareful), A052485 (not squareful), A076871 (sum of 2 squareful), A085253 (not the sum of 2 squareful).

max = 25; Complement[Range[max], Union @ Select[Total /@ Tuples[Select[Range[max], !SquareFreeQ[#] &], 2], # <= max &]] (* Amiram Eldar, Feb 24 2020 *)

A337140 Numbers m = a + b with a and b positive integers whose product a*b = k^2 is a square.

Original entry on oeis.org

2, 4, 5, 6, 8, 10, 12, 13, 14, 15, 16, 17, 18, 20, 22, 24, 25, 26, 28, 29, 30, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 48, 50, 51, 52, 53, 54, 55, 56, 58, 60, 61, 62, 64, 65, 66, 68, 70, 72, 73, 74, 75, 76, 78, 80, 82, 84, 85, 86, 87, 88, 89, 90, 91

Offset: 1

Hein van Winkel, Aug 18 2020

Related to Heron triangles with a partition point on one of the sides. Calculations become quite different when the partition a + b = m gives the perfect square k^2 = a*b.
These numbers coincide with the numbers > 1 not in A004614.
Let m = 2^t * p_1^a_1 * p_2^a_2 * ... * p_r^a_r * q_1^b_1 * q_2^b_2 * ... * q_s^b_s with t >= 0, a_i >= 0 for i=1..r, where p_i == 1 (mod 4) for i=1..r and q_j == -1 (mod 4) for j=1..s.
Even numbers (A005843) belong to this sequence: m = 2*k and p = k^2.
Numbers divisible by a prime q congruent to 1 (mod 4) (cf. A004613) belong to this sequence: m = q * m_1 = (u^2 + v^2) * m_1 and p = (u*v*q)^2.
The other numbers are divisible only by primes congruent to 3 (mod 4) (cf. A004614).
If a term m is not in the union of A005843 and A004613, then m = q_1^b_1 * q_2^b_2 * ... * q_s^b_s is a term of A018825 (numbers not the sum of two nonzero squares) = q_i * m_1 = q_i *(u^2 + v) and p = q_i^2 * u^2 * v for all u^2 < m_1 and v nonsquare. And so m is not a term: A contradiction.

			Even numbers m = 2*k give a = b = k. For example, 94 = 47+47 and k^2 = 47^2.
Numbers which are divisible by a prime q congruent to 1 (mod 4) give m = q*m' = (u^2 + v^2)*m' and p = (u*v*m')^2. For example, 87 = 3*29 = 3*(25 + 4) = (5*4*3)^2 = 60^2.

Cf. A004614.

Cf. A004613, A005843, A018825.

Select[Range[100], Length @ Select[Times @@@ IntegerPartitions[#, {2}], IntegerQ @ Sqrt[#1] &] > 0 &] (* Amiram Eldar, Aug 26 2020 *)

upto(n) = { my(res = List(vector(n\2, i, 2*i))); forstep(i = 1, n, 2, c = core(i); for(k = 1, sqrtint((n-i)\c), listput(res, i + c*k^2); ) ); listsort(res, 1); res } \\ David A. Corneth, Aug 26 2020

is(n) = for(i = 1, n\2 + 1, if(issquare(i * (n-i)), return(n>1))); 0 \\ David A. Corneth, Aug 26 2020

from itertools import count, islice
from sympy import primefactors
def A337140_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda n: n&1^1 or not all(p&2 for p in primefactors(n>>(~n & n-1).bit_length())), count(max(startvalue,2)))
A337140_list = list(islice(A337140_gen(),30)) # Chai Wah Wu, Aug 21 2024

A267967 Integers n such that n^n is the sum of two nonzero squares while n is not.

Original entry on oeis.org

30, 60, 70, 78, 102, 110, 120, 140, 150, 156, 174, 182, 190, 204, 210, 220, 222, 230, 238, 240, 246, 270, 280, 286, 300, 310, 312, 318, 330, 348, 350, 364, 366, 374, 380, 390, 406, 408, 420, 430, 438, 440, 444, 460, 470, 476, 480, 492, 494, 510, 518, 534, 540, 546, 550, 560

Offset: 1

Altug Alkan, Jan 22 2016

nonn

Terms that are not divisible by 10 are 78, 102, 156, 174, 182, 204, 222, 238, 246, 286, 312, 318, 348, 364, 366, 374, 406, 408, 438, 444, 476, 492, 494, 518, 534, 546, 572, 574, 582, 598, 606, 624, 636, 638, 646, 654, 678, 696, 728, ...

If k^2 is the sum of 2 nonzero squares, (2*k)^(2*k) is. (2*k)^(2*k) = 2^(2*k) * k^(2*k) = (2^k)^2 * k^2 * k^(2*k-2) = (2^k)^2 * k^2 * (k^(k-1))^2. So if k^2 = a^2 + b^2, then (2*k)^(2*k) = (k^(k-1)*2^k*a)^2 + (k^(k-1)*2^k*b)^2. And if k^2 = a^2 + b^2 and k is not the sum of 2 nonzero squares, 2*k is not the sum of 2 nonzero squares. So 2 * A162592(n) appears in this sequence. Note that all terms appear as even numbers.

			30 is a term because 30 is not the sum of 2 nonzero squares and 30^30 = 8609344200000000000000^2 + 11479125600000000000000^2.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000312, A000404, A018825, A162592.

Select[Range@ 120, SquaresR[2, #] == 0 && Resolve[Exists[{a, b}, Reduce[#^# == (a^2 + b^2), {a, b}, Integers], a > b > 0]] &] (* Michael De Vlieger, Jan 24 2016 *)

isA000404(n) = {for( i=1, #n=factor(n)~%4, n[1, i]==3 && n[2, i]%2 && return); n && ( vecmin(n[1, ])==1 || (n[1, 1]==2 && n[2, 1]%2));}
for(n=1, 1e3, if(isA000404(n^n) && !isA000404(n), print1(n, ", ")));

A331801 Integers that are sum of two nonsquarefree numbers.

Original entry on oeis.org

8, 12, 13, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85

Offset: 1

Bernard Schott, Jan 26 2020

nonn

Proposition: All integers > 23 are terms of this sequence (see link Prime Curios!).
Proof by exhaustion:
1) For numbers {4*k} with k>=6, then 4*k = 4*(k-1) + 4 is a term as 4*(k-1) and 4 are nonsquarefree;
2) For numbers {4*k+1} with k>=6, then 4*k+1 = 4*(k-2) + 9 is a term as 4*(k-2) and 9 are nonsquarefree;
3) For numbers {4*k+2} with k>=6, then 4*k+2 = 4*(k-4) + 18 is a term as 4*(k-4) and 18 are nonsquarefree;
4) For numbers {4*k+3}; with k=6, 27 = 9+18 is a term as 9 and 18 are nonsquarefree, and with k>=7, 4*k+3 = 4*(k-6) + 27 is also a term as 4*(k-6) and 27 are nonsquarefree.
Conclusion: every integer > 23 is sum of two nonsquarefree numbers (QED).

			13 = 4 + 9 and 21 = 9 + 12 are terms of this sequence as 4, 9 and 12 are nonsquarefree numbers.

G. L. Honaker, Jr. and Chris K. Caldwell, Prime Curios! 23 (Rupinski)

Cf. A005117 (squarefree), A013929 (nonsquarefree), A331802 (complement).
Cf. A000404 (sum of 2 nonzero squares), A018825 (not the sum of 2 nonzero squares).
Cf. A001694 (squareful), A052485 (not squareful), A076871 (sum of 2 squareful), A085253 (not the sum of 2 squareful).

max = 85; Union @ Select[Total /@ Tuples[Select[Range[max], !SquareFreeQ[#] &], 2], # <= max &] (* Amiram Eldar, Feb 04 2020 *)
Join[{8,12,13,16,17,18,20,21,22},Range[24,100]] (* or *) Complement[Range[100],{1,2,3,4,5,6,7,9,10,11,14,15,19,23}] (* Harvey P. Dale, Dec 04 2024 *)

isok(m) = {for (i=1, m-1, if (!issquarefree(i) && !issquarefree(m-i), return (1));); return(0);} \\ Michel Marcus, Jan 31 2020

A350864 Nonprimes that are not the sum of 2 nonzero squares.

Original entry on oeis.org

1, 4, 6, 12, 14, 15, 21, 22, 24, 27, 28, 30, 33, 35, 38, 39, 42, 44, 46, 48, 51, 54, 55, 56, 57, 60, 62, 63, 66, 69, 70, 75, 76, 77, 78, 84, 86, 87, 88, 91, 92, 93, 94, 95, 96, 99, 102, 105, 108, 110, 111, 112, 114, 115, 118, 119, 120, 123, 124, 126, 129, 132

Offset: 1

Robert P. P. McKone, Jan 19 2022

nonn

a(1) = 1 and a(2) = 4 are squares, but are not the sum of two nonzero squares. If zero were allowed, then 1 and 4 would not be terms, since 1 = 1 + 0 and 4 = 4 + 0.

Intersection of A018252 and A018825.

Cf. A001481.

a[n_] := Complement[Table[m, {m, 1, n}], Table[Prime[m], {m, 1, PrimePi[n]}], Select[Range[5, n], SquaresR[2, #] != 0 &]]; a[132]

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A084349 Squarefree numbers that are not the sum of two squares.

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A214329 Complement of A214328.

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A331802 Integers having no representation as sum of two nonsquarefree numbers.

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A337140 Numbers m = a + b with a and b positive integers whose product a*b = k^2 is a square.

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A267967 Integers n such that n^n is the sum of two nonzero squares while n is not.

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A331801 Integers that are sum of two nonsquarefree numbers.

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A350864 Nonprimes that are not the sum of 2 nonzero squares.

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