A009003 -id:A009003 - cp's OEIS frontend

A164927 Sum of the odd prime divisors of numbers with all odd prime divisors of the form 4k+1.

5, 5, 13, 17, 5, 5, 13, 29, 17, 37, 5, 41, 5, 13, 53, 29, 61, 18, 17, 73, 37, 5, 41, 22, 89, 97, 5, 101, 13, 53, 109, 113, 29, 61, 5, 18, 17, 137, 34, 73, 37, 149, 157, 5, 41, 13, 22, 173, 89, 181, 42, 193, 97, 197, 5, 101, 46, 13, 53, 109, 30, 113, 229, 29, 233, 241, 61, 5, 257

Offset: 1

Jonathan Vos Post, Aug 31 2009

We define a sequence b(n) = 5, 10, 13, 17, 20, 25, 26, 29, 34, 37, 40, 41, 50, 52, 53, 58, 61, 65, 68, 73, ... to consist of those numbers where all odd prime factors are primes contained in A002144, and which have at least one prime factor in this class.
b(n) differs from A009003 which also contains numbers like 30=2*3*5 or 39=3*13, 3 not being in A002144.
b(n) essentially contains elements of A004613 multiplied by powers of 2.
a(n) is the sum of the distinct odd prime factors of b(n), where "distinct" means that the multiplicity (exponent) in the prime factorization of b(n) is ignored.
Sum of distinct Pythagorean prime divisors of integers whose only odd prime divisors are Pythagorean primes A002144.
Analogous sequence for primes of form 4k+3 is A164928.
Analogous sequence for primes of form 6k+1 is A164929.
Analogous sequence for primes of form 6k+5 is A164930.

			a(18) = 18 because b(18) = 65 = 5*13, and 5+13 = 18.
The smallest number, all of whose prime factors are of form 4n+1, whose sum of distinct prime factors is prime: 1885 = 5 * 13 * 29; and 5 + 13 + 29 = 47.

Cf. A000040, A002144, A009003, A164927-A164930.

isb := proc(n) fs := numtheory[factorset](n) minus {2} ; if fs = {} then RETURN(false); else for f in fs do if op(1,f) mod 4 <> 1 then RETURN(false) ; fi; od: RETURN(true) ; fi; end:
b := proc(n) if n = 1 then 5; else for a from procname(n-1)+1 do if isb(a) then RETURN(a) ; fi; od: fi; end:
A164927 := proc(n) local f; numtheory[factorset]( b(n)) minus {2} ; add(f,f=%) ; end: seq(A164927(n),n=1..120) ; # R. J. Mathar, Sep 09 2009

Edited, definition clarified by R. J. Mathar, Sep 08 2009

A164928 Sum of the odd prime divisors of numbers whose odd prime divisors are all of the form 4k+3.

Original entry on oeis.org

3, 3, 7, 3, 11, 3, 7, 3, 19, 10, 11, 23, 3, 3, 7, 31, 14, 3, 19, 10, 43, 11, 23, 47, 3, 7, 3, 7, 22, 59, 31, 10, 14, 67, 26, 71, 3, 19, 18, 79, 3, 83, 10, 43, 11, 23, 34, 47, 3, 7, 14, 103, 107, 3, 7, 22, 59, 11, 31, 10, 127, 46, 131, 14, 26, 67, 26, 139, 50, 71, 3, 10, 151, 19, 18

Offset: 1

Jonathan Vos Post, Aug 31 2009

We define a sequence b(n) = 3, 6, 7, 9, 11, 12, 14, 18, 19, 21, 22, 23, ... to consist of those numbers where all odd prime factors are primes contained in A002145, and which have at least one prime factor in this class; b(n) is basically A004144 without the powers of 2.
a(n) is the sum of the distinct odd prime factors of b(n), where "distinct" means that the multiplicity (exponent) in the prime factorization of b(n) is ignored.
Analogous sequence for primes of form 4k+1 is A164927.
Analogous sequence for primes of form 6k+1 is A164929.
Analogous sequence for primes of form 6k+5 is A164930.

			a(11) = 10 because b(11) = 21 = 3*7, and 3+7 = 10.
The smallest nonprime number, all of whose prime factors are of form 4n+3, whose sum of distinct prime factors is prime: b(181) = 3*7*19 = 399; 3+7+19 = 29.

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

Cf. A000040, A002145, A009003, A164927-A164930.

isb := proc(n) fs := numtheory[factorset](n) minus {2} ; if fs = {} then RETURN(false); else for f in fs do if op(1,f) mod 4 <> 3 then RETURN(false) ; fi; od: RETURN(true) ; fi; end:
b := proc(n) if n = 1 then 3; else for a from procname(n-1)+1 do if isb(a) then RETURN(a) ; fi; od: fi; end:
A164928 := proc(n) local f; numtheory[factorset]( b(n)) minus {2} ; add(f,f=%) ; end: seq(A164928(n),n=1..120) ; # R. J. Mathar, Sep 08 2009

sopd[n_]:=Module[{ff=Select[Transpose[FactorInteger[n]][[1]],OddQ]},If[ And@@ (Mod[#,4]==3&/@ff),Total[ff],0]]; Select[Array[sopd,200],#>0&] (* Harvey P. Dale, Dec 16 2013 *)

Edited and extended by R. J. Mathar, Sep 08 2009

A187811 Numbers having at least one prime factor of form 4*k+3.

Original entry on oeis.org

3, 6, 7, 9, 11, 12, 14, 15, 18, 19, 21, 22, 23, 24, 27, 28, 30, 31, 33, 35, 36, 38, 39, 42, 43, 44, 45, 46, 47, 48, 49, 51, 54, 55, 56, 57, 59, 60, 62, 63, 66, 67, 69, 70, 71, 72, 75, 76, 77, 78, 79, 81, 83, 84, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99

Offset: 1

Reinhard Zumkeller, Jan 07 2013

nonn

A005091(a(n)) > 0. - Reinhard Zumkeller, Jan 07 2013

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A072437 (complement); A002145, A009003.

import Data.List (findIndices)
a187811 n = a187811_list !! (n-1)
a187811_list = map (+ 1) $ findIndices (> 0) a005091_list
-- Reinhard Zumkeller, Jan 07 2013

pfQ[n_]:=AnyTrue[Transpose[FactorInteger[n]][[1]],Mod[#,4]==3&]; Select[ Range[100],pfQ] (* The program uses the AnyTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jan 26 2016 *)

is(n)=if(n%4==3,return(1)); my(f=factor(n)[,1]%4); for(i=1,#f, if(f[i]==3, return(1))); 0 \\ Charles R Greathouse IV, Sep 01 2015

a(n) ~ n. - Charles R Greathouse IV, Sep 01 2015

A217248 Numbers whose square is the sum of two nonnegative cubes.

Original entry on oeis.org

0, 1, 3, 4, 8, 24, 27, 32, 64, 81, 98, 108, 125, 168, 192, 216, 228, 256, 312, 343, 375, 500, 512, 525, 588, 648, 671, 729, 784, 847, 864, 1000, 1014, 1029, 1183, 1225, 1261, 1323, 1331, 1344, 1372, 1536, 1728, 1824, 2048, 2187, 2197, 2496, 2646, 2744, 2888

Offset: 1

Christian N. K. Anderson & Kevin L. Schwartz, Mar 20 2013

nonn

Numbers N such that N^2 = x^3 + y^3 where x and y are nonnegative integers. First case with 2 solutions is 77976^2 = 228^3 + 1824^3 = 1026^3 + 1710^3, see A051302. - Zak Seidov, Mar 21 2013

			312 is in the sequence because 312^2 = 2^3 + 46^3.

Chai Wah Wu, Table of n, a(n) for n = 1..5000

This sequence with only positive (nonzero) cubes: A050801, and that sequence squared: A050802

A natural extension of the hypotenuse numbers A009003.

m = 2888; Sort[Reap[Do[If[IntegerQ[c = Sqrt[a^3 + b^3]], Sow[c]], {a, 0, m^(2/3)}, {b, a, (m^2 - a^3)^(1/3)}]][[2, 1]]] (* Zak Seidov, Mar 21 2013 *)

is(n)=n*=n;for(k=ceil((n/2-.5)^(1/3)),(n+.5)^(1/3),if(ispower(n-k^3,3),return(1)));0 \\ Charles R Greathouse IV, Mar 20 2013

y=c(); maxsol=3000 #All solutions x)==as.integer(x))y=c(y,x)
sort(y)

Offset and a(35) corrected and a(36)-a(51) from Giovanni Resta, Mar 20 2013

A309778 a(n) is the greatest integer such that, for every positive integer k <= a(n), n^2 can be written as the sum of k positive square integers.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 155, 1, 211, 1, 275, 1, 1, 2, 1, 1, 1, 1, 611, 662, 1, 1, 827, 886, 1, 1, 1, 1142, 1211, 1, 1355, 1, 1507, 2, 1667, 1, 1, 1, 2011, 1, 1, 1, 1, 2486, 2587, 2690, 2795, 1, 3011, 1, 1, 3350, 1, 3586, 3707, 1, 1, 1

Offset: 1

Bernard Schott, Aug 17 2019

nonn

The idea for this sequence comes from the 6th problem of the 2nd day of the 33rd International Mathematical Olympiad in Moscow, 1992 (see link).
There are four cases to examine and three possible values for a(n).
a(n) = 1 iff n is a nonhypotenuse number or iff n is in A004144.
a(n) >= 2 iff n is a hypotenuse number or iff n is in A009003.
a(n) = 2 iff n^2 is the sum of two positive squares but not the sum of three positive squares or iff n^2 is in A309779.
a(n) = n^2 - 14 iff n^2 is the sum of two and three positive squares or iff n^2 is in A231632.
Theorem: a square n^2 is the sum of k positive squares for all 1 <= k <= n^2 - 14 iff n^2 is the sum of 2 and 3 positive squares (proof in Kuczma). Consequently: A231632 = A018820.

			1 = 1^2, 4 = 2^2 and a(1) = a(2) = 1.
25 = 5^2 = 3^2 + 4^2 and a(5) = 2.
The first representations of 169 are 13^2 = 12^2 + 5^2 = 12^2 + 4^2 + 3^2 = 11^2 + 4^2 + 4^2 + 4^2 =  6^2 + 6^2 + 6^2 + 6^2 + 5^2  = 6^2 + 6^2 + 6^2 + 6^2 + 4^2 + 3^2 = ... and a(13) = 13^2 - 14 = 155.

Marcin E. Kuczma, International Mathematical Olympiads, 1986-1999, The Mathematical Association of America, 2003, pages 76-79.

IMO, 1992, Moscow, Second day. Problem 6

Cf. A018820, A004144, A009003, A231632, A309779.

A350056 Perimeter of integer-sided triangle with hypotenuse A084645(n).

Original entry on oeis.org

12, 24, 30, 36, 40, 48, 60, 70, 72, 80, 84, 84, 90, 96, 90, 108, 120, 120, 126, 132, 140, 144, 132, 160, 168, 176, 168, 180, 192, 180, 210, 208, 216, 210, 228, 234, 220, 240, 240, 252, 252, 260, 264, 252, 240, 276, 280, 270, 280, 288, 264, 270, 324, 320, 330, 336, 330, 352, 336

Offset: 1

Wesley Ivan Hurt, Dec 11 2021

nonn

Winston de Greef, Table of n, a(n) for n = 1..10000
Wikipedia, Integer Triangle

Cf. A009003.

Cf. A084645 (hypotenuses), A350057 (short leg lengths), A350058 (long leg lengths).

A350057 Short leg lengths of integer-sided triangles with hypotenuse A084645(n).

Original entry on oeis.org

3, 6, 5, 9, 8, 12, 10, 20, 18, 16, 21, 12, 15, 24, 9, 27, 24, 20, 28, 33, 40, 36, 11, 32, 42, 48, 24, 30, 48, 18, 60, 39, 54, 35, 57, 65, 20, 48, 40, 63, 56, 60, 66, 36, 15, 69, 80, 45, 56, 72, 22, 27, 81, 64, 88, 84, 55, 96, 48, 51, 72, 93, 60, 85, 84, 96, 36, 99, 52, 120, 78, 108

Offset: 1

Wesley Ivan Hurt, Dec 11 2021

nonn

Winston de Greef, Table of n, a(n) for n = 1..10000
Wikipedia, Integer Triangle

Cf. A009003.

Cf. A084645 (hypotenuses), A350056 (perimeters), A350058 (long leg lengths).

A359225 Numbers that can be expressed as (a^3 + b^3)/(a*b) with b > a >= 1.

Original entry on oeis.org

9, 18, 27, 28, 35, 36, 45, 54, 56, 63, 65, 70, 72, 81, 84, 90, 91, 99, 105, 108, 112, 117, 126, 130, 133, 135, 140, 144, 152, 153, 162, 168, 171, 175, 180, 182, 189, 195, 196, 198, 207, 210, 216, 217, 224, 225, 234, 243, 245, 252, 260, 261, 266, 270, 273, 279, 280, 288, 297

Offset: 1

Zhining Yang, Dec 22 2022

Numbers k such that k*a*b = a^3 + b^3 has integer solutions with b > a >= 1.

Numbers of the form r*(s^3 + t^3) with r >= 1 and s > t >= 1, by a = r*s*t^2, b = r*s^2*t.

			63 can be expressed as (14^3 + 28^3)/(14*28) so 63 is a term.

Antti Karttunen, Table of n, a(n) for n = 1..22000

Cf. A009003, A024670 (subsequence), A373973 (characteristic function).

Positions of positive terms in A373974.

function a = A359225( max_n )
    OneToN = [1:max_n]; a = [];
    for n = 1:max_n-1
        A = (OneToN(1:n)'*ones(1,max_n-n)).^3 ...
          + (ones(n,1)*OneToN(n+1:end)).^3;
        a = unique([a reshape(A(:),1,numel(A))]);
        a = a(1:min(length(a),max_n));
    end
    A = a'*OneToN;
    a = unique(A(:)); a = a(1:min(length(a),max_n))';
end

n = 300; Union@
 Sort@Flatten@
   Table[r*(s^3 + t^3), {r, 1, n/9}, {s, 1,
     CubeRoot[n/(2*r) - 1]}, {t, s + 1, CubeRoot[n/r - s^3]}]

isA359225 = A373973; \\ Antti Karttunen, Jun 24 2024

def aupto(limit):
    c=[k**3 for k in range(1,limit) if k**3<=limit]
    s=set()
    for i in range(len(c)):
        for j in range(i+1,len(c)):
            t=(c[i]+c[j])
            for r in range(1, limit//t+1) :
                s.add(r*t)
    return(sorted(s))
print(aupto(500))

A071821 Numbers whose largest prime factor is of the form 4k+1.

Original entry on oeis.org

5, 10, 13, 15, 17, 20, 25, 26, 29, 30, 34, 37, 39, 40, 41, 45, 50, 51, 52, 53, 58, 60, 61, 65, 68, 73, 74, 75, 78, 80, 82, 85, 87, 89, 90, 91, 97, 100, 101, 102, 104, 106, 109, 111, 113, 116, 117, 119, 120, 122, 123, 125, 130, 135, 136, 137, 143, 145, 146, 148, 149

Offset: 1

Benoit Cloitre, Jun 07 2002

Subsequence of A009003. - M. F. Hasler, Feb 06 2009

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

Cf. A004431, A006530, A009000, A009003, A073503, A083025.

filter:= proc(n)
  max(numtheory:-factorset(n)) mod 4 = 1
end proc:
select(filter, [$1..200]); # Robert Israel, Sep 11 2020

Select[Range[2, 150], Mod[FactorInteger[#][[-1,1]], 4] == 1 &] (* Amiram Eldar, May 04 2022 *)

for(n=2, 200, if((component(component(factor(n), 1), omega(n))-1)%4==0, print1(n, ", ")))

for( n=2,99, vecmax(factor(n)[,1])%4==1 && print1(n",")) \\ M. F. Hasler, Feb 06 2009

Numbers k such that A006530(k) == 1 (mod 4).

A072592 Even numbers with at least one prime factor of form 4*k+1.

Original entry on oeis.org

10, 20, 26, 30, 34, 40, 50, 52, 58, 60, 68, 70, 74, 78, 80, 82, 90, 100, 102, 104, 106, 110, 116, 120, 122, 130, 136, 140, 146, 148, 150, 156, 160, 164, 170, 174, 178, 180, 182, 190, 194, 200, 202, 204, 208, 210, 212, 218, 220, 222, 226, 230, 232, 234, 238

Offset: 1

Reinhard Zumkeller, Jun 23 2002

nonn

Conjecture: this is exactly the sequence whose terms are twice those of A009003. (This has been verified for all terms<=500.) Compare A009003. - John W. Layman, Mar 12 2008

The conjecture is true. See comments on A008846 and A004613. - Lambert Herrgesell (zero815(AT)googlemail.com), Apr 24 2008

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

Cf. A005843, A002144, A072591, A009003, A008846, A004613.

opfQ[n_]:=Count[Transpose[FactorInteger[n]][[1]],?(IntegerQ[ (#-1)/4]&)]>0; Select[Range[2,250,2],opfQ] (* _Harvey P. Dale, Jun 02 2012 *)

from itertools import count, islice
from sympy import primefactors
def A072592_gen(startvalue=1): # generator of terms >= startvalue
    return (n<<1 for n in count(max(startvalue,1)) if any(p&3==1 for p in primefactors(n)))
A072592_list = list(islice(A072592_gen(),20)) # Chai Wah Wu, Jul 07 2024

A072591(a(n)) = 0.

cp's OEIS Frontend

A164927 Sum of the odd prime divisors of numbers with all odd prime divisors of the form 4k+1.

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A164928 Sum of the odd prime divisors of numbers whose odd prime divisors are all of the form 4k+3.

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A187811 Numbers having at least one prime factor of form 4*k+3.

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A217248 Numbers whose square is the sum of two nonnegative cubes.

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R

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A309778 a(n) is the greatest integer such that, for every positive integer k <= a(n), n^2 can be written as the sum of k positive square integers.

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A350056 Perimeter of integer-sided triangle with hypotenuse A084645(n).

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A350057 Short leg lengths of integer-sided triangles with hypotenuse A084645(n).

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A359225 Numbers that can be expressed as (a^3 + b^3)/(a*b) with b > a >= 1.

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