A004144 -id:A004144 - cp's OEIS frontend

A097626 Numbers k that are the hypotenuse of exactly 67 distinct integer-sided right triangles, i.e., k^2 can be written as a sum of two squares in 67 ways.

160225, 204425, 226525, 292825, 320450, 337025, 348725, 386425, 403325, 408850, 416585, 453050, 456025, 480675, 491725, 493025, 499525, 505325, 531505, 535925, 544765, 558025, 574925, 585650, 588965, 602225, 613275, 624325, 637325, 640900

Offset: 1

James R. Buddenhagen, Sep 20 2004

nonn

If m is a term, then 2*m and p*m are terms where p is any prime of the form 4j+3. - Chai Wah Wu, Feb 29 2016

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Chai Wah Wu)

Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084648 (4), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A290500 (9), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40).

r[a_]:={b, c}/.{ToRules[Reduce[0Vincenzo Librandi, Mar 01 2016 *)

More terms from Ray Chandler, Sep 21 2004

A290499 Hypotenuses for which there exist exactly 8 distinct integer triangles.

Original entry on oeis.org

390625, 781250, 1171875, 1562500, 2343750, 2734375, 3125000, 3515625, 4296875, 4687500, 5468750, 6250000, 7031250, 7421875, 8203125, 8593750, 8984375, 9375000, 10546875, 10937500, 12109375, 12500000, 12890625, 14062500, 14843750, 16406250, 16796875, 17187500

Offset: 1

Hamdi Sahloul, Aug 04 2017

nonn

Numbers whose square is decomposable in 8 different ways into the sum of two nonzero squares: these are those with only one prime divisor of the form 4k+1 with multiplicity eight.

			a(1) = 390625 = 5^8, a(5) = 2343750 = 2*3*5^8, a(101) = 75000000 = 2^6*3*5^8.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Hamdi Sahloul)

Cf. A002144, A006339, A046080, A046109, A083025.

Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084648 (4), A084649 (5), A097219 (6), A097101 (7), A290500 (9), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

r[a_]:={b, c}/.{ToRules[Reduce[0Vincenzo Librandi, Mar 01 2016 *)

Terms are obtained by the product A004144(k)*A002144(p)^8 for k, p > 0 ordered by increasing values.

A290500 Hypotenuses for which there exist exactly 9 distinct integer triangles.

Original entry on oeis.org

1953125, 3906250, 5859375, 7812500, 11718750, 13671875, 15625000, 17578125, 21484375, 23437500, 27343750, 31250000, 35156250, 37109375, 41015625, 42968750, 44921875, 46875000, 52734375, 54687500, 60546875, 62500000, 64453125, 70312500, 74218750, 82031250

Offset: 1

Hamdi Sahloul, Aug 04 2017

nonn

Numbers whose square is decomposable in 9 different ways into the sum of two nonzero squares: these are those with only one prime divisor of the form 4k+1 with multiplicity nine.

			a(1) = 1953125 = 5^9, a(5) = 11718750 = 2*3*5^9, a(101) = 375000000 = 2^6*3*5^9.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Hamdi Sahloul)

Cf. A002144, A006339, A046080, A046109, A083025.

Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084648 (4), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

r[a_]:={b, c}/.{ToRules[Reduce[0Vincenzo Librandi, Mar 01 2016 *)

Terms are obtained by the product A004144(k)*A002144(p)^9 for k, p > 0 ordered by increasing values.

A290501 Hypotenuses for which there exist exactly 11 distinct integer triangles.

Original entry on oeis.org

48828125, 97656250, 146484375, 195312500, 292968750, 341796875, 390625000, 439453125, 537109375, 585937500, 683593750, 781250000, 878906250, 927734375, 1025390625, 1074218750, 1123046875, 1171875000, 1318359375, 1367187500, 1513671875, 1562500000, 1611328125

Offset: 1

Hamdi Sahloul, Aug 04 2017

nonn

Numbers whose square is decomposable in 11 different ways into the sum of two nonzero squares: these are those with only one prime divisor of the form 4k+1 with multiplicity eleven.

			a(1) = 48828125 = 5^11, a(5) = 292968750 = 2*3*5^11, a(101) = 9375000000 = 2^6*3*5^11.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Hamdi Sahloul)

Cf. A002144, A006339, A046080, A046109, A083025.

Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084648 (4), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A290500 (9), A097225 (10), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

r[a_]:={b, c}/.{ToRules[Reduce[0Vincenzo Librandi, Mar 01 2016 *)

Terms are obtained by the product A004144(k)*A002144(p)^11 for k, p > 0 ordered by increasing values.

A290502 Hypotenuses for which there exist exactly 14 distinct integer triangles.

Original entry on oeis.org

6103515625, 12207031250, 18310546875, 24414062500, 36621093750, 42724609375, 48828125000, 54931640625, 67138671875, 73242187500, 85449218750, 97656250000, 109863281250, 115966796875, 128173828125, 134277343750, 140380859375, 146484375000, 164794921875

Offset: 1

Hamdi Sahloul, Aug 04 2017

nonn

Numbers whose square is decomposable in 14 different ways into the sum of two nonzero squares: these are those with only one prime divisor of the form 4k+1 with multiplicity fourteen.

			a(1) = 6103515625 = 5^14, a(5) = 36621093750 = 2*3*5^14, a(101) = 1171875000000 = 2^6*3*5^14.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Hamdi Sahloul)

Cf. A002144, A006339, A046080, A046109, A083025.

Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084648 (4), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A290500 (9), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

r[a_]:={b, c}/.{ToRules[Reduce[0Vincenzo Librandi, Mar 01 2016 *)

Terms are obtained by the product A004144(k)*A002144(p)^14 for k, p > 0 ordered by increasing values.

A290503 Hypotenuses for which there exist exactly 15 distinct integer triangles.

Original entry on oeis.org

30517578125, 61035156250, 91552734375, 122070312500, 183105468750, 213623046875, 244140625000, 274658203125, 335693359375, 366210937500, 427246093750, 488281250000, 549316406250, 579833984375, 640869140625, 671386718750, 701904296875, 732421875000

Offset: 1

Hamdi Sahloul, Aug 04 2017

nonn

Numbers whose square is decomposable in 15 different ways into the sum of two nonzero squares: these are those with only one prime divisor of the form 4k+1 with multiplicity fifteen.

			a(1) = 30517578125 = 5^15, a(5) = 183105468750 = 2*3*5^15, a(101) = 5859375000000 = 2^6*3*5^15.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Hamdi Sahloul)

Cf. A002144, A006339, A046080, A046109, A083025.

Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084648 (4), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A290500 (9), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

r[a_]:={b, c}/.{ToRules[Reduce[0Vincenzo Librandi, Mar 01 2016 *)

Terms are obtained by the product A004144(k)*A002144(p)^15 for k, p > 0 ordered by increasing values.

A290504 Hypotenuses for which there exist exactly 18 distinct integer triangles.

Original entry on oeis.org

3814697265625, 7629394531250, 11444091796875, 15258789062500, 22888183593750, 26702880859375, 30517578125000, 34332275390625, 41961669921875, 45776367187500, 53405761718750, 61035156250000, 68664550781250, 72479248046875, 80108642578125, 83923339843750

Offset: 1

Hamdi Sahloul, Aug 04 2017

nonn

Numbers whose square is decomposable in 18 different ways into the sum of two nonzero squares: these are those with only one prime divisor of the form 4k+1 with multiplicity eighteen.

			a(1) = 3814697265625 = 5^18, a(5) = 22888183593750 = 2*3*5^18, a(101) = 732421875000000 = 2^6*3*5^18.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Hamdi Sahloul)

Cf. A002144, A006339, A046080, A046109, A083025.

Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084648 (4), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A290500 (9), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

r[a_]:={b, c}/.{ToRules[Reduce[0Vincenzo Librandi, Mar 01 2016 *)

Terms are obtained by the product A004144(k)*A002144(p)^18 for k, p > 0 ordered by increasing values.

A290505 Hypotenuses for which there exist exactly 19 distinct integer triangles.

Original entry on oeis.org

203125, 265625, 406250, 453125, 531250, 578125, 609375, 640625, 796875, 812500, 828125, 906250, 953125, 1062500, 1140625, 1156250, 1218750, 1281250, 1359375, 1390625, 1421875, 1515625, 1578125, 1593750, 1625000, 1656250, 1703125, 1734375, 1765625, 1812500

Offset: 1

Hamdi Sahloul, Aug 04 2017

nonn

Numbers whose square is decomposable in 19 different ways into the sum of two nonzero squares: these are those with exactly two distinct prime divisors of the form 4k+1 with one, and six respective multiplicities, or with only one prime divisor of this form with multiplicity nineteen.

			a(1) = 203125 = 5^6*13, a(5) = 531250 = 2*5^6*17, a(281) = 12796875 = 3^2*5^6*7*13.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Hamdi Sahloul)

Cf. A002144, A006339, A046080, A046109, A083025.

Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084648 (4), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A290500 (9), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

r[a_]:={b, c}/.{ToRules[Reduce[0Vincenzo Librandi, Mar 01 2016 *)

Terms are obtained by the products A004144(k)*A002144(p1)*A002144(p2)^6, or A004144(k)*A002144(p1)^19 for k, p1, p2 > 0 ordered by increasing values.

A072437 Numbers with no prime factors of form 4*k+3.

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 13, 16, 17, 20, 25, 26, 29, 32, 34, 37, 40, 41, 50, 52, 53, 58, 61, 64, 65, 68, 73, 74, 80, 82, 85, 89, 97, 100, 101, 104, 106, 109, 113, 116, 122, 125, 128, 130, 136, 137, 145, 146, 148, 149, 157, 160, 164, 169, 170, 173, 178, 181, 185, 193, 194

Offset: 1

Reinhard Zumkeller, Jun 17 2002

nonn

m is a term iff A072436(m) = m.

These numbers have density zero (Pollack).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
P. Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 119. alternate link

Cf. A004144, A002144, A002145, A004613 (odd terms).
A097706(a(n)) = 1.
Cf. A187811 (complement).

import Data.List (elemIndices)
a072437 n = a072437_list !! (n-1)
a072437_list = map (+ 1) $ elemIndices 0 a005091_list
-- Reinhard Zumkeller, Jan 07 2013

npfQ[n_]:=Count[Transpose[FactorInteger[n]][[1]],?(Mod[#,4]==3&)]==0; Select[Range[200],npfQ] (* _Harvey P. Dale, Nov 12 2013 *)

is(n)=n==1||vecmax(factor(n)[,1]%4)<3 \\ Charles R Greathouse IV, Apr 16 2012

n>0 such that A001842(n)=0. - Benoit Cloitre, Apr 24 2003
A005091(a(n)) = 0. - Reinhard Zumkeller, Jan 07 2013
A065339(a(n)) = 0 . - R. J. Mathar, Jan 28 2025

A005089 Number of distinct primes == 1 (mod 4) dividing n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 2, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 2, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1

Offset: 1

N. J. A. Sloane

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Étienne Fouvry and Peter Koymans, On Dirichlet biquadratic fields, arXiv:2001.05350 [math.NT], 2020.

Cf. A001221, A005091, A005094, A083025 (with multiplicity).

Cf. A079260, A027748, A004144, A009003.

a005089 = sum . map a079260 . a027748_row
-- Reinhard Zumkeller, Jan 07 2013

[#[p:p in PrimeDivisors(n)|p mod 4 eq 1]: n in [1..100]]; // Marius A. Burtea, Jan 16 2020

A005089 := proc(n)
    local a,pe;
    a := 0 ;
    for pe in ifactors(n)[2] do
        if modp(op(1,pe),4) =1 then
            a := a+1 ;
        end if;
    end do:
    a ;
end  proc:
seq(A005089(n),n=1..100) ; # R. J. Mathar, Jul 22 2021

f[n_]:=Length@Select[If[n==1,{},FactorInteger[n]],Mod[#[[1]],4]==1&]; Table[f[n],{n,102}] (* Ray Chandler, Dec 18 2011 *)
a[n_] := DivisorSum[n, Boole[PrimeQ[#] && Mod[#, 4] == 1]&]; Array[a, 100] (* Jean-François Alcover, Dec 01 2015 *)

for(n=1,100,print1(sumdiv(n,d,isprime(d)*if((d-1)%4,0,1)),","))

Additive with a(p^e) = 1 if p == 1 (mod 4), 0 otherwise.
From Reinhard Zumkeller, Jan 07 2013: (Start)
a(n) = Sum_{k=1..A001221(n)} A079260(A027748(n,k)).
a(A004144(n)) = 0.
a(A009003(n)) > 0. (End)

cp's OEIS Frontend

A097626 Numbers k that are the hypotenuse of exactly 67 distinct integer-sided right triangles, i.e., k^2 can be written as a sum of two squares in 67 ways.

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A290499 Hypotenuses for which there exist exactly 8 distinct integer triangles.

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A290500 Hypotenuses for which there exist exactly 9 distinct integer triangles.

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A290501 Hypotenuses for which there exist exactly 11 distinct integer triangles.

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A290502 Hypotenuses for which there exist exactly 14 distinct integer triangles.

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A290503 Hypotenuses for which there exist exactly 15 distinct integer triangles.

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A290504 Hypotenuses for which there exist exactly 18 distinct integer triangles.

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A290505 Hypotenuses for which there exist exactly 19 distinct integer triangles.

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