A004614 -id:A004614 - cp's OEIS frontend

A332531 Even numbers k such that A103230(k) is a perfect square.

442, 818, 1130, 1226, 1326, 1576, 2454, 3094, 3390, 3678, 3978, 4728, 4862, 5330, 5726, 5986, 6452, 7362, 7786, 7910, 8362, 8398, 8582, 8998, 9282, 10166, 10170, 10250, 11032, 11034, 11934, 12410, 12430, 13486, 13702, 14184, 14586, 15542, 15990, 17178, 17336

Offset: 1

Amiram Eldar, Feb 15 2020

nonn

The odd numbers k such that A103230(k) is a perfect square are the numbers that are divisible only by primes congruent to 3 mod 4 (A004614).

			442 is a term since A103230(442) = 2433600 = 1560^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A004614, A006532, A103230.

Select[2 * Range[9000], IntegerQ @ Sqrt[Abs[DivisorSigma[1, #, GaussianIntegers -> True]]^2] &]

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

A350402 Numbers k such that binomial(k, 2) divides binomial(2^k-2, 2).

Original entry on oeis.org

2, 3, 7, 11, 19, 31, 43, 127, 163, 211, 271, 311, 331, 379, 487, 571, 631, 811, 883, 991, 1459, 1471, 1747, 2311, 2531, 2647, 2791, 2971, 3079, 3631, 3943, 4091, 5171, 5419, 6571, 7591, 8863, 8911, 9199, 9791, 9931, 10891, 11827, 11971, 13591, 14407, 15391, 16759, 17011, 18523, 19531, 21871, 22111, 23431, 24967

Offset: 1

Juri-Stepan Gerasimov, Dec 29 2021

nonn

Conjecture: aside from the first term, this is a subsequence of A094179 (numbers congruent to 3 mod 4 which are divisible only by primes congruent to 3 mod 4).

The conjecture is false: a(2295) = 508606771 = 19531 * 26041 is not in A094179, nor even A004614. - Charles R Greathouse IV, Jan 22 2022

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Supersequence of A069051.

Cf. A069051 (binomial(k,2) divides binomial(2^k-1, 2)?), A094179, A350176.

[n: n in [2..25000] |  IsZero(Binomial(2^n-2, 2) mod Binomial(n, 2))];

Select[Range[2, 25000], Divisible[Binomial[2^# - 2, 2], Binomial[#,2]] &] (* Amiram Eldar, Dec 29 2021 *)

isok(n) = (n>1) && ((binomial(2^n-2, 2) % binomial(n, 2)) == 0); \\ Michel Marcus, Jan 04 2022

is(n)=my(m=n^2-n,t=Mod(2,m)^n-2); t*(t-1)==0 \\ Charles R Greathouse IV, Jan 20 2022

A354070 Lesser of an amicable pair in which both members are divisible only by primes congruent to 3 (mod 4).

Original entry on oeis.org

294706414233, 518129600373, 749347913853, 920163589191, 1692477265941, 2808347861781, 3959417614383, 4400950312143, 9190625896683, 10694894578137, 12615883061859, 15028451404659, 18971047742031, 21981625463259, 29768959571967, 37423211019579, 54939420064683, 69202873206621

Offset: 1

Amiram Eldar, May 16 2022

nonn

Since the factorization of numbers that are divisible only by primes congruent to 3 (mod 4) is the same also in Gaussian integers, these pairs are also Gaussian amicable pairs.
There are 4197267 lesser members of amicable pairs below 10^20 and only 1565 are in this sequence.
The least pair, (294706414233, 305961592167), was discovered by Herman J. J. te Riele in 1995.
The larger counterparts are in A354071.

			294706414233 is a term since (294706414233, 305961592167) is an amicable pair: A001065(294706414233) = 305961592167 and A001065(305961592167) = 294706414233, 294706414233 = 3^4 * 7^2 * 11 * 19 * 47 * 7559, and 3, 7, 11, 19, 47 and 7559 are all congruent to 3 (mod 4), and 305961592167 = 3^4 * 7 * 11 * 19 * 971 * 2659, and 3, 7, 11, 19, 971 and 2659 are all congruent to 3 (mod 4).

Amiram Eldar, Table of n, a(n) for n = 1..1565
Ranthony Ashley Clark, Gaussian Amicable Pairs, Thesis, Eastern Kentucky University, 2013.
Patrick Costello and Ranthony Clark, Gaussian Amicable Pairs: "Friendly Imaginary Numbers", 2013.
Patrick Costello and Ranthony A. C. Edmonds, Gaussian Amicable Pairs, Missouri Journal of Mathematical Sciences, Vol. 30, No. 2 (2018), pp. 107-116.
Wikipedia, Gaussian integer.

Subsequence of A002025 and A004614.

Cf. A001065, A063990, A262623, A262625, A354071.

A354071 Larger of an amicable pair in which both members are divisible only by primes congruent to 3 (mod 4).

Original entry on oeis.org

305961592167, 523630799307, 758052380547, 964086778809, 1697959925739, 2961402044139, 4049489137617, 4475588004657, 9309948700437, 10759267751463, 12799047697821, 15133576811661, 21200708842929, 22067361672741, 30807498770433, 38260957786821, 56250902008917, 70669851785379

Offset: 1

Amiram Eldar, May 16 2022

nonn

The terms are ordered according to their lesser counterparts (A354070).

See A354070 for more details.

Amiram Eldar, Table of n, a(n) for n = 1..1565

Subsequence of A002046 and A004614.

Cf. A063990, A262623, A262625, A354070.

A371010 Powerful numbers that are the sum of 2 squares.

Original entry on oeis.org

1, 4, 8, 9, 16, 25, 32, 36, 49, 64, 72, 81, 100, 121, 125, 128, 144, 169, 196, 200, 225, 256, 288, 289, 324, 361, 392, 400, 441, 484, 500, 512, 529, 576, 625, 648, 676, 729, 784, 800, 841, 900, 961, 968, 1000, 1024, 1089, 1125, 1152, 1156, 1225, 1296, 1352, 1369

Offset: 1

Amiram Eldar, Mar 08 2024

Each term can be decomposed in a unique way as 2^m * i * j^2 where m >= 2, i is a powerful number whose prime factors are all of the form 4*k + 1 (A369563), and j is a number whose prime factors are all of the form 4*k + 3 (A004614).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, Generalizations of Mertens's Formula and k-Free and s-Full Numbers with Prime Divisors in Arithmetic Progression, ResearchGate, 2024.

Intersection of A001481 and A001694.
A371011 is a subsequence.
Cf. A004614, A020893, A369563.

Select[Range[1500], SquaresR[2, #] > 0 && (# == 1 || Min[FactorInteger[#][[;; , 2]]] > 1) &]

is(n) = {my(f=factor(n)); for(i=1, #f~, if(f[i, 2] == 1 || (f[i, 2]%2 && f[i, 1]%4 == 3), return(0))); 1;}

The number of terms that do not exceed x is ~ c * sqrt(x), where c = (6/Pi^2) * (1 + 1/(3*(sqrt(2)-1))) * Product_{primes p == 1 (mod 4)} (1 + 1/((sqrt(p)-1)*(p+1))) * Product_{primes p == 3 (mod 4)} (1 + 1/(p^2-1)) = 1.58769... (Jakimczuk, 2024, Theorem 4.7, p. 50).

Sum_{n>=1} 1/a(n) = (3/2) * Product_{primes p == 1 (mod 4)} (1 + 1/(p*(p-1))) * Product_{primes p == 3 (mod 4)} (1 + 1/(p^2-1)) = (3*Pi^2/16) * A334424 = 1.86676402705119927669... .

A371014 The number of divisors of n that are the sum of 2 squares.

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 1, 4, 2, 4, 1, 3, 2, 2, 2, 5, 2, 4, 1, 6, 1, 2, 1, 4, 3, 4, 2, 3, 2, 4, 1, 6, 1, 4, 2, 6, 2, 2, 2, 8, 2, 2, 1, 3, 4, 2, 1, 5, 2, 6, 2, 6, 2, 4, 2, 4, 1, 4, 1, 6, 2, 2, 2, 7, 4, 2, 1, 6, 1, 4, 1, 8, 2, 4, 3, 3, 1, 4, 1, 10, 3, 4, 1, 3, 4, 2, 2

Offset: 1

Amiram Eldar, Mar 08 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001481, A004614, A046951, A072437, A167181, A371015.

f[p_, e_] := If[Mod[p, 4] == 3, Floor[e/2] + 1, e + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 1]%4 == 3, f[i, 2]\2 + 1, f[i, 2] + 1));}

Multiplicative with a(p^e) = floor(e/2) + 1 if p == 3 (mod 4), and e+1 otherwise.
a(n) = A000005(n) if and only if n is in A072437.
a(n) = A046951(n) if and only if n is in A004614.
a(n) = 1 if and only if n is in A167181.

A090780 a(n) = n*Product_{p prime, p|n} (p - 1)/2.

Original entry on oeis.org

1, 1, 3, 2, 10, 3, 21, 4, 9, 10, 55, 6, 78, 21, 30, 8, 136, 9, 171, 20, 63, 55, 253, 12, 50, 78, 27, 42, 406, 30, 465, 16, 165, 136, 210, 18, 666, 171, 234, 40, 820, 63, 903, 110, 90, 253, 1081, 24, 147, 50, 408, 156, 1378, 27, 550, 84, 513, 406, 1711, 60, 1830, 465, 189

Offset: 1

Benoit Cloitre, Feb 12 2004

a(2n+1) is the conjectured value of the length of period of sequence of Genocchi number of first kind read modulo (2n + 1) (cf. A001469).

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

Cf. A023900, A034444. - R. J. Mathar, Feb 08 2011

Cf. A299822, A000010, A007947, A001221.

A023900 := proc(n) add( d*numtheory[mobius](d),d=numtheory[divisors](n)) ; end proc:
A001221 := proc(n) nops(numtheory[factorset](n)) ; end proc:
A076479 := proc(n) (-1)^A001221(n) ; end proc:
A034444 := proc(n) 2^A001221(n) ;end proc:
A090780 := proc(n) n/A076479(n)/A034444(n) *A023900(n); end proc:
seq(A090780(n),n=1..20) ; # R. J. Mathar, Apr 14 2011

a[n_] := Module[{f, p, e}, fun[p_, e_] := (p - 1)*p^e/2;
If[n == 1, 1, Times @@ (fun @@@ FactorInteger[n])]]; Array[a, 50] (* Amiram Eldar, Nov 23 2018 *)

a(n) = my(f=factor(n)[,1]); n*prod(k=1, #f, (f[k]-1)/2); \\ Michel Marcus, May 26 2019

a(n) = eulerphi(n)*factorback(factorint(n)[, 1]/2) \\ Jianing Song, Aug 11 2023

a(n) = (n/(-2)^omega(n))*(Sum_{d|n} d*mu(d)) = n*A023900(n)/(A076479(n)*A034444(n)).
a(n) = n*A173557(n)/2. - R. J. Mathar, Apr 14 2011
From Jianing Song, Nov 22 2018: (Start)
Multiplicative with a(p^e) = (p - 1)*p^e/2 = A000217(p-1)*p^(e-1).
a(n) = A299822(n)/2^A001221(n).
a(prime(n)) = A034953(n).
a(n) is odd if and only if n = A004614(k) or 2*A004614(k). (End)
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + 2/(p-1)^2) = 5.72671092223951683002237367406848393189560038246828458038126468772919585... - Vaclav Kotesovec, Sep 20 2020
From Jianing Song, Aug 11 2023: (Start)
a(n) = phi(n) * Product_{p|n, p prime} (p/2), where phi = A000010.
Equals A000010(n)*A007947(n)/2^A001221(n). (End)

A260872 Squarefree numbers k such that k+1 has no primes of the form 4*m-1 and at most one 2 in its prime factorization.

Original entry on oeis.org

1, 33, 57, 73, 105, 129, 145, 177, 193, 201, 217, 249, 273, 313, 337, 345, 385, 393, 409, 457, 465, 481, 537, 553, 561, 577, 609, 633, 649, 673, 697, 705, 745, 753, 777, 793, 817, 849, 865, 889, 897, 913, 921, 969, 985, 1009, 1041, 1065, 1081, 1113, 1129

Offset: 1

J. Lowell, Aug 01 2015

nonn

An even number k is congruent to either 0 or 2 mod 4. If congruent to 0, it is divisible by 4 and thus not squarefree. If k is congruent to 2, k+1 will be one less than a multiple of 4, and thus at least one prime factor of k+1 will be one less than a multiple of 4. Thus, there are no even numbers in this sequence.

From the author's comment above, all sequence terms must be odd, so k+1 must always be even and k+1 will always be singly even. - Ray Chandler, Aug 03 2015

			41 + 1 = 42 = 2*3*7 and both 3 and 7 are prime numbers of the form 4*n-1, so 41 is not a term of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002144, A002145, A004613, A004614, A005117, A016825.

Select[Range[1100],SquareFreeQ[#]&&IntegerExponent[#+1,2]<2&&Select[First/@FactorInteger[#+1],Mod[#,4]==3&]=={}&] (* Ray Chandler, Aug 02 2015 *)

A269354 Numbers k such that 10k - 3, 10k - 1, 10k + 1 and 10k + 3 are divisible only by primes congruent to 3 mod 4.

Original entry on oeis.org

8, 13, 21, 44, 50, 75, 89, 99, 133, 146, 150, 245, 254, 289, 319, 327, 395, 468, 500, 517, 579, 601, 608, 691, 704, 761, 764, 878, 1011, 1098, 1125, 1199, 1266, 1298, 1313, 1315, 1414, 1495, 1544, 1716, 1723, 1752, 1762, 1781, 1844, 2043, 2073, 2186, 2281, 2291, 2309, 2360, 2444, 2455, 2457

Offset: 1

Juri-Stepan Gerasimov, Dec 23 2015

Prime terms: 13, 89, 601, 691, 761, 1723, 2281, 2309, 2693, 5437, 5821, 6199, ...

			8 is a term because 10*8 - 3 = 77 = 7*11, 10*8 - 1 = 79, 10*8 + 1 = 81 = 3^4 and 10*8 + 3 = 83 are divisible only by primes congruent to 3 mod 4.

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

Cf. A004614.

[n : n in [1..3000] | forall{d: d in PrimeDivisors(10*n-3) | d mod 4 eq 3}
and forall{d: d in PrimeDivisors(10*n-1) | d mod 4 eq 3}
and forall{d: d in PrimeDivisors(10*n+1) | d mod 4 eq 3}
and forall{d: d in PrimeDivisors(10*n+3) | d mod 4 eq 3}] ;

filter:= n ->
   andmap(t -> numtheory:-factorset(t) mod 4 = {3},[10*n-3,10*n-1,10*n+1,10*n+3]):
select(filter, [$1..10000]); # Robert Israel, Feb 25 2016

pc3m4Q[n_]:=AllTrue[Flatten[FactorInteger[10 n+{-3,-1,1,3}],1][[All,1]], Mod[#,4]==3&]; Select[Range[2500],pc3m4Q] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 30 2018 *)

cp's OEIS Frontend

A332531 Even numbers k such that A103230(k) is a perfect square.

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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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A350402 Numbers k such that binomial(k, 2) divides binomial(2^k-2, 2).

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A354070 Lesser of an amicable pair in which both members are divisible only by primes congruent to 3 (mod 4).

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A354071 Larger of an amicable pair in which both members are divisible only by primes congruent to 3 (mod 4).

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A371010 Powerful numbers that are the sum of 2 squares.

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A371014 The number of divisors of n that are the sum of 2 squares.

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A090780 a(n) = n*Product_{p prime, p|n} (p - 1)/2.

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