A248151 -id:A248151 - cp's OEIS frontend

A248150 Numbers whose sum of divisors (A000203) is divisible by 4.

3, 6, 7, 11, 12, 14, 15, 19, 21, 22, 23, 24, 27, 28, 30, 31, 33, 35, 38, 39, 42, 43, 44, 46, 47, 48, 51, 54, 55, 56, 57, 59, 60, 62, 63, 65, 66, 67, 69, 70, 71, 75, 76, 77, 78, 79, 83, 84, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 99, 102, 103, 105, 107, 108, 110, 111, 112, 114, 115, 118

Offset: 1

M. F. Hasler, Oct 02 2014

nonn

A subsequence of A028983 (even sum of divisors) which contains all numbers but the squares and twice the squares, so no term of this sequence is of that form, either.

Any number having at least two odd prime factors to an odd power is in this sequence, therefore it has asymptotic density 1. - M. F. Hasler, Apr 26 2017

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

Cf. A000203, A028982, A028983, A248151.

First differs from A022544 by including 65.

Select[Range[200],Divisible[DivisorSigma[1,#],4]&] (* Harvey P. Dale, Feb 20 2015 *)

PARI
```
for(n=1,999,sigma(n)%4||print1(n","))
```

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

A245226 Values of n such that the equation x^2 - n*y^2 = n has integer solutions.

Original entry on oeis.org

1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26, 29, 32, 36, 37, 40, 41, 45, 49, 50, 52, 53, 58, 61, 64, 65, 68, 72, 73, 74, 80, 81, 82, 85, 89, 90, 97, 98, 100, 101, 104, 106, 109, 113, 116, 117, 121, 122, 125, 128, 130, 137, 144, 145, 148, 149, 153, 157

Offset: 1

Colin Barker, Jul 14 2014

nonn

All the squares are in this sequence.

Differs from A001481 \ {0} and A248151 from a(17) = 36 on. The number 0 is in the sequence according to its definition. - M. F. Hasler, Oct 22 2019

			5 is in the sequence because x^2-5*y^2=5 has integer solutions, including (x,y) = (5,2) and (85,38).

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

Cf. A001481, A248151.

Select[Range[1,200],Solve[x^2==#*(1+y^2),{x,y},Integers]!={}&] (* Vaclav Kotesovec, Jul 15 2014 *)

isok(m) = if (issquare(4*m), 1, #qfbsolve(Qfb(1, 0, -m), m, 2)); \\ Michel Marcus, May 17 2025

A274397 Positive integers m such that sigma(m) is divisible by 5.

Original entry on oeis.org

8, 19, 24, 27, 29, 38, 40, 54, 56, 57, 58, 59, 72, 76, 79, 87, 88, 89, 95, 104, 108, 109, 114, 116, 118, 120, 128, 133, 135, 136, 139, 145, 149, 152, 158, 168, 171, 174, 177, 178, 179, 184, 189, 190, 199, 200, 203, 209, 216, 218, 228, 229, 232, 236, 237, 239, 247, 248, 261, 264, 266, 267, 269, 270, 278, 280, 285, 290, 295, 296, 297

Offset: 1

M. F. Hasler, Jul 02 2016

nonn

See the subsequence A274685 of odd terms for a remark on frequent pairs of the form (30k-3, 30k-1).
If m is in the sequence and gcd(k,m)=1, then k*m is also in the sequence. One might call "primitive" those terms which are not of this form, i.e., not a "coprime" multiple of an earlier term. The primitive terms are the primes and powers of primes within the sequence, cf. below.
Integers m > 0 where an integer k exists such that A000203(m) = A008587(k). - Felix Fröhlich, Jul 02 2016
For any prime p <> 5 there is an exponent k in {1, 3, 4} (depending on whether p is in A030433, A003631 or A030430) such that p^k is in this sequence. Given these p^k, the sequence consists of all numbers of the form n*p^(q*(k+1)-1) where n is coprime to p and q >= 1. Otherwise said, all numbers m which have some prime factor p with multiplicity q*(k+1)-1, where k = k(p) in {1, 3, 4} as introduced before. - M. F. Hasler, Jul 10 2016

			Some values for a(2^k): We have a(2) = 19, a(4) = 27, a(8) = 54, a(16) = 87, a(32) = 145, a(64) = 270, a(128) = 488, a(256) = 919, a(512) = 1736, a(1024) = 3267, a(2048) = 6258, a(4096) = 12035, a(8192) = 23160, a(16384) = 44878, a(32768) = 87207, a(65536) = 169911, a(131072) = 332009, a(262144) = 650031, a(524288) = 1274569, a(1048576) = 2503510, a(2097152) = 4924370, a(4194304) = 9697475, a(8388608) = 19116191.

Robert Israel, Table of n, a(n) for n = 1..10000
Tewodros Amdeberhan, Victor H. Moll, Vaishavi Sharma, and Diego Villamizar, Arithmetic properties of the sum of divisors, arXiv:2007.03088 [math.NT], 2020. See p. 20.
N. J. A. Sloane, Needed: smallest number k with sigma(sigma(k)) = 5k, SeqFan list, Jul 02 2016.

Cf. A000203, A028983 (sigma even), A087943 (sigma = 3k), A248150 (sigma = 4k); A028982 (sigma is odd), A248151 (sigma is not divisible by 4); A272930 (sigma(sigma(k)) = nk).

select(t -> numtheory:-sigma(t) mod 5 = 0, [$1..1000]); # Robert Israel, Jul 12 2016

Select[Range[300], Divisible[DivisorSigma[1, #], 5]&] (* Jean-François Alcover, Apr 09 2019 *)

PARI
```
is(n)=sigma(n)%5==0
```

is(n)=for(i=1,#n=factor(n)~,n[1,i] != 5 && (n[2,i]+1) % [5,4,4,2][n[1,i]%5] == 0 && return(1))

lim_{n->oo} a(k)/k = 2 (conjectured; cf. Examples).

Edited by M. F. Hasler, Jul 10 2016

A274685 Odd numbers n such that sigma(n) is divisible by 5.

Original entry on oeis.org

19, 27, 29, 57, 59, 79, 87, 89, 95, 109, 133, 135, 139, 145, 149, 171, 177, 179, 189, 199, 203, 209, 229, 237, 239, 247, 261, 267, 269, 285, 295, 297, 319, 323, 327, 343, 349, 351, 359, 377, 379, 389, 395, 399, 409, 413, 417, 419, 435, 437, 439, 445, 447, 449, 459, 475, 479, 493, 499

Offset: 1

M. F. Hasler, Jul 02 2016

nonn

The subsequence of odd terms in A274397.
If n is in the sequence and gcd(n,m)=1 for some odd m, then n*m is also in the sequence. One might call "primitive" those terms which are not of this form, i.e., not a "coprime" multiple of an earlier term. The list of these primitive terms is (19, 27, 29, 59, 79, 89, 109, 139, 149, 179, 199, 229, 239, 269, 343, 349, 359, 379, 389, 409, 419, 439, 449, 479, 499, ...). The primitive terms are the primes and powers of primes within the sequence. If a prime power p^k (k >= 1) is in the sequence, then p^(m(k+1)-1) is in the sequence for any m >= 1, since 1+p+...+p^(m(k+1)-1) = (1+p+...+p^k)(1+p^(k+1)+...+p^((m-1)*(k+1))). For example, with the prime p=19 we also have all odd powers 19^3, 19^5, ..., and with 27 = 3^3, we also have 27^5, 27^9, ... in the sequence.
On the other hand, for any prime p <> 5 there is an exponent k in {1, 3, 4} such that p^k is in this sequence (and therewith all higher powers of the form given above).
One may notice that there are many pairs of the form (30k-3, 30k-1), e.g., 27,29; 57,59; 87,89; 177,179; 237,239; 295,299; ... Indeed, it is likely that 30k-1 is prime and in this case, if 10k-1 is also prime, then sigma(30k-3) = 40k is divisible by 5 and sigma(30k-1) = 30k is also divisible by 5.

			Some values of a(2^k): a(2) = 27, a(4) = 57, a(8) = 89, a(16) = 171, a(32) = 297, a(64) = 545, a(128) = 1029, a(256) = 1937, a(512) = 3625, a(1024) = 6939, a(2048) = 13257, a(4096) = 25483, a(8192) = 49319, a(16384) = 95695, a(32768) = 185991, a(65536) = 362725, a(131072) = 708887, a(262144) = 1388367, a(524288) = 2722639, a(1048576) = 5346681, a(2097152) = 10514679, a(4194304) = 20698531, a(8388608) = 40790203.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000203 (sigma), A028983 (sigma even), A087943 (sigma = 3k), A248150 (sigma = 4k); A028982 (sigma is odd), A248151 (sigma is not divisible by 4); A272930(sigma(sigma(k)) = nk).

Select[Range[1, 500, 2], Divisible[DivisorSigma[1, #], 5] &] (* Michael De Vlieger, Jul 16 2016 *)

is_A274685(n)=sigma(n)%5==0&&bittest(n,0)

forstep(n=1,999,2,sigma(n)%5||print1(n","))

a(n) ~ 2n. - Charles R Greathouse IV, Jul 16 2016

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A248150 Numbers whose sum of divisors (A000203) is divisible by 4.

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A245226 Values of n such that the equation x^2 - n*y^2 = n has integer solutions.

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A274397 Positive integers m such that sigma(m) is divisible by 5.

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A274685 Odd numbers n such that sigma(n) is divisible by 5.

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