A023197 -id:A023197 - cp's OEIS frontend

A307122 3-deficient numbers with increasing abundancy: Numbers k such that sigma(m)/m < sigma(k)/k < 3 for all numbers m < k such that sigma(m)/m < 3.

1, 2, 4, 6, 12, 24, 36, 48, 60, 168, 252, 300, 336, 630, 2268, 2310, 5472, 6804, 20412, 47424, 61236, 161304, 183708, 486096, 551124, 1215216, 1653372, 4081104, 4960116, 14880348, 44641044, 133923132, 401769396, 1205308188, 1631268870, 3615924564, 10847773692

Offset: 1

Amiram Eldar, Mar 26 2019

nonn

Analogous to A259312 with 3-deficient numbers instead of 3-abundant numbers.
Analogous to A228450 with ratio 3 instead of 2.
The values of sigma(a(n))/a(n) are 1, 1.5, 1.75, 2, 2.333..., 2.5, 2.527..., 2.583..., 2.8, ...

Cf. A071927, A023197, A228450, A259312.

sm=0; seq={}; Do[s=DivisorSigma[1,n]/n; If[s<3 && s>sm, sm=s; AppendTo[seq, n]], {n,1,100000}]; seq

A387163 Numbers k such that sigma(k) >= 3*k and A324644(k)/A324198(k) = 3.

Original entry on oeis.org

10065440, 12794600, 22862840, 24806600, 27399680, 30692480, 33904640, 41629280, 41851040, 46803680, 54625760, 54842480, 70384160, 81915680, 83545280, 87311840, 91571480, 93964640, 95221280, 98030240, 101978240, 103527200, 106719200, 110116160, 121983680, 122904320, 137106200, 137359040, 143195360, 143638880, 144491200

Offset: 1

Antti Karttunen, Aug 28 2025

This sequence contains all 3-perfect numbers (A005820) that are not multiples of three: 459818240 (= a(99)), 51001180160, and also any such hypothetical triperfects of the form 4u+2, when 2u+1 is not multiple of 3. See comments in A351458.

Antti Karttunen, Table of n, a(n) for n = 1..1268 (terms < 2^33).
Index entries for sequences related to primorial base
Index entries for sequences related to sigma(n)

Intersection of A023197 and A387161.
Cf. A000203, A005820, A276086, A324198, A324644, A351458.
Cf. also A387165.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
is_A387163(n) = if(sigma(n)<3*n, 0, my(u=A276086(n)); (gcd(sigma(n), u)==3*gcd(n, u)));

{k | sigma(k) >= 3*k, A324644(k) = 3*A324198(k)}.

A206025 Triangular numbers k whose divisors can be partitioned into three disjoint sets whose sums are all sigma(k)/3.

Original entry on oeis.org

120, 780, 2016, 3240, 4560, 5460, 7140, 7260, 9180, 10296, 10440, 12720, 19110, 21528, 23220, 26796, 28680, 28920, 32640, 34980, 37128, 39060, 41328, 49770, 51360, 56280, 61776, 64620, 64980, 73920, 79800, 97020, 100128, 103740, 107880, 114960, 115440, 122760

Offset: 1

Jaroslav Krizek, Feb 03 2012

nonn

Divisors of triangular number k = 120 can be partitioned into three disjoint sets whose sums are all sigma(k)/3 and this value is triangular numbers (=120). Are there other such triangular numbers?

			Triangular number 780 is in sequence because sigma(780)/3 = 784 = 4+780 = 2+5+6+10+12+13+15+20+26+30+39+52+60+65+78+156+195 = 1+3+130+260+390 (summands are all divisors of 780).

Amiram Eldar, Table of n, a(n) for n = 1..398

Intersection of A000217 and A204830.
Subsequence of A023197.
Cf. A000203.

A306476 Numbers k, with sigma(k) >= 3k and sigma(k) divisible by 3, that are not in A204830.

Original entry on oeis.org

10556208, 10578672, 10589904, 10612368, 10657296, 10690992, 10702224, 10747152, 10825776, 10859472, 10870704, 10881936, 10938096, 10949328, 10971792, 10983024, 11005488, 11039184, 11050416, 11095344, 11117808, 11196432, 11207664, 11252592, 11286288, 11319984

Offset: 1

Giovanni Resta, Feb 18 2019

nonn

From an idea of Amiram Eldar. Analogous sequence to A171641. The divisors of the listed terms k cannot be arranged in three disjoint sets each of them adding to sigma(k)/3.

Cf. A023197, A083207, A171641, A204830, A319186.

A319186 Total number of three disjoint subsets of divisors of k, each of ones adding to sigma(k)/3, where k are the terms listed in A204830.

Original entry on oeis.org

1, 1, 5, 77, 53, 33, 14, 21, 21, 8, 1, 1940, 2, 8217, 230, 4894

Offset: 1

Paolo P. Lava, Dec 17 2018

			a(1) = 1 because A204830(1) = 120, divisors of 120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120 and 1+2+3+4+5+6+8+10+12+15+24+30 = 20+40+60 = 120 = sigma(120)/3. Of course also 1+2+3+5+8+12+15+20+24+30 = 4+6+10+40+60 = 120 = sigma(120)/3 but these three subsets share {120} with the previous ones and therefore they are not disjoint.
a(13) = 2 because A204830(13) = 780, divisors of 780 are 1, 2, 3, 4, 5, 6, 10, 12, 13, 15, 20, 26, 30, 39, 52, 60, 65, 78, 130, 156, 195, 260, 390, 780: 1+3+52+78+260+390 = 2+5+6+10+12+13+15+20+26+30+39+60+65+130+156+195 = 4+780 = sigma(780)/3 and 5+6+10+12+13+15+26+39+52+60+65+130+156+195 = 2+4+20+30+78+260+390 = 1+3+780 = sigma(780)/3.
a(3) = 5 because A204830(3) = 240, divisors of 240 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240 and sigma(240)/3 = 248: it is easy to see that the total number of three disjoint subset is 5 because the only subsets containing 240 that sum to 248 are {1,2,5,240}, {1,3,4,240}, {2,6,240}, {3,5,240} and {8,240}.

Cf. A023197, A083206, A204830.

with(numtheory): with(combstruct): P:=proc(q,h) local a,b,c,d,f,k,n,r;
for n from 1 to q do a:=sigma(n); b:=op(divisors(n));
if a mod h=0 and a>=h*n then k:=0; c:=1/h*a-n;
r:=select(m->m<=c,[b]); f:=iterstructs(Combination(r));
while not finished(f) do if c=add(d,d=nextstruct(f)) then k:=k+1; fi; od; lprint(n,k); fi; od; end: P(10^4,3);

cp's OEIS Frontend

A307122 3-deficient numbers with increasing abundancy: Numbers k such that sigma(m)/m < sigma(k)/k < 3 for all numbers m < k such that sigma(m)/m < 3.

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A387163 Numbers k such that sigma(k) >= 3*k and A324644(k)/A324198(k) = 3.

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A206025 Triangular numbers k whose divisors can be partitioned into three disjoint sets whose sums are all sigma(k)/3.

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A306476 Numbers k, with sigma(k) >= 3k and sigma(k) divisible by 3, that are not in A204830.

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A319186 Total number of three disjoint subsets of divisors of k, each of ones adding to sigma(k)/3, where k are the terms listed in A204830.

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Maple