Zdenek Cervenka - cp's OEIS frontend

A368242 Numbers k whose number of proper divisors is greater than sqrt(k).

6, 8, 12, 18, 20, 24, 30, 36, 40, 42, 48, 60, 72, 80, 84, 90, 96, 108, 120, 144, 168, 180, 210, 216, 240, 252, 288, 336, 360, 420, 480, 504, 720, 840

Offset: 1

Zdenek Cervenka, Dec 18 2023

Cf. A025487, A032741, A034884.

Select[Range[1, 10000000], Length[Divisors[#]] - 1 > Sqrt[#] &]

for(k=1,10^6,if(numdiv(k)-1 > sqrtint(k), print1(k,", "))) \\ Joerg Arndt, Jan 06 2024

12 is a term since it has 5 proper divisors (1,2,3,4,6), and 5 > sqrt(12).

A356410 Numbers k for which k^3 is divisible by sigma(k).

Original entry on oeis.org

1, 6, 28, 30, 84, 102, 120, 364, 420, 496, 672, 840, 1080, 1092, 1320, 1428, 1488, 1782, 2280, 2716, 2760, 3276, 3360, 3444, 3472, 3480, 3720, 4452, 5640, 7080, 7392, 7440, 7560, 8128, 8148, 8736, 8910, 9240, 9480, 10416, 10920, 11880, 12400, 15456, 15960

Offset: 1

Zdenek Cervenka, Aug 05 2022

nonn

			30 is a term, because 30^3 = 27000, sigma(30) = 72 and 27000 / 72 = 375.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2000 from Robert Israel)

Cf. A000578, A000203, A090777.

select(t -> t^3 mod numtheory:-sigma(t) = 0, [$1..20000]); # Robert Israel, Sep 16 2022

Select[Range[16000], Divisible[#^3, DivisorSigma[1, #]] &]
Select[Range[16000],PowerMod[#,3,DivisorSigma[1,#]]==0&] (* Harvey P. Dale, Sep 04 2024 *)

for(k=1,10^6,if(k^3%sigma(k)==0,print1(k,", "))) \\ Alexandru Petrescu, Sep 10 2022

A351842 Numbers whose sum of digits and number of proper divisors are equal.

Original entry on oeis.org

21, 32, 50, 70, 111, 162, 168, 201, 212, 232, 250, 308, 322, 380, 384, 405, 416, 430, 456, 546, 610, 650, 690, 740, 744, 812, 832, 870, 980, 1004, 1011, 1015, 1053, 1101, 1105, 1222, 1316, 1352, 1365, 1460, 1464, 1482, 1510, 1518, 1550, 1554, 1590, 1608, 1752

Offset: 1

Zdenek Cervenka, Feb 21 2022

			21 is a term since its digits sum to 2 + 1 = 3 and it has three proper divisors (1, 3, and 7).

Cf. A007953, A032741, A057531.

S := n -> add(convert(n, base, 10)):
PD := n -> nops(NumberTheory[Divisors](n)) - 1:
a := n -> select(x -> S(x) = PD(x), [seq(1..n)])

Select[Range[1, 1700], Total[IntegerDigits[#]] == Length[Divisors[#]] - 1 &]

isok(m) = sumdigits(m) == numdiv(m) - 1; \\ Michel Marcus, Feb 21 2022

list(nn) = forcomposite(n=1, nn, if (sumdigits(n) == (numdiv(n) - 1), print1(n, ", ")));
list(1700);

from sympy import divisor_count
def ok(n): return sum(map(int, str(n))) == divisor_count(n) - 1
print([k for k in range(1753) if ok(k)]) # Michael S. Branicky, Feb 21 2022

A344347 Numbers k such that sigma(k)^2 is divisible by k-1.

Original entry on oeis.org

2, 3, 5, 10, 33, 55, 82, 129, 136, 145, 261, 351, 385, 406, 442, 513, 649, 897, 1090, 2241, 4726, 5185, 8650, 13601, 17101, 17641, 18241, 26625, 26937, 29697, 29953, 32896, 34561, 35841, 38417, 44955, 46081, 46593, 51985, 63505, 65703, 66249, 84376, 93313, 97903

Offset: 1

Zdenek Cervenka, May 15 2021

nonn

			For k=10, sigma(10)^2 / (10-1) = 18^2 / 9 = 324 / 9 = 36.

Cf. A072861, A263928.

Select[Range[2, 10^5], Divisible[DivisorSigma[1, #]^2, # - 1] &] (* Amiram Eldar, May 15 2021 *)

list(nn) = for(n=2, nn, if (sigma(n)^2 % (n-1) == 0, print1(n, ", ")))
list(100000)

A341693 Numbers k whose sum of digits divides sigma(k)-k.

Original entry on oeis.org

1, 6, 10, 14, 20, 24, 26, 30, 38, 42, 44, 60, 78, 90, 100, 102, 106, 110, 112, 114, 120, 121, 132, 150, 153, 176, 182, 190, 198, 202, 204, 210, 220, 222, 224, 240, 244, 258, 260, 264, 268, 270, 272, 280, 285, 294, 298, 306, 312, 314, 330, 332, 334, 360, 361, 393, 395

Offset: 1

Zdenek Cervenka, May 24 2021

			k=10 -> sigma(k)=1+2+5+10=18 sum_digits(k)=1+0=1 -> 18/1 = 18.
k=42 -> sigma(k)=1+2+3+6+7+14+21+42=96 sum_digits(k)=4+2=6 -> 96/6 = 16.

Cf. A007953, A001065, A132628.

isA341693 := proc(n)
    if modp(numtheory[sigma](n)-n,digsum(n)) =0 then
        true;
    else
        false;
    end if
end proc:
for n from 1 to 395 do
    if isA341693(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jun 04 2021

Select[Range[400], Divisible[DivisorSigma[1, #] - #, Plus @@ IntegerDigits[#]] &] (* Amiram Eldar, May 24 2021 *)

list(nn) = for(n=1, nn, if ((sigma(n)-n) % sumdigits(n) == 0, print1(n, ", ")))
list(1000)

A298563 Numbers k such that k - 2 | sigma(k).

Original entry on oeis.org

1, 3, 5, 6, 14, 44, 110, 152, 884, 2144, 8384, 18632, 116624, 8394752, 15370304, 73995392, 536920064, 2147581952, 34360131584, 27034175140420610, 36028797421617152, 576460753914036224

Offset: 1

Zdenek Cervenka, Jan 21 2018

Similar to A055708.
Sequence includes every number of the form 2^(j-1)*(2^j+3) such that 2^j+3 is prime (i.e., j is a term in A057732); terms of this form are 5, 14, 44, 152, 2144, 8384, 8394752, 536920064, 2147581952, 34360131584, ... - Jon E. Schoenfield, Jan 22 2018
Superset of A125246. - Giovanni Resta, Jan 23 2018
Contains 2 times odd terms of A191363. Also, if m is a term of A056006 and q := (sigma(m) + 2)/m is coprime to m, them q*m is a term. - Max Alekseyev, May 25 2025

			For k=44, sigma(k)/(k-2) = sigma(44)/(44-2) = 84/42 = 2, so 44 belongs to the sequence;
for k=110, sigma(k)/(k-2) = sigma(110)/(110-2) = 216/108 = 2, so 110 is also a term.

Cf. A055708, A057732, A125246.

[n: n in [3..10^7]| DivisorSigma(1, n) mod (n-2) eq 0]; // Vincenzo Librandi, Jan 22 2018

Select[Range[10^6], Divisible[DivisorSigma[1, #], # - 2] &] (* Michael De Vlieger, Jan 21 2018 *)

isok(k) = (k!=2) && !(sigma(k) % (k-2)); \\ Michel Marcus, Jan 22 2018

a(17)-a(18) from Robert G. Wilson v, Jan 21 2018
a(19) from Giovanni Resta, Jan 23 2018
a(20)-a(22) from Max Alekseyev, May 27 2025

A294149 Numbers k such that the sum of divisors of k is divisible by the sum of nontrivial divisors of k (that is, excluding 1 and k).

Original entry on oeis.org

15, 20, 35, 95, 104, 119, 143, 207, 209, 287, 319, 323, 377, 464, 527, 559, 650, 779, 899, 923, 989, 1007, 1023, 1189, 1199, 1343, 1349, 1519, 1763, 1919, 1952, 2015, 2159, 2507, 2759, 2911, 2915, 2975, 3239, 3599, 3827, 4031, 4199, 4607, 5183, 5207, 5249

Offset: 1

Zdenek Cervenka, Oct 23 2017

nonn

Numbers k such that sigma(k)/(sigma(k)-k-1) is a positive integer.

			15 is in the sequence since sigma(15)/(sigma(15)-15-1) = 24/8 = 3.

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

Subsequence of A002808 (composite numbers).
Cf. A000203, A048050.
Cf. A088831 (k=2), A063906 (k=3).

Quiet@ Select[Range[2, 5300], And[IntegerQ[#], # > 1] &[#2/(#2 - #1 - 1)] & @@ {#, DivisorSigma[1, #]} &] (* Michael De Vlieger, Oct 24 2017 *)

lista(nn) = forcomposite(n=1, nn, if (denominator(sigma(n)/(sigma(n)-n-1)) == 1, print1(n, ", "))); \\ Michel Marcus, Oct 24 2017

list(lim)=my(v=List(),s,t); forfactored(n=9,lim\1, s=sigma(n); t=s-n[1]-1; if(t && s%t==0, listput(v, n[1]))); Vec(v) \\ Charles R Greathouse IV, Nov 11 2017

This sequence gives all numbers a(n) in increasing order which satisfy A000203(a(n))/A048050(a(n)) = A000203(a(n))/(A000203(a(n)) - (a(n)+1)) = k(n), with a positive integer k(n) for n >= 1. - Wolfdieter Lang, Nov 10 2017

Edited by Wolfdieter Lang, Nov 10 2017

Name corrected by Michel Marcus, Nov 12 2017

A293992 Numbers k such that sigma(k) - k - 1 is a perfect number.

Original entry on oeis.org

8, 115, 187, 1375, 2455, 8143, 13543, 18261, 21103, 23479, 40615, 41623, 43279, 49183, 49441, 51703, 56743, 61063, 61279, 61423, 89287, 95551, 137887, 214303, 331567, 379807, 476071, 715471, 1422871, 1515967, 1793527, 1977127, 2431087, 3098527, 3663871

Offset: 1

Zdenek Cervenka, Oct 21 2017

nonn

			sigma(8) - 8 - 1 = 6, a perfect number, so 8 is a term;
sigma(115) - 115 - 1 = 28, a perfect number, so 115 is a term.

Cf. A000396, A048050.

With[{pn=PerfectNumber[Range[10]]},Select[Range[37*10^5],MemberQ[pn, DivisorSigma[ 1,#]-#-1]&]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Feb 04 2019 *)

A226954 Numbers n such that there are seven distinct triples (k, k+n, k+2n) of squares.

Original entry on oeis.org

12671122464000, 50684489856000, 114040102176000, 202737959424000, 316778061600000, 456160408704000, 620885000736000, 810951837696000, 1026360919584000, 1267112246400000, 1533205818144000

Offset: 1

Zdenek Cervenka, Jun 26 2013

nonn

For the first 11 terms we have a(n) = n^2 * a(1). Are there any other primitive terms other than a(1)?

			These 7 triples of squares have a common difference of 12671122464000: (676403^2, 3623347^2, 5079347^2), (911820^2, 3674580^2, 5116020^2), (2615340^2, 4417140^2, 5672940^2), (4885860^2, 6045060^2, 7015260^2), (5664815^2, 6690385^2, 7578415^2), (10873380^2, 11441220^2, 11982180^2) and (11985330^2, 12502770^2, 12999630^2).

Cf. A198387, A222154, A222155, A214155, A226858.

A226858 Numbers n such that there are six distinct triples (k, k+n, k+2n) of squares.

Original entry on oeis.org

258594336000, 1034377344000, 2327349024000, 4137509376000, 6464858400000, 9309396096000, 12671122464000, 16550037504000, 20946141216000, 25859433600000

Offset: 1

Zdenek Cervenka, Jun 20 2013

For the first 10 terms we have a(n) = n^2 * a(1). Are there any other primitive terms other than a(1)?

			These 6 triples have a common difference of 9309396096000: (579774^2, 3105726^2, 4353726^2), (781560^2, 3149640^2, 4385160^2), (2241720^2, 3786120^2, 4862520^2), (4187880^2, 5181480^2, 6013080^2), (9320040^2, 9806760^2, 10270440^2), and (10273140^2, 10716660^2, 11142540^2).

Cf. A198387, A222154, A222155, A214155, A226954.

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User: Zdenek Cervenka

A368242 Numbers k whose number of proper divisors is greater than sqrt(k).

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A356410 Numbers k for which k^3 is divisible by sigma(k).

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Maple

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A351842 Numbers whose sum of digits and number of proper divisors are equal.

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A344347 Numbers k such that sigma(k)^2 is divisible by k-1.

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A341693 Numbers k whose sum of digits divides sigma(k)-k.

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Maple

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A298563 Numbers k such that k - 2 | sigma(k).

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Magma

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A294149 Numbers k such that the sum of divisors of k is divisible by the sum of nontrivial divisors of k (that is, excluding 1 and k).

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A293992 Numbers k such that sigma(k) - k - 1 is a perfect number.