cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A152215 Numbers k such that sigma_2(k)/(sigma_1(k)*sigma_0(k)) = c, c an integer.

Original entry on oeis.org

1, 4, 25, 100, 121, 256, 289, 484, 529, 841, 1156, 1600, 1681, 2116, 2209, 2809, 3025, 3364, 3481, 5041, 6400, 6724, 6889, 7225, 7921, 8836, 10201, 11236, 11449, 12100, 12769, 13225, 13924, 17161, 18225, 18496, 18769, 20164, 21025, 22201, 27556, 27889, 28900
Offset: 1

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Author

Ctibor O. Zizka, Nov 29 2008

Keywords

Comments

k : A001157(k)/(A000203(k)*A000005(k)) = c, c an integer.

Links

Crossrefs

Cf. A000005, A000203, A001157, A003601, A094471, A140480, A003601, A020486, A020487

Programs

  • Mathematica
    Select[Range[50000],IntegerQ[DivisorSigma[2,#]/(DivisorSigma[1,#] DivisorSigma[ 0,#])]&] (* Harvey P. Dale, Feb 12 2013 *)
  • PARI
    isok(k) = denominator(sigma(k,2)/(sigma(k, 1)*sigma(k,0))) == 1; \\ Michel Marcus, Jul 15 2019

Extensions

More terms from Harvey P. Dale, Feb 12 2013

A279814 Numbers n such that the average of the squares of the proper divisors of n is an integer.

Original entry on oeis.org

2, 3, 5, 7, 8, 9, 10, 11, 13, 14, 17, 19, 22, 23, 25, 26, 29, 31, 34, 35, 37, 38, 41, 43, 46, 47, 49, 50, 53, 55, 58, 59, 61, 62, 65, 67, 71, 73, 74, 75, 77, 79, 80, 81, 82, 83, 85, 86, 89, 91, 94, 95, 97, 101, 103, 106, 107, 109, 113, 115, 118, 119, 121, 122, 125, 127, 131, 133, 134, 137, 139, 140, 142, 143, 145, 146, 149
Offset: 1

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Author

Ilya Gutkovskiy, Dec 19 2016

Keywords

Comments

Numbers n such that number of proper divisors of n (A032741) divides sum of squares of proper divisors of n (A067558).
All the prime numbers are in this sequence.

Examples

			8 is in the sequence because 8 has 3 proper divisors {1,2,4}, 1^2 + 2^2 + 4^2 = 21 and 3 divides 21.

Links

Crossrefs

Cf. A020486, A023884, A032741, A067558.

Programs

  • Mathematica
    Select[Range[150], Mod[DivisorSigma[2, #1] - #1^2, DivisorSigma[0, #1] - 1] == 0 &]
Select[Range[200],IntegerQ[Mean[Most[Divisors[#]]^2]]&] (* Harvey P. Dale, Jul 26 2019 *)
  • PARI
    is(n)=my(d=divisors(n)); d=apply(k->k^2, d[1..#d-1]); n>1 && sum(i=1,#d,d[i])%#d==0 \\ Charles R Greathouse IV, Dec 19 2016

A279815 Numbers n such that the average of the squares of the numbers less than n that do not divide n is an integer.

Original entry on oeis.org

3, 4, 7, 13, 16, 19, 20, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139, 151, 157, 163, 181, 188, 193, 199, 211, 223, 229, 241, 271, 277, 283, 307, 313, 331, 337, 349, 367, 373, 379, 397, 409, 421, 433, 439, 457, 463, 487, 499, 523, 541, 547, 571, 577, 601, 607, 613, 619, 631, 643, 661, 673, 691, 709, 727, 733, 739, 751, 757, 769, 787
Offset: 1

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Author

Ilya Gutkovskiy, Dec 19 2016

Keywords

Comments

Numbers n such that A049820(n) divides A276984(n).

Examples

			7 is in the sequence because 7 has 2 divisors {1,7} therefore 5 non-divisors {2,3,4,5,6}, 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 90 and 5 divides 90.

Links

Crossrefs

Cf. A020486, A049820, A140826, A276984.
Superset of A045375(m) (m > 1) ?

Programs

  • Mathematica
    Select[Range[800], Mod[#1 (#1 + 1) ((2 #1 + 1)/6) - DivisorSigma[2, #1], #1 - DivisorSigma[0, #1]] == 0 & ]
  • PARI
    is(n)=my(f=factor(n)); n>2 && ((2*n^3+3*n^2+n)/6-sigma(f,2))%(n-numdiv(f))==0 \\ Charles R Greathouse IV, Dec 19 2016

A361683 a(n) is the least k such that tau(k) divides sigma_n(k) but not sigma(k), or -1 if no such k exists.

Original entry on oeis.org

4, 64, 4, 7168, 4, 606528, 4, 64, 4, 4194304, 4
Offset: 2

Views

Author

Mohammed Yaseen, Mar 20 2023

Keywords

Comments

a(13) <= 31525197391593472. - David A. Corneth, Mar 20 2023
From Thomas Scheuerle, Mar 22 2023: (Start)
a(17) <= 15211807202738752817960438464512 and a(19) <= 2^190*11.
Conjecture: a(n) is of the form 2^b*p1^c*p2^d*...*pk^j with b > 0 and A020639(n) divides b*(c+1)*(d+1)*...*(j+1). (p1, p2, ..., pk are distinct odd prime numbers). (End)

Crossrefs

Cf. A003601, A020486, A046839, A046840.
Cf. A000005, A000203, A001157, A001158.
Cf. A020639, A010709 (bisection).

Programs

  • Mathematica
    a[n_] := Module[{k = 1, d}, While[Divisible[DivisorSigma[1, k], (d = DivisorSigma[0, k])] || !Divisible[DivisorSigma[n, k], d], k++]; k]; Array[a, 11, 2] (* Amiram Eldar, Mar 20 2023 *)
  • PARI
    isok(k, n) = my(f=factor(k), nd=numdiv(f)); (sigma(f) % nd) && !(sigma(f,n) % nd);
a(n) = my(k=1); while (!isok(k,n), k++); k; \\ Michel Marcus, Mar 20 2023

Formula

a(2*m) = 4 for m >= 1.
a(6*m-3) = 64 for m >= 1.
From Thomas Scheuerle, Mar 22 2023: (Start)
a(m) <= a(A020639(m)) if a(A020639(m)) <> -1.
Conjecture: For primes q > p, a(q) > a(p). If true, we could replace "<=" with "=" in the above formula. (End)
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