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.

Showing 1-5 of 5 results.

A229087 a(n) = sigma(n) mod n - antisigma(n) mod n, where sigma(n) = A000203(n) = sum of divisor of n, antisigma(n) = A024816(n) = sum of non-divisors of n.

Original entry on oeis.org

0, 1, -1, 0, -3, -3, -5, 2, -1, 1, -9, 2, -11, -1, 3, 6, -15, -3, -17, -6, 1, -5, -21, 12, -13, -7, -1, -14, -27, 9, -29, 14, -3, -11, -9, -16, -35, -13, -5, 0, -39, 3, -41, 14, 21, -17, -45, -16, -33, 11, -9, 14, -51, -3, -21, -12, -11, -23, -57, 6, -59, -25
Offset: 1

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Author

Jaroslav Krizek, Oct 24 2013

Keywords

Comments

Sequence contains anomalous increased frequency of values 14 (see A229115), a(n) = 14 for n = 32, 44, 52, 68, 76, 92, ... ).

Examples

			For n = 32; a(32 ) = sigma(32) mod 32 - antisigma(32) mod 32 = 63 mod 32 - 465 mod 32 = 31 - 17 = 14.

Links

Crossrefs

Cf. A000203 (sigma(n)), A024816 (antisigma(n)).
Cf. A054024 (sigma(n) mod n), A229110(antisigma(n) mod n).
Cf. A229088 (numbers n such that sigma(n) mod n = antisigma(n) mod n).
Cf. A229089 (numbers n such that sigma(n) mod n < antisigma(n) mod n).
Cf. A229090 (numbers n such that sigma(n) mod n > antisigma(n) mod n).

Formula

a(n) = A000203(n) mod n - A024816(n) mod n = A054024(n) - A229110(n).

A229090 Numbers n such that sigma(n) mod n > antisigma(n) mod n, where sigma(n) = A000203(n) = sum of divisors of n, antisigma(n) = A024816(n) = sum of non-divisors of n.

Original entry on oeis.org

2, 8, 10, 12, 15, 16, 21, 24, 30, 32, 42, 44, 45, 50, 52, 60, 63, 64, 68, 75, 76, 80, 92, 99, 105, 110, 116, 117, 124, 126, 128, 130, 135, 136, 140, 144, 147, 148, 150, 152, 153, 154, 160, 164, 165, 168, 170, 171, 172, 182, 184, 188, 189, 190, 195, 198, 200
Offset: 1

Views

Author

Jaroslav Krizek, Oct 24 2013

Keywords

Comments

Numbers n such that A229087(n) = A000203(n) mod n - A024816(n) mod n = A054024(n) - A229110(n) > 0.
Complement of union A229088 and A229089 with respect to A000027, where A229088 = numbers n such that sigma(n) mod n = antisigma(n) mod n, A229089 = numbers n such that sigma(n) mod n < antisigma(n) mod n.

Examples

			Number 12 is in sequence because sigma(12) mod 12 = 28 mod 12 = 4 > antisigma(12) mod 12 = 50 mod 12 = 2.

Links

Crossrefs

Cf. A000203 (sigma(n)), A024816 (antisigma(n)).
Cf. A054024 (sigma(n) mod n), A229110 (antisigma(n) mod n).

Programs

  • Mathematica
    smQ[n_]:=Module[{sig=DivisorSigma[1,n]},Mod[sig,n]>Mod[(n(n+1))/2-sig,n]]; Select[Range[200],smQ] (* Harvey P. Dale, Dec 23 2013 *)

A229089 Numbers n such that sigma(n) mod n < antisigma(n) mod n, where sigma(n) = A000203(n) = sum of divisor of n, antisigma(n) = A024816(n) = sum of non-divisors of n.

Original entry on oeis.org

3, 5, 6, 7, 9, 11, 13, 14, 17, 18, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 33, 34, 35, 36, 37, 38, 39, 41, 43, 46, 47, 48, 49, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 72, 73, 74, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 93
Offset: 1

Views

Author

Jaroslav Krizek, Oct 24 2013

Keywords

Comments

Numbers n such that A229087(n) = A000203(n) mod n - A024816(n) mod n =A054024(n) - A229110(n) < 0.
Complement of union A229088 and A229090 with respect to A000027, where
A229088 = numbers n such that sigma(n) mod n = antisigma(n) mod n,
A229090 = numbers n such that sigma(n) mod n > antisigma(n) mod n.

Examples

			Number 11 is in sequence because sigma(11) mod 11 = 12 mod 11 = 1 < antisigma(11) mod 11 = 54 mod 11 = 10.

Links

Crossrefs

Cf. A000203 (sigma(n)), A024816 (antisigma(n)), A229110 (antisigma(n) mod n), A054024 (sigma(n) mod n).

Programs

  • Mathematica
    Select[Range[100],Mod[Total[Complement[Range[#],Divisors[#]]],#]> Mod[ DivisorSigma[ 1,#],#]&] (* Harvey P. Dale, Jan 24 2022 *)

A067237 Numbers k such that gcd(sigma(k),k) = k/5.

Original entry on oeis.org

5, 10, 15, 30, 60, 90, 140, 420, 1170, 2480, 3360, 6200, 7440, 8190, 18600, 40640, 114660, 121920, 131040, 297600, 997920, 2618880, 5059200, 64995840, 72602880, 95472000, 102136320, 167751680, 197308800, 433305600, 503255040, 668304000, 714954240, 1307124000, 1381161600, 1502582400
Offset: 1

Views

Author

Benoit Cloitre, Feb 20 2002

Keywords

Comments

Also numbers k such that denominator(sigma(k)/k) = 5. - David A. Corneth, Oct 15 2023

Examples

			30 is in the sequence as gcd(sigma(30), 30) = gcd(72, 30) = 6 = 30/5. - _David A. Corneth_, Oct 15 2023

Links

Crossrefs

Cf. A000203.
Cf. similar sequences with A017666(n)=k: A159907 (k=2), A245775 (k=3), A229088 (k=4), A262359 (k=6).

Programs

Extensions

More terms from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Nov 15 2004

A262359 Numbers k such that denominator(sigma(k)/k) = 6.

Original entry on oeis.org

18, 22932, 14520576, 1610563584, 1907020800, 2836487808, 6399679104, 70912195200, 82819376640, 159991977600, 2732372020224, 6164773235712, 68309300505600, 148068998977536, 154119330892800, 264727305267840, 649657533767040, 3701724974438400, 42503412523106304, 220312341220608000
Offset: 1

Views

Author

Michel Marcus, Sep 19 2015

Keywords

Examples

			sigma(18)/18 = 13/6, hence 18 is a term.

Links

Crossrefs

Cf. similar sequences with A017666(n)=k: A159907 (k=2), A245775 (k=3), A229088 (k=4), A067237 (k=5).
Cf. A000203.

Programs

  • PARI
    isok(n) = denominator(sigma(n, -1)) == 6;
  • PARI
    lista(nn) = {k = 6; nb = 0; while (nb != nn, if (denominator(sigma(k,-1)) == 6, print1(k, ", "); nb++); k += 6;);}

Extensions

a(10)-a(12) from Jud McCranie, Oct 14 2023
More terms from David A. Corneth, Oct 15 2023
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