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 1-6 of 6 results.

A007691 Multiply-perfect numbers: n divides sigma(n).

Original entry on oeis.org

1, 6, 28, 120, 496, 672, 8128, 30240, 32760, 523776, 2178540, 23569920, 33550336, 45532800, 142990848, 459818240, 1379454720, 1476304896, 8589869056, 14182439040, 31998395520, 43861478400, 51001180160, 66433720320, 137438691328, 153003540480, 403031236608
Offset: 1

Views

Author

N. J. A. Sloane, Robert G. Wilson v

Keywords

Comments

sigma(n)/n is in A054030.
Also numbers such that the sum of the reciprocals of the divisors is an integer. - Harvey P. Dale, Jul 24 2001
Luca's solution of problem 11090, which proves that for k>1 there are an infinite number of n such that n divides sigma_k(n), does not apply to this sequence. However, it is conjectured that this sequence is also infinite. - T. D. Noe, Nov 04 2007
Numbers k such that sigma(k) is divisible by all divisors of k, subsequence of A166070. - Jaroslav Krizek, Oct 06 2009
A017666(a(n)) = 1. - Reinhard Zumkeller, Apr 06 2012
Bach, Miller, & Shallit show that this sequence can be recognized in polynomial time with arbitrarily small error by a probabilistic Turing machine; that is, this sequence is in BPP. - Charles R Greathouse IV, Jun 21 2013
Conjecture: If n is such that 2^n-1 is in A066175 then a(n) is a triangular number. - Ivan N. Ianakiev, Aug 26 2013
Conjecture: Every multiply-perfect number is practical (A005153). I've verified this conjecture for the first 5261 terms with abundancy > 2 using Achim Flammenkamp's data. The even perfect numbers are easily shown to be practical, but every practical number > 1 is even, so a weak form says every even multiply-perfect number is practical. - Jaycob Coleman, Oct 15 2013
Numbers such that A054024(n) = 0. - Michel Marcus, Nov 16 2013
Numbers n such that k(n) = A229110(n) = antisigma(n) mod n = A024816(n) mod n = A000217(n) mod n = (n(n+1)/2) mod n = A142150(n). k(n) = n/2 for even n; k(n) = 0 for odd n (for number 1 and eventually odd multiply-perfect numbers n > 1). - Jaroslav Krizek, May 28 2014
The only terms m > 1 of this sequence that are not in A145551 are m for which sigma(m)/m is not a divisor of m. Conjecture: after 1, A323653 lists all such m (and no other numbers). - Antti Karttunen, Mar 19 2021

Examples

			120 is OK because divisors of 120 are {1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120}, the sum of which is 360=120*3.

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 22.
  • J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 176.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • I. Stewart, L'univers des nombres, "Les nombres multiparfaits", Chapter 15, pp. 82-88, Belin-Pour La Science, Paris 2000.
  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 141-148.
  • David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, pp. 135-136.

Links

Crossrefs

Complement is A054027. Cf. A000203, A054030.
Cf. A000396, A005820, A027687, A046060, A046061, for subsequences of terms with quotient sigma(n)/n = 2..6.
Other subsequences: A046985, A046986, A046987, A047728, A065997, A066289, (A076231, A076233, A076234), A088844, A088845, A088846, A091443, A114887, A166069, A245782, A260508, A306667, (A325021 U A325022), (A325023 U A325024), (A325025 U A325026), A325637, A323653, A330532, (A330533 U A331724), A336702, A341045.
Subsequence of the following sequences: A011775, A071707, A083865, A089748 (after the initial 1), A102783, A166070, A175200, A225110, A226476, A237719, A245774, A246454, A259307, A263928, A282775, A323652, A336745, A340864. Also conjectured to be a subsequence of A005153, of A307740, and after 1 also of A295078.
Various number-theoretical functions applied to these numbers: A088843 [tau], A098203 [phi], A098204 [gcd(a(n),phi(a(n)))], A134665 [2-adic valuation], A307741 [sigma], A308423 [product of divisors], A320024 [the odd part], A134740 [omega], A342658 [bigomega], A342659 [smallest prime not dividing], A342660 [largest prime divisor].
Positions of ones in A017666, A019294, A094701, A227470, of zeros in A054024, A082901, A173438, A272008, A318996, A326194, A341524. Fixed points of A009194.
Cf. A069926, A330746 (left inverses, when applied to a(n) give n).
Cf. A007358, A189000, A327158, A332318/A332319 (for analogs) and A046762, A046763, A046764, A055715, A056006, A081756, A214842, A227302, A227306, A245775, A300906, A325639 (other variants).
Cf. (other related sequences) A007539, A066135, A066961, A093034, A094467, A134639, A145551, A019278, A194771 [= 2*a(n)], A219545, A229110, A262432, A335830, A336849, A341608.

Programs

  • Haskell
    a007691 n = a007691_list !! (n-1)
a007691_list = filter ((== 1) . a017666) [1..]
-- Reinhard Zumkeller, Apr 06 2012
  • Mathematica
    Do[If[Mod[DivisorSigma[1, n], n] == 0, Print[n]], {n, 2, 2*10^11}] (* or *)
Transpose[Select[Table[{n, DivisorSigma[-1, n]}, {n, 100000}], IntegerQ[ #[[2]] ]& ] ][[1]]
(* Third program: *)
Select[Range[10^6], IntegerQ@ DivisorSigma[-1, #] &] (* Michael De Vlieger, Mar 19 2021 *)
  • PARI
    for(n=1,1e6,if(sigma(n)%n==0, print1(n", ")))
  • Python
    from sympy import divisor_sigma as sigma
def ok(n): return sigma(n, 1)%n == 0
print([n for n in range(1, 10**4) if ok(n)]) # Michael S. Branicky, Jan 06 2021

Extensions

More terms from Jud McCranie and then from David W. Wilson.
Incorrect comment removed and the crossrefs-section reorganized by Antti Karttunen, Mar 20 2021

A325634 a(n) = A091255(n, sigma(n)).

Original entry on oeis.org

1, 1, 1, 1, 3, 6, 1, 1, 1, 6, 1, 4, 1, 2, 3, 1, 3, 3, 1, 2, 1, 2, 3, 12, 1, 2, 5, 28, 3, 6, 1, 1, 3, 10, 1, 1, 1, 2, 1, 10, 1, 2, 1, 4, 5, 6, 1, 4, 1, 1, 3, 2, 3, 10, 1, 8, 5, 6, 1, 4, 1, 2, 1, 1, 21, 6, 1, 6, 1, 14, 3, 9, 1, 2, 1, 4, 3, 2, 1, 2, 1, 2, 7, 28, 5, 6, 1, 4, 3, 2, 1, 4, 1, 2, 5, 12, 1, 1, 5, 1, 3, 10, 1, 2, 3
Offset: 1

Views

Author

Antti Karttunen, May 21 2019

Keywords

Links

Crossrefs

Cf. A000203, A009194, A091255, A169813, A325632, A325633, A325635, A325639 (fixed points, n such that a(n) = n).

Programs

  • PARI
    A091255sq(a,b) = fromdigits(Vec(lift(gcd(Pol(binary(a))*Mod(1, 2),Pol(binary(b))*Mod(1, 2)))),2);
A325634(n) = A091255sq(n, sigma(n));

Formula

a(n) = A091255(n, A000203(n)).
a(n) = A091255(n, A169813(n)) = A091255(A000203(n), A169813(n)).

A379114 Numbers k for which A379113(k) > 1, i.e., k that have a proper unitary divisor d > 1 such that A048720(A065621(sigma(d)),sigma(k/d)) is equal to sigma(k).

Original entry on oeis.org

6, 10, 12, 14, 15, 20, 21, 22, 24, 26, 28, 30, 33, 35, 39, 40, 42, 44, 46, 48, 50, 51, 52, 55, 56, 57, 58, 60, 62, 63, 65, 66, 69, 70, 72, 75, 76, 77, 78, 80, 84, 86, 87, 88, 91, 92, 93, 94, 96, 100, 102, 104, 105, 108, 111, 112, 114, 115, 116, 118, 119, 120, 122, 123, 124, 126, 129, 132, 133, 136, 138, 140, 141, 143, 144
Offset: 1

Views

Author

Antti Karttunen, Dec 17 2024

Keywords

Links

Crossrefs

Positions of terms > 1 in A379113.
Cf. A000396 (subsequence, at least the even terms are), A379118 (characteristic function).
Cf. also A325638, A325639 (not subsequences).

Programs

A325632 a(n) = gcd(n, A325634(n)) = gcd(n, A091255(n, sigma(n))).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 4, 1, 2, 3, 1, 1, 3, 1, 2, 1, 2, 1, 12, 1, 2, 1, 28, 1, 6, 1, 1, 3, 2, 1, 1, 1, 2, 1, 10, 1, 2, 1, 4, 5, 2, 1, 4, 1, 1, 3, 2, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 1, 1, 6, 1, 2, 1, 14, 1, 9, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 28, 5, 2, 1, 4, 1, 2, 1, 4, 1, 2, 5, 12, 1, 1, 1, 1, 1, 2, 1, 2, 3
Offset: 1

Views

Author

Antti Karttunen, May 21 2019

Keywords

Links

Crossrefs

Cf. A000203, A091255, A325634, A325639 (fixed points).
Differs from A009194 for the first time at n=42, where a(42) = 2, while A009194(42) = 6.
Differs from A325633 and A325640 for the first time at n=45, where a(45) = 5, while A325633(45) = 1 and A325640(45) = 3.

Programs

Formula

a(n) = gcd(n, A325634(n)) = gcd(n, A091255(n, A000203(n))).

A325638 Numbers m such that sigma(m) can be obtained as the base-2 carryless product of 2m and some k.

Original entry on oeis.org

6, 28, 456, 496, 6552, 8128, 30240, 31452, 32760, 429240, 2178540, 7505976, 23569920, 33550336, 45532800, 142990848, 1379454720
Offset: 1

Views

Author

Antti Karttunen, May 21 2019

Keywords

Comments

Numbers m such that A000203(m) = A048720(2m, k) for some k.
Numbers m for which A091255(2m, sigma(m)) = 2m.
Conjecture: all terms are even. If this is true, then there are no odd perfect numbers. See also conjectures in A325639 and in A325808.

Links

Crossrefs

Cf. A000203, A091255, A325635, A325637, A325808.
Subsequence of A325639.
Cf. A000396 (a subsequence).

Programs

  • PARI
    A091255sq(a,b) = fromdigits(Vec(lift(gcd(Pol(binary(a))*Mod(1, 2),Pol(binary(b))*Mod(1, 2)))),2);
A325635(n) = A091255sq(n+n, sigma(n));
isA325638(n) = ((n+n)==A325635(n));

Extensions

a(17) from Amiram Eldar, Jun 26 2024

A325808 Numbers n such that sigma(n) can be obtained as the base-3 carryless product of 2n and some k.

Original entry on oeis.org

1, 6, 28, 120, 259, 496, 8128, 18990, 667296, 1858939, 2097414, 2383279, 4843717, 33550336, 150588313, 186695863, 188908297
Offset: 1

Views

Author

Antti Karttunen, May 22 2019

Keywords

Comments

Numbers n that satisfy A000203(n) = A325820(2n, k) for some k.
Numbers n such that polynomial p divides polynomial q over GF(3), where p and q are obtained from the base-3 representations of 2n and sigma(n). (See the examples).
Conjecture: If we select only those n of these for which sigma(n) >= 2n, then we get a subsequence which contains only even terms: 6, 28, 120, 496, 8128, 18990, 667296, 2097414, 33550336, etc. If this is true, then there are no odd perfect numbers. See also conjectures in A325638 and A325639.

Examples

			2*120 has ternary representation (A007089) 22220_3, thus it encodes polynomial 2*x^4 + 2*x^3 + 2*x^2 + 2*x, while sigma(120) = 360 = 111100_3, encodes polynomial x^5 + x^4 + x^3 + x^2 which is a multiple of the former as it is equal to 2x(x^4 + x^3 + x^2 + x) when polynomial multiplication is done over GF(3). Thus 120 is included in this sequence.
2*259 = 201012_3 encodes polynomial 2*x^5 + x^3 + x + 2, while sigma(259) = 304 = 102021_3 encodes polynomial x^5 + 2*x^3 + 2*x + 1 = 2(2*x^5 + x^3 + x + 2), thus 259 is included.
2*18990 = 1221002200_3 encodes polynomial x^9 + 2*x^8 + 2*x^7 + x^6 + 2*x^3 + 2*x^2, while sigma(18990) = 49608 = 2112001100_3 encodes polynomial 2*x^9 + x^8 + x^7 + 2*x^6 + x^3 + x^2 = 2(x^9 + 2*x^8 + 2*x^7 + x^6 + 2*x^3), thus 18990 is included.
2*667296 = 2111210201100_3 encodes polynomial 2*x^12 + x^11 + x^10 + x^9 + 2*x^8 + x^7 + 2*x^5 + x^3 + x^2, while sigma(667296) = 2175264 = 11002111220100_3 encodes polynomial x^13 + x^12 + 2*x^9 + x^8 + x^7 + x^6 + 2*x^5 + 2*x^4 + x^2 = (2*x + 1)(2*x^12 + x^11 + x^10 + x^9 + 2*x^8 + x^7 + 2*x^5 + x^3 + x^2) [when polynomial multiplication is done over GF(3)], thus 667296 is included.

Links

Crossrefs

Cf. A000203, A325638, A325820, A325825, A325827.
Cf. A000396 (a subsequence).

Programs

  • PARI
    isA325808(n) = { my(p=Pol(digits(n+n,3))*Mod(1, 3), q=Pol(digits(sigma(n),3))*Mod(1, 3)); !(q%p); };
Showing 1-6 of 6 results.

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