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 1-10 of 11 results. Next

A204824 Duplicate of A156903.

Original entry on oeis.org

18, 36, 72, 100, 144, 162, 196, 200, 288, 324, 392, 400, 450, 576, 648, 784, 800, 882, 900, 968, 1152, 1296, 1352, 1458, 1568, 1600, 1764, 1800, 1936, 2178, 2304, 2450, 2500, 2592, 2704, 2916, 3042, 3136, 3200, 3528, 3600, 3872, 4050, 4356, 4608, 4624, 4900, 5000
Offset: 1

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Author

Keywords

A171641 Non-deficient numbers with even sigma which are not Zumkeller.

Original entry on oeis.org

738, 748, 774, 846, 954, 1062, 1098, 1206, 1278, 1314, 1422, 1494, 1602, 1746, 1818, 1854, 1926, 1962, 2034, 2286, 2358, 2466, 2502, 2682, 2718, 2826, 2934, 3006, 3114, 3222, 3258, 3438, 3474, 3492, 3546, 3582, 3636, 3708, 3798, 3852, 3924, 4014, 4068, 4086
Offset: 1

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Author

Peter Luschny, Dec 14 2009

Keywords

Comments

Numbers which are non-deficient (sigma(n) >= 2n) [A023196] such that sigma(n) [A000203] is even but which are not Zumkeller numbers [A083207], i.e., the positive factors of n cannot be partitioned into two disjoint parts so that the sums of the two parts are equal.

Crossrefs

Setwise difference A083211 \ A156903.
Positions of even negative terms in A378600.

Programs

  • Maple
    with(NumberTheory):
    isA171641 := proc(n) local s, p, i, P;
        s := SumOfDivisors(n);
        if s::odd or s < n*2 then false else
        P := mul(1 + x^i, i in Divisors(n));
        0 = coeff(P, x, s/2) fi end:
    select(isA171641, [seq(1..4100)]);  # Peter Luschny, Oct 19 2024
  • Mathematica
    Reap[For[n = 2, n <= 4000, n = n+2, sigma = DivisorSigma[1, n]; If[sigma >= 2n && EvenQ[sigma] && Coefficient[ Times @@ (1 + x^Divisors[n]) // Expand, x, sigma/2] == 0, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jul 26 2013 *)
  • Python
    from sympy import divisors
    import numpy as np
    A171641 = []
    for n in range(2,10**6):
        d = divisors(n)
        s = sum(d)
        if not s % 2 and 2*n <= s:
            d.remove(n)
            s2, ld = int(s/2-n), len(d)
            z = np.zeros((ld+1,s2+1),dtype=int)
            for i in range(1,ld+1):
                y = min(d[i-1],s2+1)
                z[i,range(y)] = z[i-1,range(y)]
                z[i,range(y,s2+1)] = np.maximum(z[i-1,range(y,s2+1)],z[i-1,range(0,s2+1-y)]+y)
            if z[ld,s2] != s2:
                A171641.append(n)
    # Chai Wah Wu, Aug 19 2014

A156942 Odd abundant numbers whose abundance is odd.

Original entry on oeis.org

11025, 99225, 245025, 275625, 342225, 540225, 893025, 1334025, 1863225, 2205225, 2480625, 3080025, 3186225, 3980025, 4601025, 4862025, 5832225, 6125625, 6890625, 7868025, 8037225, 8555625, 9272025, 9828225, 10595025, 10989225
Offset: 1

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Author

Robert G. Wilson v, Feb 18 2009

Keywords

Comments

Number of terms <10^n: 0, 0, 0, 0, 2, 7, 24, 83, 250, 792, 2484, 7988, 25383, 80082, ..., . Not all are a multiple of 25, i.e.; 81162081 = 9009^2 = (9*7*11*13)^2. See A156943.
Any term must be an odd square. Square roots are in A174830.
Indeed, the sum of divisors of any number isn't odd unless it's a square or twice a square (A028982), and to get the abundance, twice the number is subtracted, so the parity remains the same. - M. F. Hasler, Jan 26 2020
Question: Is this a subsequence of A379503? (Is A379504(a(n)) > 0 for all n? See A379951). The first 15000 terms are all included there. - Amiram Eldar and Antti Karttunen, Jan 06 2025
Question 2: Is A379505(a(n)) > 1 for all n, especially if there are no quasiperfect numbers (numbers k such that sigma(k) = 2k+1)? - Antti Karttunen, Jan 06 2025
From Amiram Eldar, Jan 16 2025: (Start)
The least term that is not divisible by 5 is a(75) = 81162081.
The least term that is not divisible by 3 is a(296889) = 1382511906801025.
The least term that is coprime to 15 is 15285071557677427358507559514565648611799881. (End)

Crossrefs

Subsequences: A156943, A325311 (thus also A379490), A347890, A379949 (terms that are primitive abundant).

Programs

  • Mathematica
    fQ[n_] := Block[{ds = DivisorSigma[1, n] - 2 n}, ds > 0 && OddQ@ ds]; Select[ Range[1, 12006223, 2], fQ @# &]
  • PARI
    is(n)=my(s=sigma(n)); n%2 && s>2*n && (s-2*n)%2 \\ Charles R Greathouse IV, Feb 21 2017

Formula

a(n) = A174830(n)^2. - M. F. Hasler, Jan 26 2020

Extensions

Edited by Robert G. Wilson v at the suggestion of T. D. Noe, Mar 30 2010

A083211 Abundant numbers (A005101) with no subset of their divisors such that the complement has the same sum.

Original entry on oeis.org

18, 36, 72, 100, 144, 162, 196, 200, 288, 324, 392, 400, 450, 576, 648, 738, 748, 774, 784, 800, 846, 882, 900, 954, 968, 1062, 1098, 1152, 1206, 1278, 1296, 1314, 1352, 1422, 1458, 1494, 1568, 1600, 1602, 1746, 1764, 1800, 1818, 1854, 1926, 1936, 1962, 2034, 2178, 2286, 2304, 2358, 2450, 2466, 2500, 2502, 2592, 2682, 2704, 2718, 2826, 2916, 2934, 3006, 3042
Offset: 1

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Author

Reinhard Zumkeller, Apr 22 2003

Keywords

Comments

A083206(a(n)) = 0; subsequence of A083210.
All [abundant] numbers with an odd sum of divisors (either a square or twice a square, A028982) must be terms because for these numbers the two subsets will be of opposite parity. - Robert G. Wilson v, Apr 01 2010, clarified by Antti Karttunen, Dec 05 2024

Examples

			Divisors of n=18: {1,2,3,6,9,18}; 18 is pseudo-perfect (A005835): 18=9+6+3, but there exist no two complementary subsets of divisors having the same sum, therefore 18 is a term.
		

Crossrefs

Intersection of A005101 and A083210.
Disjoint union of A156903 and A171641. - Amiram Eldar, Jun 20 2020
Positions of negative terms in A378600.
Cf. A000203, A028982, A083206, A156942 (odd terms), A378661 (characteristic function).

Programs

  • Mathematica
    fQ[n_] := Block[{d = Divisors[n], t, ds, x}, ds = Total[d]; If[Mod[ds, 2] > 0, False, t = CoefficientList[Product[1 + x^i, {i, d}], x]; t[[1 + ds/2]] > 0]]; Select[Range[3042], And[DivisorSigma[1, #] > 2 #, ! fQ[#]] &] (* Michael De Vlieger, Dec 04 2024, after T. D. Noe at A083207 *)
  • PARI
    A083206(n) = { my(s=sigma(n),p=1); if(s%2 || s < 2*n, 0, fordiv(n, d, p *= ('x^d + 'x^-d)); (polcoeff(p, 0)/2)); };
    is_A083211(n) = ((sigma(n)>2*n) && (0==A083206(n))); \\ Antti Karttunen, Dec 04 2024

Formula

{k such that sigma(k) > 2*k and A083206(k) = 0}. - Antti Karttunen, Dec 04 2024

Extensions

a(21)-a(46) from Robert G. Wilson v, Apr 01 2010
Many missing terms inserted, first ones at a(29) = 1206 and a(30) = 1278 - Antti Karttunen, Dec 04 2024

A378600 Signed variant of Zumkeller deficiency: a(n) = signum(A033879(n)) * A103977(n).

Original entry on oeis.org

1, 1, 2, 1, 4, 0, 6, 1, 5, 2, 10, 0, 12, 4, 6, 1, 16, -1, 18, 0, 10, 8, 22, 0, 19, 10, 14, 0, 28, 0, 30, 1, 18, 14, 22, -1, 36, 16, 22, 0, 40, 0, 42, 4, 12, 20, 46, 0, 41, 7, 30, 6, 52, 0, 38, 0, 34, 26, 58, 0, 60, 28, 22, 1, 46, 0, 66, 10, 42, 0, 70, -1, 72, 34, 26, 12, 58, 0, 78, 0, 41, 38, 82, 0, 62, 40, 54, 0, 88, 0, 70
Offset: 1

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Author

Antti Karttunen, Dec 04 2024

Keywords

Comments

If n is abundant, then negate the value of A103977(n), otherwise use as it is.

Crossrefs

Cf. A005100 (positions of terms > 0), A083207 (positions of 0's), A083211 (positions of negative terms), A156903 (positions of odd negative terms), A171641 (of even negative terms).

Programs

  • PARI
    A033879(n) = (n+n-sigma(n));
    nonzerocoefpositions(p) = { my(v=Vec(p), lista=List([])); for(i=1,#v,if(v[i], listput(lista,i))); Vec(lista); };
    A103977(n) = { my(p=1); fordiv(n, d, p *= (1 + 'x^d)); my(plist=nonzerocoefpositions(p), m = #plist, d); if(!(m%2), plist[1+(m/2)]-plist[m/2], d = plist[(m+1)/2]-plist[(m-1)/2]; if(1==d,0,d)); };
    A378600(n) = { my(d=A033879(n)); if(d>=0, d, -A103977(n)); };

Formula

If A033879(n) >= 0, a(n) = A033879(n), otherwise a(n) = -A103977(n).

A204825 Abundant numbers with even sum of divisors.

Original entry on oeis.org

12, 20, 24, 30, 40, 42, 48, 54, 56, 60, 66, 70, 78, 80, 84, 88, 90, 96, 102, 104, 108, 112, 114, 120, 126, 132, 138, 140, 150, 156, 160, 168, 174, 176, 180, 186, 192, 198, 204, 208, 210, 216, 220, 222, 224, 228, 234, 240, 246, 252, 258, 260, 264, 270, 272, 276, 280
Offset: 1

Views

Author

Jaroslav Krizek, Jan 22 2012

Keywords

Comments

Numbers m from A005101 such that sigma(m) = A000203(m) is even.
Complement of A156903 with respect to A005101 (abundant numbers).

Examples

			Abundant number 12 is in sequence because sigma(12) = 28 (even number).
		

Crossrefs

Cf. A156903 (abundant numbers with odd sum of divisors), A204826 (deficient numbers with odd sum of divisors), A204827 (deficient numbers with even sum of divisors).

Programs

  • Mathematica
    aQ[n_] := EvenQ[(s = DivisorSigma[1, n])] && s > 2n; Select[Range[280], aQ] (* Amiram Eldar, Sep 02 2019 *)

A204826 Deficient numbers with odd sum of divisors.

Original entry on oeis.org

1, 2, 4, 8, 9, 16, 25, 32, 49, 50, 64, 81, 98, 121, 128, 169, 225, 242, 256, 289, 338, 361, 441, 484, 512, 529, 578, 625, 676, 722, 729, 841, 961, 1024, 1058, 1089, 1156, 1225, 1250, 1369, 1444, 1521, 1681, 1682, 1849, 1922, 2025, 2048, 2116, 2209, 2312, 2401
Offset: 1

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Author

Jaroslav Krizek, Jan 22 2012

Keywords

Comments

Numbers m from A005100 such that sigma(m) = A000203(m) is odd.
Complement of A204827 with respect to A005100 (deficient numbers).

Examples

			Deficient number 16 is in sequence because sigma(16) = 13 (odd number).
		

Crossrefs

Cf. A156903 (abundant numbers with odd sum of divisors), A204825 (abundant numbers with even sum of divisors), A204827 (deficient numbers with even sum of divisors).

Programs

  • Mathematica
    aQ[n_] := OddQ[(s = DivisorSigma[1, n])] && s < 2n; Select[Range[2401], aQ] (* Amiram Eldar, Sep 02 2019 *)

A204827 Deficient numbers with even sum of divisors.

Original entry on oeis.org

3, 5, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 27, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 82, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 99, 101, 103
Offset: 1

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Author

Jaroslav Krizek, Jan 22 2012

Keywords

Comments

Numbers m from A005100 such that sigma(m) = A000203(m) is even.
Complement of A204826 with respect to A005100 (deficient numbers).

Examples

			Deficient number 15 is in sequence because sigma(15) = 24 (even number).
		

Crossrefs

Cf. A156903 (abundant numbers with odd sum of divisors), A204825 (abundant numbers with even sum of divisors), A204826 (deficient numbers with odd sum of divisors).

Programs

  • Mathematica
    dnesQ[n_]:=Module[{s=DivisorSigma[1,n]},s<2n&&EvenQ[s]]; Select[ Range[ 120],dnesQ] (* Harvey P. Dale, Nov 23 2014 *)

A140863 Odd numbers k such that sigma(m) = 2m+k has a solution in m, where sigma is the sum-of-divisors function A000203.

Original entry on oeis.org

3, 7, 17, 19, 31, 39, 41, 51, 59, 65, 71, 89, 115, 119, 127, 161, 185, 199, 215, 243, 251, 259, 265, 269, 299, 309, 353, 363, 399, 401, 455, 459, 467, 499, 519, 593, 635, 713, 737, 815, 831, 845, 899, 921, 923, 965, 967, 983, 1011, 1021, 1025, 1049, 1053, 1055
Offset: 1

Views

Author

Lekraj Beedassy, Jul 20 2008

Keywords

Comments

From M. F. Hasler and Farideh Firoozbakht, Nov 26 2009: (Start)
The sequence of Mersenne primes, A000668 is a subsequence of this sequence.
Because if k=2^p-1 is prime then n=2^(p-1)*(2^p-1)^2 is a solution of the equation sigma(x)=2x+k. The proof is easy. (End)
The definition is equivalent to asking for a number m with abundance A033880(m) = k. - M. F. Hasler, Mar 10 2025

References

  • J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 196, p. 58, Ellipses, Paris 2008.

Crossrefs

Cf. A000668. - M. F. Hasler and Farideh Firoozbakht, Nov 26 2009
Cf. A156903. - Robert G. Wilson v, Dec 09 2018
Cf. A000203 (sigma), A033880 (abundance: sigma(n)-2n).
Cf. A380866 (smallest solutions m to the given equation).

Formula

A033880(A156903), image of A156903 under A033880, or range of A033880 restricted to A156903, where A033880 is the abundance sigma(x)-2x, and A156903 are numbers with odd positive abundance. - M. F. Hasler, Mar 10 2025

Extensions

a(13)-a(54) from Donovan Johnson, Dec 09 2008

A156943 Odd abundant numbers not divisible by 25 whose abundancy is odd.

Original entry on oeis.org

81162081, 138791961, 173369889, 441882441, 730458729, 1091179089, 1249127649, 1524043521, 1560329001, 1744649361, 2286465489, 2606204601, 3255501249, 3976941969, 4770526761, 5567099769, 6574128561, 6800806089, 7584145569
Offset: 1

Views

Author

Robert G. Wilson v, Feb 18 2009

Keywords

Comments

Number of terms < 10^n: 0, 0, 0, 0, 0, 0, 0, 1, 5, 25, 63, ..., .

Crossrefs

Subsequence of A156903 and A156942.

Programs

  • Mathematica
    q[n_] := Module[{ab = DivisorSigma[1, n] - 2*n}, ab > 0 && OddQ[ab]]; Select[Range[1, 7775536039, 2], ! Divisible[#, 25] && q[#] &] (* corrected by Amiram Eldar, Oct 10 2023 *)

Extensions

Name corrected by Amiram Eldar, Oct 10 2023
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