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.

Previous Showing 11-18 of 18 results.

A332572 Numbers that are norm-deficient in Gaussian integers.

Original entry on oeis.org

1, 3, 7, 8, 11, 16, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 57, 59, 61, 67, 69, 71, 73, 77, 79, 83, 89, 93, 97, 101, 103, 107, 109, 113, 121, 127, 128, 129, 131, 133, 137, 139, 141, 149, 151, 152, 157, 161, 163, 167, 173, 177, 179, 181, 184, 191, 193, 197
Offset: 1

Views

Author

Amiram Eldar, Feb 16 2020

Keywords

Comments

Numbers k such that N(sigma(k)) < 2*N(k) = 2*k^2, where sigma(k) = A103228(k) + i*A103229(k) is the sum of divisors of k in Gaussian integers (i is the imaginary unit), and N(z) = Re(z)^2 + Im(z)^2 is the norm of the complex number z.
The number of terms not exceeding 10^k for k = 1, 2, ... is 4, 30, 289, 2998, 30075, 298919, 2983713, 29925997, 299442606, 2992921174, ... Apparently this sequence has an asymptotic density of ~0.3.

Examples

			3 is norm-deficient since sigma(3) = 4 and N(4) = 4^2 = 16 < 2 * 3^2 = 18.
8 is norm-deficient since sigma(8) = -8 - 7*i and N(-8 - 7*i) = (-8)^2 + (-7)^2 = 113 < 2 * 8^2 = 128.

Links

Crossrefs

Cf. A005100, A103228, A103229, A103230, A332570.

Programs

  • Mathematica
    normDefQ[z_] := Abs[DivisorSigma[1, z, GaussianIntegers -> True]]^2 < 2*Abs[z]^2; Select[Range[200], normDefQ]

A102574 a(n) is the sum of the distinct norms of the divisors of n over the Gaussian integers.

Original entry on oeis.org

1, 7, 10, 31, 31, 70, 50, 127, 91, 217, 122, 310, 183, 350, 310, 511, 307, 637, 362, 961, 500, 854, 530, 1270, 781, 1281, 820, 1550, 871, 2170, 962, 2047, 1220, 2149, 1550, 2821, 1407, 2534, 1830, 3937, 1723, 3500, 1850, 3782, 2821, 3710, 2210, 5110, 2451
Offset: 1

Views

Author

Yasutoshi Kohmoto, Feb 25 2005

Keywords

Comments

Also sum of divisors of n^2 which are the sum of two squares (A001481). For example the divisors of 3^2 are 1, 3, 9 of which only 1 and 9 are in A001481 and a(3) = 1 + 9 = 10. - Jianing Song, Aug 03 2018

Examples

			Let ||i|| denote the norm of i.
a(2) = 1 + ||1+i|| + 2^2 = 1 + 2 + 4 = 7.
a(5) = 1 + ||1+2i|| + 5^2 = 1 + 5 + 25 = 31. Note that ||1+2i|| = ||2+i|| so their norm (5) is only counted once.

Links

Crossrefs

Cf. A000203 (sigma), A001157 (sigma_2), A001481, A097706, A103230, A243380.

Programs

  • Mathematica
    b[n_] := Product[{p, e} = pe; If[Mod[p, 4] == 3, p^e, 1], {pe, FactorInteger[n]}];
a[n_] := With[{r = b[n]}, DivisorSigma[2, r] DivisorSigma[1, (n/r)^2]];
a /@ Range[50] (* Jean-François Alcover, Sep 20 2019, from PARI *)
  • PARI
    \\ here b(n) is A097706.
b(n)={my(f=factor(n)); my(r=prod(i=1, #f~, my([p,e]=f[i,]); if(p%4==3, p^e, 1))); r}
a(n)={my(r=b(n)); sigma(r,2)*sigma((n/r)^2)} \\ Andrew Howroyd, Aug 03 2018
  • Python
    from math import prod
from sympy import factorint
def A102574(n): return prod((q := int(p & 3 == 3))*(p**(2*(e+1))-1)//(p**2-1) + (1-q)*(p**(2*e+1)-1)//(p-1) for p, e in factorint(n).items()) # Chai Wah Wu, Jun 28 2022

Formula

a(n) = sigma_2(A097706(n)) * sigma((n/A097706(n))^2). - Andrew Howroyd, Aug 03 2018
Multiplicative with a(p^e) = sigma(p^(2e)) = (p^(2e+1) - 1)/(p - 1) if p = 2 or p == 1 (mod 4); sigma_2(p^e) = (p^(2e+2) - 1)/(p^2 - 1) if p == 3 (mod 4). - Jianing Song, Aug 03 2018
Sum_{k=1..n} a(k) ~ c * n^3, where c = (5/12) * zeta(3) * A243380 = 0.52812367275583317729... . - Amiram Eldar, Feb 13 2024

Extensions

Corrected and extended by David Wasserman, Apr 08 2008
Keyword:mult added by Andrew Howroyd, Aug 03 2018
Name clarified by Jianing Song, Aug 03 2018

A332571 Numbers that are primitive norm-abundant in Gaussian integers.

Original entry on oeis.org

5, 9, 13, 21, 33, 119, 187, 203, 287, 543, 699, 807, 831, 843, 879, 939, 951, 1011, 1047, 1059, 1119, 1167, 1191, 1263, 1299, 1311, 1347, 1383, 1563, 1671, 1767, 1769, 1961, 2117, 2139, 2173, 2257, 2451, 2501, 2581, 2679, 2813, 2929, 2967, 2993, 3161, 3233, 3243
Offset: 1

Views

Author

Amiram Eldar, Feb 16 2020

Keywords

Comments

Numbers that are norm-abundant (A332570) in Gaussian integers and having no norm-abundant proper divisor.

Examples

			5 is primitive norm-abundant since it is norm-abundant, sigma(5) = 4 + 8*i and N(4 + 8*i) = 4^2 + 8^2 = 80 > 2 * 5^2 = 50, and none of the proper divisors of 5, {1, 1 + 2*i, 2 + i}, are norm-abundant: N(sigma(1)) = 1 < 2 * 1^2, N(sigma(1 + 2*i)) = N(2 + 2*i) = 8 < 2 * N(1 + 2*i) = 10, and N(sigma(2 + i)) = N(3 + i) = 10 = 2 * N(2 + i). (sigma(k) = A103228(k) + i*A103229(k) is the sum of divisors of k in Gaussian integers, i is the imaginary unit, and N(z) = Re(z)^2 + Im(z)^2 is the norm of the complex number z.)

References

  • Miriam Hausman, On Norm Abundant Gaussian Integers, The Journal of the Indian Mathematical Society, Vol. 49 (1987), pp. 119-123.
  • József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, chapter III, page 120.

Links

Crossrefs

Cf. A091191, A103228, A103229, A103230, A332570.

Programs

  • Mathematica
    normAbQ[z_] := normAbQ[z] = Abs[DivisorSigma[1, z, GaussianIntegers -> True]]^2 > 2*Abs[z]^2; primNormAbQ[z_] := normAbQ[z] && !AnyTrue[Most[Divisors[z, GaussianIntegers -> True]], normAbQ]; Select[Range[1000], primNormAbQ]

A332573 Numbers k such that k and k + 1 are both norm-deficient in Gaussian integers (A332572).

Original entry on oeis.org

7, 16, 127, 128, 151, 248, 256, 343, 472, 536, 568, 631, 632, 751, 752, 823, 856, 943, 1048, 1111, 1136, 1207, 1303, 1327, 1328, 1336, 1432, 1527, 1528, 1591, 1687, 1688, 1711, 1712, 1783, 1816, 1912, 2031, 2032, 2047, 2048, 2103, 2167, 2263, 2416, 2487, 2488
Offset: 1

Views

Author

Amiram Eldar, Feb 16 2020

Keywords

Examples

			7 is a term since both 7 and 7 + 1 = 8 are norm-deficient in Gaussian integers (A332572).

Links

Crossrefs

Cf. A005100, A103228, A103229, A103230, A317047, A332572.

Programs

  • Mathematica
    normDefQ[z_] := Abs[DivisorSigma[1, z, GaussianIntegers -> True]]^2 < 2*Abs[z]^2; Select[Range[2500], normDefQ[#] && normDefQ[# + 1] &]

A332315 Numbers k such that k and k + 1 have the same norm of the sum of divisors in Gaussian integers.

Original entry on oeis.org

30514, 36777, 43978, 3474262, 5745125, 10628554, 16567494, 40831527, 58008301, 111798477, 142981839, 288834504, 392413941, 580867202, 650141557, 944224497, 967593411, 1874210882, 6306287377, 6442064745, 7377567197, 8121464245
Offset: 1

Views

Author

Amiram Eldar, Feb 09 2020

Keywords

Comments

The first term, 30514, is also a number k such that k and k + 1 have the sum divisors in Gaussian integers: -54720 + 48960*i (where i is the imaginary unit). What is the next term with this property?
No more terms below 1.5*10^10.

Examples

			30514 is a term since A103230(30514) = A103230(30515) = 5391360000.

Crossrefs

Cf. A002961, A064125, A103228, A103229, A103230, A293183, A306985.

Programs

  • Mathematica
    csigma[n_] :=(Abs @ DivisorSigma[1, n, GaussianIntegers -> True])^2; seq = {}; n1 = csigma[1]; Do[n2 = csigma[n]; If[n1 == n2, AppendTo[seq, n - 1]]; n1 = n2, {n, 2, 5*10^5}]; seq

A332317 Numbers k that are harmonic in Gaussian integers: k * A062327(k) is divisible by A103228(k) + i*A103229(k) (where i is the imaginary unit).

Original entry on oeis.org

1, 5, 130, 390, 585, 3250, 31980, 133250, 223860, 799500, 7195500, 13591500, 122323500, 258238500, 394153500, 405346500, 910630500, 1345558500, 2025133500, 8195674500
Offset: 1

Views

Author

Amiram Eldar, Feb 09 2020

Keywords

Comments

Analogous to harmonic numbers (A001599), with the number and sum of divisors functions generalized for Gaussian integers (A062327, A103228, A103229) instead of the number and sum of divisors functions (A000005, A000203).
No more terms below 10^10.

Examples

			5 is a term since 5 * A062327(5)/(A103228(5) + i*A103229(5)) = 5 * 4 /(4 + 8*i) = 1 - 2*i is a Gaussian integer.

Crossrefs

Cf. A000005, A000203, A001599, A062327, A103228, A103229, A103230.

Programs

  • Mathematica
    Select[Range[10^4], Divisible[# * DivisorSigma[0, #, GaussianIntegers -> True], DivisorSigma[1, #, GaussianIntegers -> True]] &]

A332574 Numbers k such that k, k + 1 and k + 2 are all norm-deficient in Gaussian integers (A332572).

Original entry on oeis.org

127, 631, 751, 1327, 1527, 1687, 1711, 2031, 2047, 2487, 2647, 3207, 3271, 3351, 3511, 3831, 4567, 4791, 4911, 5127, 6007, 6087, 6711, 7431, 8247, 8367, 8391, 8407, 8551, 8751, 8871, 9031, 9447, 9991, 10407, 10551, 10887, 10927, 11631, 12471, 12567, 12631, 13807
Offset: 1

Views

Author

Amiram Eldar, Feb 16 2020

Keywords

Examples

			127 is a term since 127, 128 and 129 are all norm-deficient in Gaussian integers (A332572).

Links

Crossrefs

Cf. A005100, A103228, A103229, A103230, A317047, A332572, A332573.

Programs

  • Mathematica
    normDefQ[z_] := Abs[DivisorSigma[1, z, GaussianIntegers -> True]]^2 < 2*Abs[z]^2; Select[Range[10^4], AllTrue[# + Range[0, 2], normDefQ] &]

A332575 Least start of a run of exactly n consecutive numbers that are norm-abundant in Gaussian integers (A332570).

Original entry on oeis.org

2, 9, 4, 12, 24, 185, 114, 1649, 692, 4977, 1412, 416345, 22624, 72233, 199892, 25262152, 1351880, 130824185, 16305324, 1688906313, 9412730, 10393378914, 721753400
Offset: 1

Views

Author

Amiram Eldar, Feb 16 2020

Keywords

Examples

			a(2) = 9 since 9 and 10 are the least pair of 2 consecutive numbers that are norm-abundant in Gaussian integers, and 8 and 11 are not norm-abundant.

Crossrefs

Cf. A005101, A096399, A096536, A103228, A103229, A103230, A332570.

Programs

  • Mathematica
    normAbQ[z_] := Abs[DivisorSigma[1, z, GaussianIntegers -> True]]^2 > 2*Abs[z]^2; n = 1; count = 0; max = 15; seq = Table[0, {max}]; While[count < max, n1 = n; If[normAbQ[n], While[normAbQ[++n1]]; d = n1 - n; If[d <= max && seq[[d]] == 0, count++; seq[[d]] = n]]; n = n1 + 1]; seq
Previous Showing 11-18 of 18 results.

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