A332321
Numbers k that are norm-superabundant in Gaussian integers, i.e., A103230(m)/m^2 < A103230(k)/k^2 for all m < k.
Original entry on oeis.org
1, 2, 6, 10, 30, 90, 130, 210, 390, 1170, 2730, 5850, 6630, 19890, 46410, 99450, 139230, 192270, 576810, 1345890, 2884050, 4037670, 7883070, 12113010, 20188350, 23649210, 44414370, 49797930, 55181490, 118246050, 149393790, 165544470, 496633410, 746968950, 827722350
Offset: 1
The first 6 terms of A103230 are 1, 13, 16, 41, 80, 208. The corresponding values of A103230(n)/n^2 are 1, 3.25, 1.777..., 2.5625, 3.2, 5.777... and the record values occur at n = 1, 2, 6, the first 3 terms of this sequence.
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r[n_] := Abs[DivisorSigma[1, n, GaussianIntegers -> True]]^2/n^2; rm = 0; seq = {}; Do[r1 = r[n]; If[r1 > rm, rm = r1; AppendTo[seq, n]], {n, 1, 6*10^5}]; seq
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
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.)
- 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.
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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
7 is a term since both 7 and 7 + 1 = 8 are norm-deficient in Gaussian integers (A332572).
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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
30514 is a term since A103230(30514) = A103230(30515) = 5391360000.
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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
A332532
Even numbers k such that the sum of divisors of k in Gaussian integers is a real number.
Original entry on oeis.org
10250, 30750, 40400, 71750, 92250, 112750, 121200, 194750, 215250, 235750, 276750, 282800, 308050, 317750, 338250, 363600, 440750, 444400, 467860, 481750, 502250, 584250, 604750, 645750, 686750, 707250, 727750, 767600, 789250, 809750, 830250, 848400, 850750
Offset: 1
10250 is a term since its sum of divisors in Gaussian integers is 41600 which is a real number.
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Select[Range[2, 10^6, 2], Im[DivisorSigma[1, #, GaussianIntegers -> True]] == 0 &]
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
127 is a term since 127, 128 and 129 are all norm-deficient in Gaussian integers (A332572).
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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
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.
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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
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