A322256
Numbers k such that t(k) = t(k+1) where t(k) = tau(k) + sigma(k) = A007503(k) is the number of subgroups of the dihedral group of order 2k.
Original entry on oeis.org
14, 1334, 1634, 2685, 33998, 42818, 64665, 84134, 109214, 122073, 166934, 289454, 383594, 440013, 544334, 605985, 649154, 655005, 792855, 845126, 1642154, 2284814, 2305557, 2913105, 3571905, 3682622, 4701537, 5181045, 6431732, 6444873, 6771405, 10074477
Offset: 1
- Amiram Eldar, Table of n, a(n) for n = 1..400
- David W. Jensen and Michael K. Keane, A Number-Theoretic Approach to Subgroups of Dihedral Groups, USAFA-TR-90-2, Air Force Academy Colorado Springs, Colorado, 1990.
- David W. Jensen and Eric R. Bussian, A Number-Theoretic Approach to Counting Subgroups of Dihedral Groups, The College Mathematics Journal, Vol. 23, No. 2 (1992), pp. 150-152.
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[n: n in [1..2*10^6] | (NumberOfDivisors(n) + SumOfDivisors(n)) eq (NumberOfDivisors(n+1) + SumOfDivisors(n+1))]; // Vincenzo Librandi, Dec 08 2018
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t[n_] := DivisorSigma[0, n] + DivisorSigma[1, n]; tQ[n_] := t[n] == t[n + 1]; Select[Range[1000000], tQ]
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isok(n) = (numdiv(n)+sigma(n)) == (numdiv(n+1)+sigma(n+1)); \\ Michel Marcus, Dec 04 2018
A350800
Numbers k such that k and k+1 have the same number and sum of divisors but a different number of distinct prime factors.
Original entry on oeis.org
64665, 109214, 2305557, 4701537, 6444873, 10118654, 32225337, 33876117, 70282053, 105967784, 149205914, 187434621, 268890218, 279113505, 334925577, 357340922, 391392134, 424942604, 575712494, 610752933, 612863198, 641703842, 701792234, 743194142, 800679495
Offset: 1
64665 is a term of this sequence since tau(64665) = tau(64666) = 8 and sigma(64665) = sigma(64666) = 2160, but omega(64665) = 4 and omega(64666) = 3.
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Select[Range[10^7], DivisorSigma[{0, 1}, #] == DivisorSigma[{0, 1}, # + 1] && PrimeNu[#] != PrimeNu[# + 1] &] (* Amiram Eldar, Jan 20 2022 *)
A353033
Numbers m such that tau(m) = 2 * tau(m - 1) and simultaneously sigma(m) = 2 * sigma(m - 1), where tau(k) = A000005(k) and sigma(k) = A000203(k).
Original entry on oeis.org
6, 47796, 111684, 123498, 224562, 228378, 384858, 773016, 1096824, 1174542, 2351240, 2529414, 3320472, 3332616, 3650376, 4605096, 4838838, 4978476, 5014842, 5788662, 6023928, 6302724, 7658024, 8298978, 9287240, 9967974, 10950024, 12677496, 14036694, 14120360, 14927990
Offset: 1
tau(6) = 4 = 2 * tau(5) = 2 * 2, sigma(6) = 12 = 2 * sigma(5) = 2 * 6.
A353034
Numbers m such that tau(m) = 2 * tau(m + 1) and simultaneously sigma(m) = 2 * sigma(m + 1), where tau(k) = A000005(k) and sigma(k) = A000203(k).
Original entry on oeis.org
20118, 20712, 79338, 103410, 203898, 267630, 570342, 907710, 1093026, 1228062, 1263918, 1663752, 2322760, 3268782, 3468486, 3527250, 5483418, 6277038, 6500442, 7637980, 9181578, 9297078, 17708178, 18638646, 25274946, 25364526, 25768302, 25909254, 31118664
Offset: 1
tau(20118) = 16 = 2 * tau(20119) = 2 * 8, sigma(20118) = 46080 = 2 * sigma(20119) = 2 * 23040.
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