A074827 Numbers n such that tau(n) > tau(n+1) where tau(x) = A000005(x).
4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 28, 30, 32, 36, 40, 42, 45, 46, 48, 50, 52, 54, 56, 58, 60, 64, 66, 68, 70, 72, 76, 78, 80, 81, 82, 84, 88, 90, 92, 96, 100, 102, 105, 106, 108, 110, 112, 114, 117, 120, 124, 126, 128, 130, 132, 136, 138, 140, 144, 148, 150, 152
Offset: 1
Keywords
Links
- Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
- P. Erdős, On a problem of Chowla and some related problems, Proc. Cambridge Philos. Soc. 32 (1936), pp. 530-540.
Crossrefs
Cf. A074775 (tau(n)
Programs
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Mathematica
Select[Range@ 152, DivisorSigma[0, #] > DivisorSigma[0, # + 1] &] (* Michael De Vlieger, Jul 03 2015 *) Position[Partition[DivisorSigma[0,Range[200]],2,1],?(#[[1]]>#[[2]]&),{1},Heads->False]//Flatten (* _Harvey P. Dale, Jan 17 2017 *)
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PARI
is(n)=numdiv(n) > numdiv(n+1) \\ Charles R Greathouse IV, Dec 05 2012
Formula
a(n) seems to be asymptotic to d*n with d=2.2... - Benoit Cloitre, Sep 07 2002
In fact, Erdős proved that a(n) ~ 2n. - Charles R Greathouse IV, Dec 05 2012
Extensions
Corrected and extended by Robert G. Wilson v, Sep 06 2002
A074772 Numbers k such that tau(k) < tau(k+1) and phi(k) < phi(k+1).
62, 74, 134, 146, 188, 206, 254, 274, 278, 284, 356, 362, 386, 398, 404, 422, 428, 454, 458, 482, 494, 538, 554, 566, 614, 626, 662, 674, 692, 746, 758, 764, 794, 818, 854, 866, 890, 914, 926, 934, 956, 998, 1004, 1028, 1034, 1052, 1070, 1082, 1084, 1094
Offset: 1
Keywords
Comments
There are few odd terms in the sequence, first one is 18015.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
- Vaclav Kotesovec, Plot of a(n)/n for n = 1..1600000
Programs
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Mathematica
Select[Range[1, 1000], DivisorSigma[0,#] < DivisorSigma[0,#+1] && EulerPhi[#] < EulerPhi[#+1]&] (* Vaclav Kotesovec, Feb 16 2019 *) Position[Partition[Table[{DivisorSigma[0,n],EulerPhi[n]},{n,1100}],2,1], ?(#[[1,1]]<#[[2,1]]&&#[[1,2]]<#[[2,2]]&),1,Heads->False]//Flatten (* _Harvey P. Dale, Dec 11 2020 *)
Formula
It seems that a(n) is asymptotic to c*n with 14<=c<=16. [This conjecture is false, see plot. - Vaclav Kotesovec, Feb 16 2019]
A172969 Numbers k such that 3*A000005(k) = 2*A000005(k+1).
3, 27, 51, 62, 74, 91, 99, 115, 123, 146, 187, 206, 235, 267, 274, 278, 291, 351, 355, 362, 386, 403, 411, 422, 427, 451, 459, 494, 538, 584, 665, 667, 721, 723, 746, 763, 771, 824, 843, 854, 866, 875, 926, 955, 987, 1003, 1027, 1034, 1057, 1070, 1082
Offset: 1
Keywords
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Programs
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Maple
for n from 1 to 1100 do if 3*numtheory[tau](n) = 2*numtheory[tau](n+1) then printf("%d,",n) ; end if; end do: -
Mathematica
Select[Range[1200], 3*DivisorSigma[0, #] == 2*DivisorSigma[0, # + 1] &] (* Amiram Eldar, Apr 09 2024 *)
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PARI
is(n) = 3 * numdiv(n) == 2 * numdiv(n+1); \\ Amiram Eldar, Apr 09 2024
Formula
{k: 3*tau(k) = 2*tau(k+1)}.
A364715 Numbers k such that d(k) < d(k+1) < d(k+2), where d(n) is the number of divisors of n.
61, 62, 73, 163, 187, 193, 194, 206, 254, 274, 277, 278, 283, 313, 355, 361, 362, 397, 398, 403, 421, 422, 427, 454, 457, 458, 482, 493, 523, 538, 583, 613, 614, 661, 673, 691, 733, 746, 757, 758, 763, 823, 853, 866, 889, 926, 934, 943, 955, 997, 998, 1003, 1027
Offset: 1
Programs
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PARI
isok(n) = numdiv(n)
A364716 Numbers k such that d(k) < d(k+1) < d(k+2) < d(k+3), where d(n) is the number of divisors of n.
61, 193, 277, 361, 397, 421, 457, 613, 757, 997, 1213, 1237, 1453, 1657, 1867, 1873, 1933, 2137, 2347, 2593, 2797, 2917, 3013, 3183, 3217, 3361, 3427, 3481, 3517, 3697, 3721, 3805, 4057, 4083, 4177, 4261, 4603, 4621, 4717, 4771, 4813, 4957, 5029, 5041, 5101, 5107, 5223
Offset: 1
Keywords
Programs
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PARI
isok(n) = numdiv(n)
A364717 Numbers k such that d(k) < d(k+1) < d(k+2) < d(k+3) < d(k+4), where d(n) is the number of divisors of n.
11371, 11372, 35521, 38281, 45613, 48121, 50821, 50822, 52321, 52322, 54421, 54422, 59341, 59342, 71821, 79621, 86873, 87181, 117841, 125737, 127852, 130021, 130022, 132051, 132206, 133396, 151082, 153221, 173221, 180001, 184973, 186481, 195541, 195542, 196171, 196172
Offset: 1
Keywords
Programs
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PARI
isok(n) = numdiv(n)
A074771 Numbers k such that tau(k) < tau(k+1), phi(k) < phi(k+1) and sigma(k) < sigma(k+1).
62, 74, 134, 146, 254, 398, 404, 458, 482, 494, 554, 566, 614, 626, 662, 674, 692, 758, 764, 794, 818, 854, 914, 998, 1034, 1094, 1124, 1214, 1238, 1286, 1322, 1394, 1454, 1514, 1538, 1646, 1658, 1682, 1826, 1844, 1874, 1934, 2078, 2114, 2174, 2234, 2246
Offset: 1
Keywords
Comments
There are few odd terms in the sequence, first one is 18015.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
- Vaclav Kotesovec, Plot of a(n)/n for n = 1..360000
Programs
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Mathematica
Select[Range[1, 3000], DivisorSigma[0,#] < DivisorSigma[0,#+1] && EulerPhi[#] < EulerPhi[#+1] && DivisorSigma[1,#] < DivisorSigma[1,#+1]&] (* Vaclav Kotesovec, Feb 16 2019 *) Position[Partition[Table[{DivisorSigma[0,n],EulerPhi[n],DivisorSigma[1,n]},{n,2300}],2,1],?(Max[#[[1]]-#[[2]]]<0&),1,Heads-> False]// Flatten (* _Harvey P. Dale, Jun 23 2019 *)
Formula
It seems that a(n) is asymptotic to c*n with 51<=c<=52
Comments