Two or three more sequences??
Richard Guy
rkg at cpsc.ucalgary.ca
Mon Feb 11 18:39:15 CET 2008
Here is a quote from the 2007 Western Number Theory
problems set, edited by Gerry Myerson.
``007:02 (Doug Iannucci) Call n a year number if
phi(n) / phi(sigma(n)) = 2 (note that 365 is a
year number, whence the terminology). Are there any
even year numbers? Are there any odd year numbers
that are not squarefree?
``Remark: If n = q_1 q_2 ... q_k is a product
of odd primes such that (q_j + 1)/2 is prime (*)
for all j, then n is a year number.
``Solution: n = 5491 = 17^2 * 19 is the
smallest non-squarefree year number. The next few
non-squarefree year numbers are 8075, 25317, 27455,
71383, 72283, 76131, 104975, 138575, 193041
``The smallest non-cubefree (and non-4th-powerfree)
year number is 295569.
``Eric Landquist found year numbers divixsible by
7^2, 7^3, and 7^4, as well as
120781449 = 3^8 * 41 * 449
The existence of even year numbers is still open,
but Eric checked all 200-smooth even integers with
a single large prime up to 10^8 and found no year
numbers among them.''
(*) I think we need `odd prime' here, since I don't
believe we can take q_1 = 3
Here are suggestions for new sequences, if anyone has
the urge to check and extend the calculations. First
note that A006872 is numbers such that
phi(n) / phi(sigma(n)) = 1.
1. Numbers such that phi(n) / phi(sigma(n)) = 2
viz., Iannucci's `year numbers':
5,13,37,61,65,73,...,185,...,305,...325,...,365
2. Non-squarefree such numbers (see above)
3. Numbers such that phi(n) / phi(sigma(n)) = 1/2
viz. 2, 6, 8, 9, 16, 28, 70, 78
Best, R.
rg> From seqfan-owner at ext.jussieu.fr Mon Feb 11 18:39:30 2008
rg> Date: Mon, 11 Feb 2008 10:39:15 -0700 (MST)
rg> From: Richard Guy <rkg at cpsc.ucalgary.ca>
rg> To: seqfans at seqfan.net, Sequence Fans <seqfan at ext.jussieu.fr>
rg> cc: "Sloane's Dream Team" <editors at seqfan.net>
rg> Subject: Two or three more sequences??
rg>
rg> Here is a quote from the 2007 Western Number Theory
rg> problems set, edited by Gerry Myerson.
rg>
rg> ``007:02 (Doug Iannucci) Call n a year number if
rg> phi(n) / phi(sigma(n)) = 2 (note that 365 is a
rg> year number, whence the terminology). Are there any
rg> even year numbers? Are there any odd year numbers
rg> that are not squarefree?
rg>
rg> ``Remark: If n = q_1 q_2 ... q_k is a product
rg> of odd primes such that (q_j + 1)/2 is prime (*)
rg> for all j, then n is a year number.
rg>
rg> ``Solution: n = 5491 = 17^2 * 19 is the
rg> smallest non-squarefree year number. The next few
rg> non-squarefree year numbers are 8075, 25317, 27455,
rg> 71383, 72283, 76131, 104975, 138575, 193041
rg>
rg> ``The smallest non-cubefree (and non-4th-powerfree)
rg> year number is 295569.
rg>
rg> ``Eric Landquist found year numbers divixsible by
rg> 7^2, 7^3, and 7^4, as well as
rg> 120781449 = 3^8 * 41 * 449
rg> The existence of even year numbers is still open,
rg> but Eric checked all 200-smooth even integers with
rg> a single large prime up to 10^8 and found no year
rg> numbers among them.''
rg>
rg> (*) I think we need `odd prime' here, since I don't
rg> believe we can take q_1 = 3
rg>
rg> Here are suggestions for new sequences, if anyone has
rg> the urge to check and extend the calculations. First
rg> note that A006872 is numbers such that
rg> phi(n) / phi(sigma(n)) = 1.
rg>
rg> 1. Numbers such that phi(n) / phi(sigma(n)) = 2
rg> viz., Iannucci's `year numbers':
rg> 5,13,37,61,65,73,...,185,...,305,...325,...,365
In PARI notation with the prime decomposition included, one per line:
5 Mat([5, 1])
13 Mat([13, 1])
37 Mat([37, 1])
61 Mat([61, 1])
65 [5, 1; 13, 1]
73 Mat([73, 1])
119 [7, 1; 17, 1]
157 Mat([157, 1])
185 [5, 1; 37, 1]
193 Mat([193, 1])
277 Mat([277, 1])
305 [5, 1; 61, 1]
313 Mat([313, 1])
365 [5, 1; 73, 1]
397 Mat([397, 1])
421 Mat([421, 1])
457 Mat([457, 1])
481 [13, 1; 37, 1]
541 Mat([541, 1])
613 Mat([613, 1])
661 Mat([661, 1])
673 Mat([673, 1])
733 Mat([733, 1])
757 Mat([757, 1])
785 [5, 1; 157, 1]
793 [13, 1; 61, 1]
877 Mat([877, 1])
949 [13, 1; 73, 1]
965 [5, 1; 193, 1]
997 Mat([997, 1])
1093 Mat([1093, 1])
1153 Mat([1153, 1])
1201 Mat([1201, 1])
1213 Mat([1213, 1])
1237 Mat([1237, 1])
1321 Mat([1321, 1])
1381 Mat([1381, 1])
1385 [5, 1; 277, 1]
1453 Mat([1453, 1])
1547 [7, 1; 13, 1; 17, 1]
1565 [5, 1; 313, 1]
1615 [5, 1; 17, 1; 19, 1]
1621 Mat([1621, 1])
1657 Mat([1657, 1])
1753 Mat([1753, 1])
1873 Mat([1873, 1])
1933 Mat([1933, 1])
1985 [5, 1; 397, 1]
1993 Mat([1993, 1])
2017 Mat([2017, 1])
2041 [13, 1; 157, 1]
2105 [5, 1; 421, 1]
2137 Mat([2137, 1])
2257 [37, 1; 61, 1]
2285 [5, 1; 457, 1]
2341 Mat([2341, 1])
2405 [5, 1; 13, 1; 37, 1]
2473 Mat([2473, 1])
....
rg> 2. Non-squarefree such numbers (see above)
with squares:
5491 [17, 2; 19, 1]
8075 [5, 2; 17, 1; 19, 1]
25317 [3, 2; 29, 1; 97, 1]
27455 [5, 1; 17, 2; 19, 1]
71383 [13, 1; 17, 2; 19, 1]
72283 [41, 2; 43, 1]
76131 [3, 2; 11, 1; 769, 1]
104975 [5, 2; 13, 1; 17, 1; 19, 1]
138575 [5, 2; 23, 1; 241, 1]
193041 [3, 2; 89, 1; 241, 1]
203167 [17, 2; 19, 1; 37, 1]
298775 [5, 2; 17, 1; 19, 1; 37, 1]
334951 [17, 2; 19, 1; 61, 1]
356915 [5, 1; 13, 1; 17, 2; 19, 1]
361415 [5, 1; 41, 2; 43, 1]
400843 [17, 2; 19, 1; 73, 1]
451535 [5, 1; 7, 2; 19, 1; 97, 1]
492275 [5, 2; 7, 1; 29, 1; 97, 1]
509575 [5, 2; 11, 1; 17, 1; 109, 1]
572975 [5, 2; 13, 1; 41, 1; 43, 1]
589475 [5, 2; 17, 1; 19, 1; 73, 1]
595975 [5, 2; 31, 1; 769, 1]
654493 [7, 2; 19, 2; 37, 1]
683757 [3, 2; 17, 1; 41, 1; 109, 1]
815975 [5, 2; 127, 1; 257, 1]
862087 [17, 2; 19, 1; 157, 1]
876627 [3, 2; 257, 1; 379, 1]
..
with cubes:
295947 [3, 3; 97, 1; 113, 1]
1586763 [3, 3; 17, 1; 3457, 1]
6820555 [5, 1; 7, 3; 41, 1; 97, 1]
7285923 [3, 3; 449, 1; 601, 1]
...
with 4th powers:
295569 [3, 4; 41, 1; 89, 1]
1964375 [5, 4; 7, 1; 449, 1]
2069469 [3, 4; 29, 1; 881, 1]
5854375 [5, 4; 17, 1; 19, 1; 29, 1]
...
with 5th powers:
1811079 [3, 5; 29, 1; 257, 1]
4473387 [3, 5; 41, 1; 449, 1]
...
with horse powers:
..
with hoarse powers:
..a'hem
rg>
rg> 3. Numbers such that phi(n) / phi(sigma(n)) = 1/2
rg>viz. 2, 6, 8, 9, 16, 28, 70, 78
rg>
2 Mat([2, 1])
6 [2, 1; 3, 1]
8 Mat([2, 3])
9 Mat([3, 2])
24 [2, 3; 3, 1]
28 [2, 2; 7, 1]
70 [2, 1; 5, 1; 7, 1]
78 [2, 1; 3, 1; 13, 1]
128 Mat([2, 7])
140 [2, 2; 5, 1; 7, 1]
222 [2, 1; 3, 1; 37, 1]
234 [2, 1; 3, 2; 13, 1]
280 [2, 3; 5, 1; 7, 1]
312 [2, 3; 3, 1; 13, 1]
350 [2, 1; 5, 2; 7, 1]
366 [2, 1; 3, 1; 61, 1]
384 [2, 7; 3, 1]
438 [2, 1; 3, 1; 73, 1]
496 [2, 4; 31, 1]
525 [3, 1; 5, 2; 7, 1]
666 [2, 1; 3, 2; 37, 1]
864 [2, 5; 3, 3]
888 [2, 3; 3, 1; 37, 1]
910 [2, 1; 5, 1; 7, 1; 13, 1]
918 [2, 1; 3, 3; 17, 1]
936 [2, 3; 3, 2; 13, 1]
942 [2, 1; 3, 1; 157, 1]
1036 [2, 2; 7, 1; 37, 1]
1098 [2, 1; 3, 2; 61, 1]
1158 [2, 1; 3, 1; 193, 1]
1232 [2, 4; 7, 1; 11, 1]
1314 [2, 1; 3, 2; 73, 1]
1400 [2, 3; 5, 2; 7, 1]
1464 [2, 3; 3, 1; 61, 1]
1575 [3, 2; 5, 2; 7, 1]
1662 [2, 1; 3, 1; 277, 1]
1708 [2, 2; 7, 1; 61, 1]
1752 [2, 3; 3, 1; 73, 1]
1824 [2, 5; 3, 1; 19, 1]
1836 [2, 2; 3, 3; 17, 1]
1878 [2, 1; 3, 1; 313, 1]
1900 [2, 2; 5, 2; 19, 1]
1938 [2, 1; 3, 1; 17, 1; 19, 1]
2044 [2, 2; 7, 1; 73, 1]
2382 [2, 1; 3, 1; 397, 1]
2480 [2, 4; 5, 1; 31, 1]
2526 [2, 1; 3, 1; 421, 1]
2590 [2, 1; 5, 1; 7, 1; 37, 1]
2664 [2, 3; 3, 2; 37, 1]
2742 [2, 1; 3, 1; 457, 1]
2826 [2, 1; 3, 2; 157, 1]
2886 [2, 1; 3, 1; 13, 1; 37, 1]
3246 [2, 1; 3, 1; 541, 1]
3474 [2, 1; 3, 2; 193, 1]
3640 [2, 3; 5, 1; 7, 1; 13, 1]
3678 [2, 1; 3, 1; 613, 1]
3768 [2, 3; 3, 1; 157, 1]
3876 [2, 2; 3, 1; 17, 1; 19, 1]
3966 [2, 1; 3, 1; 661, 1]
4038 [2, 1; 3, 1; 673, 1]
4270 [2, 1; 5, 1; 7, 1; 61, 1]
4392 [2, 3; 3, 2; 61, 1]
4396 [2, 2; 7, 1; 157, 1]
...
PARI scanner for item 1+2 (UNIX: pipe through fgrep ", 2" to get squares, for example.):
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