Search: a309307 -id:a309307
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A087893
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Number of numbers m satisfying 1 < m < n such that m^2 == m (mod n).
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+10
3
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0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 2, 0, 2, 2, 0, 0, 2, 0, 2, 2, 2, 0, 2, 0, 2, 0, 2, 0, 6, 0, 0, 2, 2, 2, 2, 0, 2, 2, 2, 0, 6, 0, 2, 2, 2, 0, 2, 0, 2, 2, 2, 0, 2, 2, 2, 2, 2, 0, 6, 0, 2, 2, 0, 2, 6, 0, 2, 2, 6, 0, 2, 0, 2, 2, 2, 2, 6, 0, 2, 0, 2, 0, 6, 2, 2, 2, 2, 0, 6, 2, 2, 2, 2, 2, 2, 0, 2, 2, 2, 0, 6, 0, 2, 6
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OFFSET
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1,6
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COMMENTS
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The number of nontrivial unitary divisors of n (i.e., excluding 1 and n). - Amiram Eldar, May 29 2020
a(n) first deviates from b(n) = 2*A079275(n) at a(210) = 14 <> b(210) = 12. - Georg Fischer, May 23 2024
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REFERENCES
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C. R. J. Singleton, "Prime Function Problem": Solution to Problem 2355, Journal of Recreational Mathematics, Vol. 29(3) pp. 232-234, 1998.
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LINKS
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FORMULA
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a(n) = 2^omega(n) - 2 (for n > 1).
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MATHEMATICA
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Join[{0}, Table[2^(PrimeNu[n]) - 2, {n, 2, 50}]] (* or *) Table[2*Module[{c = PrimeNu[n]}, (c (c - 1))/2], {n, 1, 20}] (* G. C. Greubel, May 20 2017 *)
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PROG
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(PARI) concat([0], for(n=2, 50, print1( 2^(omega(n)) - 2, ", "))) \\ G. C. Greubel, May 20 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A335268
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Numbers that are not powers of primes (A024619) whose harmonic mean of their unitary divisors that are larger than 1 is an integer.
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+10
3
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6, 15, 20, 24, 28, 30, 45, 60, 72, 90, 91, 96, 100, 112, 153, 216, 220, 240, 264, 272, 325, 352, 360, 364, 378, 496, 703, 765, 780, 816, 832, 1056, 1125, 1170, 1225, 1360, 1431, 1512, 1656, 1760, 1891, 1900, 1984, 2275, 2448, 2520, 2701, 2912, 3024, 3168, 3321
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OFFSET
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1,1
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COMMENTS
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Since the unitary divisors of a power of prime (A000961), p^e, are {1, p^e}, they are trivial terms and hence they are excluded from this sequence.
The corresponding harmonic means are 3, 5, 6, 6, 7, 5, 9, 7, 12, 7, 13, 8, 10, 14, 17, ...
Equivalently, numbers m such that omega(m) > 1 and (usigma(m)-m) | m * (2^omega(m)-1), or A063919(m) | (m * A309307(m)), where usigma is the sum of unitary divisors (A034448), and 2^omega(m) = A034444(m) is the number of the unitary divisors of m.
The squarefree terms of A335267 are also terms of this sequence.
The terms with 2 distinct prime divisors are of the form p^e * (2*p^e - 1), when the second factor is also a prime power. The least term which both of its 2 prime divisors are nonunitary (with multiplicity larger than 1) is 1225 = 5^2 * 7^2 = 5^2 * (2 * 5^2 - 1).
The unitary perfect numbers (A002827) are terms of this sequence: if m is a unitary perfect number then usigma(m)-m = m.
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LINKS
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EXAMPLE
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6 is a term since its unitary divisors other than 1 are 2, 3 and 6, and their harmonic mean, 3/(1/2 + 1/3 + 1/6) = 3, is an integer.
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MATHEMATICA
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usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); Select[Range[3000], (omega = PrimeNu[#]) > 1 && Divisible[# * (2^omega-1), usigma[#] - #] &]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A335270
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Numbers that are not powers of primes (A024619) whose harmonic mean of their proper unitary divisors is an integer.
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+10
2
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OFFSET
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1,1
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COMMENTS
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Since 1 is the only proper unitary divisor of powers of prime (A000961), they are trivial terms and hence they are excluded from this sequence.
The corresponding harmonic means are 4, 5, 5, 9, 18, 20.
Equivalently, numbers m such that omega(m) > 1 and (usigma(m)-1) | m*(2^omega(m)-1), where usigma is the sum of unitary divisors (A034448), and 2^omega(m) - 1 = A034444(m) - 1 = A309307(m) is the number of the proper unitary divisors of m.
The squarefree terms of A247077 are also terms of this sequence.
Conjecture: all terms are of the form n*(usigma(n)-1) where usigma(n)-1 is prime. - Chai Wah Wu, Dec 17 2020
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LINKS
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EXAMPLE
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228 is a term since the harmonic mean of its proper unitary divisors, {1, 3, 4, 12, 19, 57, 76} is 4 which is an integer.
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MATHEMATICA
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usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); Select[Range[10^5], (omega = PrimeNu[#]) > 1 && Divisible[# * (2^omega-1), usigma[#] - 1] &]
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CROSSREFS
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KEYWORD
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nonn,hard,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A329534
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Irregular triangle read by rows: for n >= 1 row n lists the k from [1, 2, ... , n] such that A002378(k-1) = (k-1)*k == 0 (mod n).
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+10
1
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1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 3, 4, 6, 1, 7, 1, 8, 1, 9, 1, 5, 6, 10, 1, 11, 1, 4, 9, 12, 1, 13, 1, 7, 8, 14, 1, 6, 10, 15, 1, 16, 1, 17, 1, 9, 10, 18, 1, 19, 1, 5, 16, 20, 1, 7, 15, 21, 1, 11, 12, 22, 1, 23, 1, 9, 16, 24, 1, 25
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OFFSET
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1,3
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COMMENTS
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n-th row length gives 1 for n = 1, and 2^A001221(n) for n >= 2 , that is A034444(n). [Proof: Unique lifting theorem (e.g., Apostol, 5.30 (a), p.121) for this congruence, and only two solutions 1 and p for primes p. See also the Yuval Dekel, Sep 21 2003, comment in A034444. - Wolfdieter Lang, Feb 05 2020]
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REFERENCES
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Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986.
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LINKS
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EXAMPLE
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The irregular triangle T(n,k) begins
n\k 1 2 3 4 ...
1: 1
2: 1 2
3: 1 3
4: 1 4
5: 1 5
6: 1 3 4 6
7: 1 7
8: 1 8
9: 1 9
10: 1 5 6 10
11: 1 11
12: 1 4 9 12
13: 1 13
14: 1 7 8 14
15: 1 6 10 15
16: 1 16
17: 1 17
18: 1 9 10 18
19: 1 19
20: 1 5 16 20
...
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MATHEMATICA
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Table[Select[Range@ n, Mod[-n + # (# - 1), n] == 0 &], {n, 25}] // Flatten (* Michael De Vlieger, Nov 18 2019 *)
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PROG
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(Magma) [[k: k in [1..n] | k^2 mod n eq k]: n in [1..38]];
(PARI) row(n) = select(x->(Mod(x, n) == Mod(x, n)^2), [1..n]); \\ Michel Marcus, Nov 19 2019
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CROSSREFS
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Cf. A000010, A000225, A000688, A000961, A001221, A006881, A006530, A007875, A020639, A024619, A034444, A077610, A135972, A309307.
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KEYWORD
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nonn,easy,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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