A326141 -id:A326141 - cp's OEIS frontend

A326140 a(n) = gcd(A318878(n), A318879(n)).

1, 1, 2, 1, 4, 2, 6, 1, 5, 2, 10, 2, 12, 2, 6, 1, 16, 1, 18, 2, 10, 2, 22, 2, 19, 2, 14, 6, 28, 6, 30, 1, 18, 2, 22, 1, 36, 2, 22, 2, 40, 2, 42, 2, 12, 2, 46, 2, 41, 1, 30, 6, 52, 2, 38, 2, 34, 2, 58, 6, 60, 2, 22, 1, 46, 6, 66, 2, 42, 2, 70, 1, 72, 2, 26, 6, 58, 2, 78, 2, 41, 2, 82, 2, 62, 2, 54, 2, 88, 6, 70, 2, 58, 2

Offset: 1

Antti Karttunen, Jun 09 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A033879, A083254, A318878, A318879, A326141.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326057, A326060, A326062, A326129, A326130, A326144.

A326140(n) = { my(t=0, u=0); fordiv(n,d, d -= 2*eulerphi(d); if(d<0, t -= d, u += d)); gcd(t,u); };

A318878(n) = sumdiv(n,d,d=(2*eulerphi(d))-d; (d>0)*d);
A318879(n) = sumdiv(n,d,d=d-(2*eulerphi(d)); (d>0)*d);
A326140(n) = gcd(A318878(n), A318879(n));

A326064 Odd composite numbers n, not squares of primes, such that (A001065(n) - A032742(n)) divides (n - A032742(n)), where A032742 gives the largest proper divisor, and A001065 is the sum of proper divisors.

Original entry on oeis.org

117, 775, 10309, 56347, 88723, 2896363, 9597529, 12326221, 12654079, 25774633, 29817121, 63455131, 105100903, 203822581, 261019543, 296765173, 422857021, 573332713, 782481673, 900952687, 1129152721, 3350861677, 3703086229, 7395290407, 9347001661, 9350506057

Offset: 1

Antti Karttunen, Jun 06 2019

nonn

Nineteen initial terms factored:
   n       a(n)   factorization   A060681(a(n))/A318505(a(n))
      117 =   3^2 *    13,           (3)
      775 =   5^2 *    31,          (10)
    10309 =  13^2 *    61,          (39)
    56347 =  29^2 *    67,          (58)
    88723 =  17^2 *   307,         (136)
  2896363 =  41^2 *  1723,         (820)
  9597529 =  73^2 *  1801,        (1314)
 12326221 =  59^2 *  3541,        (1711)
 12654079 = 113^2 *   991,         (904)
 25774633 =  71^2 *  5113,        (2485)
 29817121 =  97^2 *  3169,        (2328)
 63455131 =  89^2 *  8011,        (3916)
105100903 = 101^2 * 10303,        (5050)
203822581 = 157^2 *  8269,        (6123)
261019543 = 349^2 *  2143,        (2094)
296765173 = 131^2 * 17293,        (8515)
422857021 = 233^2 *  7789,        (6757)
573332713 = 331^2 *  5233,        (4965)
782481673 = 167^2 * 28057,       (13861).
Note how the quotient (in the rightmost column) seems always to be a multiple of non-unitary prime factor and less than the unitary prime factor.
For p, q prime, if p^2+p+1 = kq and k+1|p-1, then p^2*q is in this sequence. - Charlie Neder, Jun 09 2019

Subsequence of A326063.
Cf. A032742, A060681, A246282, A318505.
Cf. also A228058, A325981, A326131, A326141.

Select[Range[15, 10^6 + 1, 2], And[! PrimePowerQ@ #1, Mod[#1 - #2, #2 - #3] == 0] & @@ {#, DivisorSigma[1, #] - #, Divisors[#][[-2]]} &] (* Michael De Vlieger, Jun 22 2019 *)

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A060681(n) = (n-A032742(n));
A318505(n) = if(1==n,0,(sigma(n)-A032742(n))-n);
isA326064(n) = if((n%2)&&(2!=isprimepower(n)), my(s=A032742(n), t=sigma(n)-s); (gcd(t-n, n-A032742(n)) == t-n), 0);

More terms from Amiram Eldar, Dec 24 2020

A326131 Positive numbers n for which A000120(n) = k*A294898(n), with k < 0; numbers for which A326130(n) = sigma(n) - A005187(n).

Original entry on oeis.org

6, 28, 110, 496, 884, 8128, 18632, 85936, 116624, 15370304, 33550336, 73995392, 815634435, 3915380170, 5556840416, 6800695312, 8589869056

Offset: 1

Antti Karttunen, Jun 09 2019

No further terms below 2^31.
See also comments in A326133.
The quotients A000120(k)/(sigma(k)-A005187(k)) for these terms are: 1, 1, 5, 1, 3, 1, 5, 9, 2, 2, 1, 2, 2. Ones occur at the positions of perfect numbers.
a(18) > 10^11. - Amiram Eldar, Jan 03 2021

			110 is "1101110" in binary, thus A000120(110) = 5. Sigma(110) = 216, while A005187(110) = 215, thus as 5 = 5*(216-215), 110 is included in this sequence.

Intersection of A326132 and A326133, also of A326132 and A326138.
Cf. A000120, A000396 (subsequence), A005187, A294898, A295296, A295297, A295298, A295299, A326130.
Cf. also A325981, A326141.

q[n_] := Module[{bw = DigitCount[n, 2, 1], ab = DivisorSigma[1, n] - 2*n, sum}, (sum = ab + bw) > 0 && Divisible[bw, sum]]; Select[Range[10^5], q] (* Amiram Eldar, Jan 03 2021 *)

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
isA326131(n) = { my(t=sigma(n)-A005187(n)); (gcd(hammingweight(n), t) == t); };

a(14)-a(17) from Amiram Eldar, Jan 03 2021

A326137 Numbers with at least five distinct prime factors that satisfy Euler's criterion (A228058) for odd perfect numbers.

Original entry on oeis.org

17342325, 22678425, 31674825, 38686725, 41420925, 45090045, 49358925, 51740325, 54033525, 54695025, 67660425, 68939325, 70703325, 75818925, 76392225, 77106645, 78217425, 81375525, 92400525, 96316605, 97383825, 98750925, 99147825, 102284325, 107694405, 113656725, 115420725, 117890325, 118728225, 120536325, 127766925

Offset: 1

Antti Karttunen, Jun 12 2019

nonn

P. P. Nielsen's 2006 paper shows that any odd perfect number must have at least nine distinct prime factors, thus if such numbers exist at all, they must occur in this sequence.

I conjecture that it is eventually possible to find an easy proof that this sequence has no common terms with A325981, and/or several other sequences (A326064, A326074, A326141, A326148, etc.) listed under index entry "sequences where odd perfect numbers must occur", thus settling the question about the existence of such numbers.

Antti Karttunen, Table of n, a(n) for n = 1..1032; all terms < 2^31
Charles Greathouse and Eric W. Weisstein, MathWorld: Odd perfect number
Oliver Knill, The oldest open problem in mathematics, Handout for NEU Math Circle, December 2, 2007
P. P. Nielsen, Odd Perfect Numbers Have At Least Nine Distinct Prime Factors, arXiv:math/0602485 [math.NT], 2006.
Wikipedia, Perfect number: Odd perfect numbers
Index entries for sequences where any odd perfect numbers must occur

Subsequence of A228058.

Cf. A001221, A325981, A326064, A326074, A326141, A326148.

isA228058(n) = if(!(n%2)||(omega(n)<2),0,my(f=factor(n),y=0); for(i=1,#f~,if(1==(f[i,2]%4), if((1==y)||(1!=(f[i,1]%4)),return(0),y=1), if(f[i,2]%2, return(0)))); (y));
isA326137(n) = ((omega(n)>=5)&&isA228058(n));

A329640 Numbers n for which A329639(n) is equal to gcd(A329638(n), A329639(n)).

Original entry on oeis.org

1, 9, 18, 27, 45, 54, 70, 75, 84, 125, 135, 144, 153, 198, 279, 366, 390, 392, 423, 459, 747, 837, 855, 858, 891, 927, 1269, 1341, 1494, 1503, 1690, 1899, 2097, 2241, 2493, 2604, 2679, 2763, 2781, 2888, 2979, 2988, 3177, 3411, 3507, 3879, 4023, 4041, 4050, 4482, 4491, 4509, 4707, 5067, 5283, 5463, 5679, 5697, 5817, 5877, 5982, 6093

Offset: 1

Antti Karttunen, Nov 21 2019

nonn

After the initial 1, numbers n such that A329638(n) is a multiple of A329639(n).

Antti Karttunen, Table of n, a(n) for n = 1..85 (based on Hans Havermann's factorization of A156552)

Cf. A156552, A329638, A329639, A329641.
Cf. A324201 (a subsequence).
Cf. also A326141.

isA329640(n) = { my(u=A329639(n)); (u==gcd(A329638(n), u)); };

cp's OEIS Frontend

A326140 a(n) = gcd(A318878(n), A318879(n)).

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A326064 Odd composite numbers n, not squares of primes, such that (A001065(n) - A032742(n)) divides (n - A032742(n)), where A032742 gives the largest proper divisor, and A001065 is the sum of proper divisors.

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A326131 Positive numbers n for which A000120(n) = k*A294898(n), with k < 0; numbers for which A326130(n) = sigma(n) - A005187(n).

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A326137 Numbers with at least five distinct prime factors that satisfy Euler's criterion (A228058) for odd perfect numbers.

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A329640 Numbers n for which A329639(n) is equal to gcd(A329638(n), A329639(n)).

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