A019279 -id:A019279 - cp's OEIS frontend

A138883 Number of digits in n-th even superperfect number A061652(n).

1, 1, 2, 2, 4, 5, 6, 10, 19, 27, 32, 38, 157, 183, 385, 663, 687, 969, 1280, 1332, 2917, 2993, 3376, 6002, 6533, 6987, 13395, 25962, 33265, 39751, 65050, 227831, 258715, 378632, 420921, 895932, 909525, 2098960

Offset: 1

Omar E. Pol, Apr 08 2008

Also, number of digits in n-th superperfect number A019279(n), if there are no odd superperfect numbers.

Cf. A019279, A028335, A061193, A061652.

a(n)=A055642(A061652(n)). - R. J. Mathar, May 22 2008

More terms from R. J. Mathar, May 22 2008

A138884 Numbers that are not even superperfect numbers.

Original entry on oeis.org

1, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65

Offset: 1

Omar E. Pol, Apr 06 2008

nonn

Also, numbers that are not superperfect numbers A019279, if there are no odd superperfect numbers.

Even superperfect numbers: A061652. Cf. A019279, A132999, A133398.

A174226 Partial sums of A034897.

Original entry on oeis.org

6, 27, 55, 356, 681, 1177, 1874, 3207, 5116, 7157, 9290, 13191, 21319, 32012, 48525, 68046, 92647, 119624, 170925, 267286, 397439, 557280, 720481, 897142, 1111415, 1361736, 1637569, 1933910, 2240091, 2629684, 3116561, 3612090, 4154503

Offset: 1

Jonathan Vos Post, Mar 12 2010

nonn

Partial sums of hyperperfect numbers. The subsequence of prime values in this partial sum begins: 21319, 92647, 720481.

			a(x) = 6 + 21 + 28 + 301 + 325 + 496 + 697 + 1333 + 1909 + 2041 + 2133 + 3901 + 8128 + 10693 + 16513 + 19521 + 24601 = 92647 is prime.

Cf. A034897, A034898, A007592, A019279.

a(n) = SUM[i=1..n] A034897(i) = SUM[i=1..n] {m such that k*sigma(m) = (k+1)*m + k - 1 for some positive integer k, and sigma being the divisor function A000005; noting that k=1 gives perfect numbers}.

A181711 Numbers of the form m(2^k-1), where m = 2^(k-1)(2^k-1) is a perfect number (A000396).

Original entry on oeis.org

18, 196, 15376, 1032256, 274810802176, 1125882727038976, 72057319160283136, 4951760152529835082242850816, 6129982163463555428116476125461573244012649752219877376

Offset: 1

Vladimir Shevelev, Nov 07 2010

nonn

The associated exponents k are in A000043: 2, 3, 5, 7, 13, 17, 19 ,31, 61, ...

One can prove that, if m = 2^(k-1)*(2^k-1) is a perfect number, then m*2^k and m*(2^k-1) are both in A181595. Thus every even term in A000396 is a difference of two terms in A181595.

			With k=3, m = 2^(k-1)*(2^k - 1) = 2^2*(8 - 1) = 28 is a perfect number (A000396), so m*(2^k - 1) = 28*7 = 196 is in the sequence. - _Michael B. Porter_, Jul 19 2016

Cf. A000043, A000396, A181595, A181596, A181701, A181710.

If odd perfect numbers do not exist, then a(n) = A181710(n) - A000396(n).

a(n) = A019279(n)*(A000668(n))^2 if there are no odd superperfect numbers. - César Aguilera, Jun 13 2017

Definition condensed by R. J. Mathar, Dec 05 2010

A192275 Super anti-perfect numbers.

Original entry on oeis.org

7, 167, 507, 543, 3433, 6707, 9615, 15690, 87667, 1023971, 2054435, 5871045, 9150387, 14146891, 63538839, 169623015

Offset: 1

Paolo P. Lava, Jun 28 2011

Like A019279 but using anti-divisors. sigma*^2(n)=sigma*(sigma*(n))=2n, where sigma*(n) is the sum of the anti-divisors of n.

Cf. A019279, A066417.

for n from 1 do
        if A066417(A066417(n)) = 2*n then
                print(n);
        end if;
end do: # R. J. Mathar, Oct 02 2011

a(9)-a(16) from Donovan Johnson, Sep 22 2011

A200758 Superimperfect numbers.

Original entry on oeis.org

2, 4, 8, 128, 32768, 2147483648

Offset: 1

Laszlo Toth, Nov 22 2011

A number n is said to be superimperfect if 2*beta(beta(n)) = n, where beta is the multiplicative function defined by beta(p^e) = p^e - p^(e-1) + p^(e-2) - ... + (-1)^e for every prime power p^e. The function beta is called the alternating sum-of-divisors function. Here beta(n) is the absolute value of A061020(n). There are no other superimperfect numbers up to 10^7. The number 2^(2^k-1) is superimperfect if and only if k=1,2,3,4,5.

Laszlo Toth, A survey of the alternating sum-of-divisors function, arXiv:1111.4842 [math.NT], 2011-2014.
Laszlo Toth, The alternating sum-of-divisors function, 9th Joint Conf. on Math. and Comp. Sci., February 9-12, 2012, Siofok, Hungary.

Cf. A061020, A127724, A127725, A019279, A058891, A206369.

beta(n)=sumdiv(n,d,(-1)^bigomega(n/d)*d)
for(n=1,1e8,if(2*beta(beta(n))==n,print1(n", "))) \\ Charles R Greathouse IV, Nov 22 2011

ak(p,e)=my(s=1); for(i=1,e, s=s*p + (-1)^i); s
beta(n)=my(f=factor(n)); prod(i=1,#f~, ak(f[i,1],f[i,2]))
is(n)=my(b=beta(n)); 2*b-2 >= n && 2*beta(b)==n \\ Charles R Greathouse IV, Dec 27 2016

A283147 Number n such that there are no primes of the form sigma(n)/k where 1 < k < n is a (proper) nondivisor of n.

Original entry on oeis.org

1, 2, 4, 9, 12, 16, 25, 48, 64, 112, 192, 240, 289, 448, 729, 960, 1344, 1681, 1984, 2401, 3481, 4096, 5041, 6720, 7921, 10201, 12288, 15625, 17161, 27889, 28561, 28672, 29929, 39270, 53130, 61440, 65536, 71610, 82110

Offset: 1

Juri-Stepan Gerasimov, Mar 01 2017

nonn

Supersequence of A023194, A019279 and A061652.

Cf. A000203, A173540.

is(n)=my(s=sigma(n), p=factor(s)[,1], k); for(i=1,#p, k=s/p[i]; if(kCharles R Greathouse IV, Mar 01 2017

Corrected by Charles R Greathouse IV, Mar 01 2017

A377142 Numbers m such that phi(2m-1)/2 = phi(2m) - 1, where phi = A000010.

Original entry on oeis.org

2, 4, 5, 16, 64, 4096, 65536, 262144

Offset: 1

Juri-Stepan Gerasimov, Oct 19 2024

Conjecture 1: each term has the form p^(q-1), where p, q both some primes.
Conjecture 2: sequence is infinite.
Presumably the sequence of numbers of the form (exponent of a(n)) + (smallest divisor of a(n)) is a supersequence of Mersenne exponents.
If 2*m-1 is a Mersenne prime (A000668), then phi(2*m-1)/2 = m-1 = phi(2*m) - 1, so m is a term. - Robert Israel, Oct 20 2024

			2 is a term because phi(2*2-1)/2 = phi(3)/2 = 2/2 = 1 is equal to phi(2*2)-1 = phi(4)-1 = 2-1 = 1;
5 is a term because phi(2*5-1)/2 = phi(9)/2 = 6/2 = 3 is equal to phi(2*5)-1 = phi(10)-1 = 4-1 = 3.

Supersequence of A019279 and A061652.

Cf. A000010, A000043, A000668, A090748, A291901, A376337.

[m: m in [2..2*10^6] | EulerPhi(2*m-1)/2 eq EulerPhi(2*m)-1];

filter:= m -> numtheory:-phi(2*m-1)/2 = numtheory:-phi(2*m)-1:
select(filter, [$1..10^7]); # Robert Israel, Oct 20 2024

Select[Range[300000], EulerPhi[2*# - 1]/2 == EulerPhi[2*#] - 1 &] (* Amiram Eldar, Oct 30 2024 *)

isok(m) = eulerphi(2*m-1)/2 == eulerphi(2*m) - 1; \\ Michel Marcus, Oct 30 2024

a(n) = (A376337(n) + 1)/2.

cp's OEIS Frontend

A138883 Number of digits in n-th even superperfect number A061652(n).

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A138884 Numbers that are not even superperfect numbers.

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A174226 Partial sums of A034897.

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A181711 Numbers of the form m*(2^k-1), where m = 2^(k-1)*(2^k-1) is a perfect number (A000396).

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A192275 Super anti-perfect numbers.

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A200758 Superimperfect numbers.

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A283147 Number n such that there are no primes of the form sigma(n)/k where 1 < k < n is a (proper) nondivisor of n.

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A377142 Numbers m such that phi(2*m-1)/2 = phi(2*m) - 1, where phi = A000010.

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Magma

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A181711 Numbers of the form m(2^k-1), where m = 2^(k-1)(2^k-1) is a perfect number (A000396).

A377142 Numbers m such that phi(2m-1)/2 = phi(2m) - 1, where phi = A000010.