A294386 -id:A294386 - cp's OEIS frontend

A294682 Numbers n such that A294386(n) = 2^(k-1)(2^k - 2n - 1) for some k such that 2^k - 2*n - 1 is prime.

0, 12, 62, 121, 126, 205, 241, 877, 1021, 1022, 1645, 2041, 2424, 2761, 2791, 2965, 3355, 3445, 3541, 4021, 4081, 4094, 4165, 4825, 5071, 5191, 5251, 5593, 6151, 6385, 6631, 7465, 7765, 7884, 8137, 8188

Offset: 1

Robert Israel, Nov 06 2017

			a(3) = 62 is in the sequence because A294386(62) = 192 = 2^6*3 where 2^7 - 2*62 - 1 = 3 is prime.

Cf. A096502, A294386.

# Assuming A294386[n] has been assigned for n from 0 to N
Res:= NULL:
for n from 0 to N do
  for k from ilog2(2*n+1)+1 do
    p:= 2^k - 2*n-1;
    if 2^(k-1)*p > A294386[n] then break fi;
    if isprime(p) then
      if A294386[n] = 2^(k-1)*p then Res:= Res, n fi;
      break
    fi
  od
od:
Res;

A294347 a(n) is the smallest number whose deficiency or abundance is equal to n, or a(n) = 0 if such a number does not exist.

Original entry on oeis.org

6, 1, 3, 18, 5, 9, 7, 50, 22

Offset: 0

Omar E. Pol, Oct 29 2017

If nonzero, a(9) > 10^9. - Michel Marcus, Oct 29 2017
From Robert Israel, Oct 29 2017: (Start)
If n is odd, then a(n) must be a square or twice a square (A028982).
If nonzero, a(9) > 10^13.
Some other values: a(11)=244036, a(17)=100, a(19)=25, a(25)=98, a(31)=15376, a(37)=484, a(39)=162, a(41)=49, a(47)=225, a(51)=72. (End)
a(n) > 10^20 for n in (9, 13, 15, 21, 23, 27, 29, 33, 35, 43, 45); see the intersection of A234285 and A234286. - Michel Marcus, Oct 30 2017
For the intersection mentioned above see A294406. - Omar E. Pol, Nov 01 2017

Nichole Davis, Dominic Klyve and Nicole Kraght, On the difference between an integer and the sum of its proper divisors, Involve, Vol. 6 (2013), No. 4, 493-504; DOI: 10.2140/involve.2013.6.493.
Raven Dean, Rick Erdman, Dominic Klyve, Emily Lycette, Melissa Pidde, and Derek Wheel, Families of Values of the Excedent Function sigma(n)-2n, Missouri J. Math. Sci., Volume 27, Issue 1 (2015), 37-46.

Cf. A000203, A000396, A005100, A005101, A028982, A033879, A033880, A234285, A234286, A294386, A294393, A294406.

Table[k = 1; While[Abs[2 k - DivisorSigma[1, k]] != n, k++]; k, {n, 0, 8}] (* Michael De Vlieger, Oct 30 2017 *)

a(n) = {my(k=1); while (abs(2*k-sigma(k)) != n, k++); k;} \\ Michel Marcus, Oct 29 2017

A294393 a(n) is the smallest number whose deficiency or abundance is equal to 2*n (or 0 if such a number does not exist), minus the n-th odd number.

Original entry on oeis.org

5, 0, 0, 0, 13, 0, 0, 12, 0, 0, 25, 0, 87, 31, 0, 0, 217, 22, 0, 16, 0, 0, 49, 0, 11, 55, 0, 32, 27, 0, 0, 22, 43, 0, 73, 0, 0, 637, 81, 0, 75, 0, 320, 28, 0, 50, 313, 24, 0, 103, 0, 0, 109, 0, 0, 34, 0, 62, 301, 24, -1, 120, 67, 0, 133, 0, 128, 139, 0, 0, 433, 42, 23, 151, 0, 0, 219, 82, 0, 28, 119

Offset: 0

Omar E. Pol, Oct 30 2017

sign

Note that a(60) = -1.

			--------------------------------------
n    A294386(n) - A005408(n)  =  a(n)
--------------------------------------
0        6            1           5
1        3            3           0
2        5            5           0
3        7            7           0
4       22            9          13
...

Cf. A000203, A000396, A005100, A005101, A033879, A033880, A234285, A234286, A294347, A294386.

f(n) = abs(2*n-sigma(n));
a(n) = my(k=1); while(f(k) != 2*n, k++); k - (2*n+1); \\ Michel Marcus, Oct 31 2017

a(n) = A294386(n) - A005408(n).

cp's OEIS Frontend

A294682 Numbers n such that A294386(n) = 2^(k-1)(2^k - 2n - 1) for some k such that 2^k - 2*n - 1 is prime.

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A294347 a(n) is the smallest number whose deficiency or abundance is equal to n, or a(n) = 0 if such a number does not exist.

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PARI

A294393 a(n) is the smallest number whose deficiency or abundance is equal to 2*n (or 0 if such a number does not exist), minus the n-th odd number.

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Author

Keywords

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PARI

Formula

cp's OEIS Frontend

A294682 Numbers n such that A294386(n) = 2^(k-1)*(2^k - 2*n - 1) for some k such that 2^k - 2*n - 1 is prime.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

A294347 a(n) is the smallest number whose deficiency or abundance is equal to n, or a(n) = 0 if such a number does not exist.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A294393 a(n) is the smallest number whose deficiency or abundance is equal to 2*n (or 0 if such a number does not exist), minus the n-th odd number.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A294682 Numbers n such that A294386(n) = 2^(k-1)(2^k - 2n - 1) for some k such that 2^k - 2*n - 1 is prime.