A056075 -id:A056075 - cp's OEIS frontend

A070019 Duplicate of A056075.

1, 4, 56, 7192, 7232, 7912, 10792, 17272, 30592, 114256, 2154584, 3428368, 44375136

Offset: 1

dead

A238227 Numbers n such that if x=sigma(n)-tau(n)-n then n=sigma(x)-tau(x)-x.

1, 56, 66, 70, 992, 1012, 2260, 2516, 6042, 6902, 7192, 7210, 7232, 7750, 7912, 8178, 9086, 10792, 12198, 13706, 17272, 30592, 32778, 33352, 35032, 40166, 44034, 45010, 46670, 47710, 55374, 62296, 63688, 65570, 114256, 132916, 133892, 138244, 141236, 146804, 155572

Offset: 1

Paolo P. Lava, Feb 20 2014

nonn

If the second term (4) is not considered, A056075 is almost a subset of this sequence: it lists the fixed points of the transform n -> sigma(n)-tau(n)-n.

			Fixed points: 56, 7192, 7232, 7912, 10792, ...
sigma(66) = 144, tau(66) = 8 and 144 - 8 - 66 = 70.
sigma(70) = 144, tau(70) = 8 and 144 - 8 - 70 = 66.

Amiram Eldar, Table of n, a(n) for n = 1..1001 (terms 1..50 from Paolo P. Lava)

Cf. A000005, A000203, A056075, A238225, A238226, A238228, A238229, A238230.

with(numtheory); P:=proc(q)local a,n;
for n from 1 to q do a:=sigma(n)-tau(n)-n;
if sigma(a)-tau(a)-a=n then print(n);
fi; od; end: P(10^6);

A055681 Numbers k that divide sigma(k)-phi(k).

Original entry on oeis.org

1, 2, 12, 42, 1242, 75960, 1447488, 3506976, 6137440, 10834488, 17156160, 90288000, 431440416, 454460160, 704592000, 1385119360, 1588268480, 10674673152, 24913095480, 31103703540, 56015374080, 80767843200, 129631788000, 463308768000, 469897798656, 834460413696

Offset: 1

Robert G. Wilson v, Jun 09 2000

nonn

a(37) > 10^13. - Giovanni Resta, Jun 12 2013

Giovanni Resta, Table of n, a(n) for n = 1..36 (terms < 10^13)

Cf. A011774, A000203, A000010, A054024, A056075.

with(numtheory): A055681:=n->`if`(sigma(n)-phi(n) mod n=0,n,NULL): seq(A055681(n), n=1..10^5); # Wesley Ivan Hurt, Sep 13 2014

Do[If[Mod[DivisorSigma[1, n]-EulerPhi[n], n]==0, Print[n]], {n, 1, 10^9}]

for(n=1,10^8,if((sigma(n)-eulerphi(n))%n==0,print1(n,", "))) \\ Derek Orr, Sep 13 2014

a(16)-a(26) from Donovan Johnson, Feb 28 2012

A301975 Numbers whose abundance is divisible by its number of divisors.

Original entry on oeis.org

1, 3, 5, 6, 7, 11, 13, 14, 17, 19, 22, 23, 28, 29, 31, 37, 38, 41, 43, 45, 46, 47, 52, 53, 56, 59, 60, 61, 62, 67, 71, 73, 76, 79, 83, 86, 89, 94, 96, 97, 99, 101, 103, 107, 109, 113, 118, 124, 126, 127, 130, 131, 132, 134, 137, 139, 142, 147, 148, 149, 150, 151, 153, 157, 158, 163, 166, 167, 168, 170, 172, 173, 175, 176, 179

Offset: 1

Waldemar Puszkarz, Mar 29 2018

nonn

Numbers n such that f(n) = A033880(n)/A000005(n) is an integer.
Perfect numbers (A000396) and odd primes (A065091) are members, unified (along with 1) into a subsequence on which abs(f(n)) reaches record extrema. For perfect numbers, these are global minima, for the other terms, maxima.
Another notable subsequence is defined by f(n)=1: numbers whose abundance equals their number of divisors. They all belong to A056075. The first 3 terms are 56, 7192, 7232. There are 11 of them up to 10^9.

			11 is a term as its abundance is -10 and its number of divisors is 2, the former number being divisible by the latter.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A033880 (abundance), A000005 (number of divisors), A065091, A000396 (subsequences), A056075 (related sequence).

Select[Range[180], Divisible[DivisorSigma[1,#]-2#, DivisorSigma[0,#]]&]

for(n=1, 180, ((sigma(n)-2*n)%numdiv(n)==0) && print1(n ", "))

isok(n) = !((sigma(n)-2*n)%numdiv(n)); \\ Michel Marcus, Apr 09 2018

A344781 Numbers k such that A070313(k) = 2^k - (2*k+1) is a prime number.

Original entry on oeis.org

4, 7, 8, 28, 32, 81, 669, 1108, 1699, 1839, 2319, 9566, 14866, 30855, 35932, 56048, 70915, 72578

Offset: 1

Amiram Eldar, May 28 2021

The corresponding primes are 7, 113, 239, 268435399, 4294967231, 2417851639229258349412189, ...

If k is a term of this sequence then 2^(k-1)*(2^k-(2*k+1)) is a term of A056075 (see Farideh Firoozbakht's comment in A056075).

			4 is a term since 2^4 - (2*4+1) = 16 - 9 = 7 is a prime.
7 is a term since 2^7 - (2*7+1) = 128 - 15 = 113 is a prime.

Cf. A056075, A070313.

Select[Range[2400], PrimeQ[2^# - 2*# - 1] &]

a(16)-a(18) from Michael S. Branicky, May 07 2024

cp's OEIS Frontend

A070019 Duplicate of A056075.

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A238227 Numbers n such that if x=sigma(n)-tau(n)-n then n=sigma(x)-tau(x)-x.

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A055681 Numbers k that divide sigma(k)-phi(k).

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A301975 Numbers whose abundance is divisible by its number of divisors.

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A344781 Numbers k such that A070313(k) = 2^k - (2*k+1) is a prime number.

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