A011251 -id:A011251 - cp's OEIS frontend

A001771 Numbers k such that 7*2^k - 1 is prime.

1, 5, 9, 17, 21, 29, 45, 177, 18381, 22529, 24557, 26109, 34857, 41957, 67421, 70209, 169085, 173489, 177977, 363929, 372897

Offset: 1

N. J. A. Sloane

k is always of the form 4*j + 1.

If k is in the sequence and m=2^(k+2)*3*(7*2^k-1) then phi(m)+sigma(m)=3m (m is in the sequence A011251). The proof is easy. - Farideh Firoozbakht, Mar 04 2005

H. Riesel, "Prime numbers and computer methods for factorization", Progress in Mathematics, Vol. 57, Birkhäuser, Boston, 1985, Chap. 4, see pp. 381-384.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Wilfrid Keller, List of primes k.2^n - 1 for k < 300
H. C. Williams and C. R. Zarnke, A report on prime numbers of the forms M = (6a+1)*2^(2m-1)-1 and (6a-1)*2^(2m)-1, Math. Comp., 22 (1968), 420-422.
Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime

Cf. A050523, A003307, A002235, A046865, A079906, A046866, A005541, A056725, A046867, A079907.

Cf. A032353 (7*2^k+1 is prime).

Do[ If[ PrimeQ[7*2^n - 1], Print[n]], {n, 1, 2500}]

v=[ ]; for(n=0,2000, if(isprime(7*2^n-1),v=concat(v,n),)); v

More terms from Douglas Burke (dburke(AT)nevada.edu).

More terms from Hugo Pfoertner, Jun 23 2004

A011774 Nonprimes k that divide sigma(k) + phi(k).

Original entry on oeis.org

1, 312, 560, 588, 1400, 23760, 59400, 85632, 147492, 153720, 556160, 569328, 1590816, 2013216, 3343776, 4563000, 4695456, 9745728, 12558912, 22013952, 23336172, 30002960, 45326160, 52021242, 75007400, 113315400, 137617728, 153587720, 402831360, 699117024

Offset: 1

R. K. Guy

2*k = sigma(k) + phi(k) if and only if k is 1 or a prime.
If 7*2^j - 1 is prime then m = 2^(j+2)*3*(7*2^j - 1) is in the sequence. Because phi(m) = 2^(j+2)*(7*2^j - 2); sigma(m) = 7*2^(j+2)*(2^(j+3) - 1) so phi(m) + sigma(m) = 2^(j+2)*((7*2^j - 2) + (7*2^(j+3) - 7)) = 2^(j+2)* (63*2^(j+2) - 9) = 3*(2^(j+2)*3*(7*2^j - 1)) = 3*m, hence m is a term of A011251 and consequently m is a term of this sequence. A112729 gives such m's. - Farideh Firoozbakht, Dec 01 2005
Conjecture: For n > 1, a(n) is a Zumkeller number (A083207). Verified for all n in [2,63]. - Ivan N. Ianakiev, Jan 25 2023

			a(26) = 113315400: sigma = 426535200, phi = 26726400, quotient = 4.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B42, p. 151.
Zhang Ming-Zhi, typescript submitted to Unsolved Problems section of Monthly, 96-01-10.

Giovanni Resta, Table of n, a(n) for n = 1..63 (terms < 10^13; first 53 terms from Donovan Johnson)
Richard K. Guy, Divisors and desires, Amer. Math. Monthly, 104 (1997), 359-360.
Q.-X. Jin and M. Tang, The 4-Nicol Numbers Having Five Different Prime Divisors, J. Int. Seq. 14 (2011) # 11.7.2.
Eric Weisstein's World of Mathematics, Prime Number.

Cf. A065387, A011251, A011254, A055681, A001771, A112729.

Do[If[Mod[DivisorSigma[1, n]+EulerPhi[n], n]==0, Print[n]], {n, 1, 2*10^7}]
Do[ If[ ! PrimeQ[n] && Mod[ DivisorSigma[1, n] + EulerPhi[n], n] == 0, Print[n] ], {n, 1, 10^8} ]

sp(n)=my(f=factor(n));n*prod(i=1, #f[,1], 1-1/f[i,1]) + prod(i=1, #f[,1], (f[i,1]^(f[i,2]+1)-1)/(f[i,1]-1))
p=2;forprime(q=3, 1e6, for(n=p+1, q-1, if(sp(n)%n==0, print1(n", ")));p=q) \\ Charles R Greathouse IV, Mar 19 2012

More terms from David W. Wilson

Corrected by Labos Elemer, Feb 12 2004

A011254 Numbers k such that phi(k) + sigma(k) = 4*k.

Original entry on oeis.org

23760, 59400, 153720, 4563000, 45326160, 113315400, 402831360, 731601000, 803685120, 865950624, 919501200, 1178491680, 3504597120, 3786686400, 6429564000, 14924714400, 25310621952, 26998616736, 53138687040, 86955675840, 513969369984, 1054373308800, 1868445408960

Offset: 1

N. J. A. Sloane

If (sigma(m)-phi(m))/(4*m-sigma(m)-phi(m)) is a prime integer p not dividing m, then p*m is in the sequence. 135230346701100 is in the sequence and not divisible by 24. - Jens Kruse Andersen, Feb 17 2009

If k=80*m is in the sequence and gcd(m,10) = 1 then 200*m is also in the sequence. Proof: phi(200*m) + sigma(200*m) = phi(200)*phi(m) + sigma(200)*sigma(m) = 80*phi(m) + 465*sigma(k) = (5/2)*(32*phi(m) + 186*sigma(m)) = (5/2)*(phi(80)*phi(m) + sigma(80)*sigma(m)) = (5/2)*(phi(80*m) + sigma(80*m)) = (5/2)*(phi(k) + sigma(k)) = (5/2)*(4*k) = 5/2*(4*80*m) = 4*(200*m) so 200*m is in the sequence. - Farideh Firoozbakht, Mar 30 2009

			phi(23760) + sigma(23760) = 5760 + 89280 = 4*23760, so 23760 is in the sequence.

David Wells, Prime Numbers: The Most Mysterious Figures in Math, Hoboken, New Jersey, John Wiley & Sons (2005), p. 75.
Zhang Ming-Zhi (typescript submitted to Unsolved Problems section of Monthly, Oct 01 1996.

Donovan Johnson, Table of n, a(n) for n = 1..25 (terms < 5*10^12)
Richard K. Guy, Divisors and desires, Amer. Math. Monthly, 104 (1997), 359-360.
Kelley Harris, On the classification of integers n that divide phi(n)+sigma(n), J. Num. Theory 129 (2009) 2093-2110

Cf. A000010, A000203, A011251, A011774, A015704.

Select[Range[1000000], DivisorSigma[1, #] + EulerPhi[#] == 4 # &] (* David Nacin, Feb 28 2012 *)

is(n)=eulerphi(n)+sigma(n)==4*n \\ Charles R Greathouse IV, Nov 27 2013

More terms from Jud McCranie
1178491680 from Farideh Firoozbakht, Jan 31 2006
2 more terms from Jud McCranie, Jan 31 2006
24 divides all known terms of the sequence. If this is true for the next five terms then they are 6429564000, 14924714400, 25310621952, 26998616736 and 53138687040. - Farideh Firoozbakht, Mar 11 2006
More terms from Jens Kruse Andersen, Feb 17 2009
a(21) from Donovan Johnson, Feb 28 2012
a(22)-a(23) from Donovan Johnson, Apr 04 2012

A015704 a(n) is the smallest number m such that phi(m) + sigma(m) = n*m.

Original entry on oeis.org

1, 312, 23760, 336280120525440

Offset: 2

Robert G. Wilson v

The offset is 2, because for all numbers m, phi(m)+sigma(m) >= 2*m, so there is no number a(1) such that phi(a(1))+sigma(a(1))=1*a(1). - Farideh Firoozbakht, Jan 22 2008
a(5) >= 2*10^9. - Farideh Firoozbakht, Jan 22 2008
10^13 < a(5) <= 336280120525440. Charles R Greathouse IV showed that 6 divides a(5). 336280120525440 and 60493590969525342720 are the only m values I found such that phi(m) + sigma(m) = 5*m. - Donovan Johnson, Sep 11 2012

Richard K. Guy, Divisors and desires, Amer. Math. Monthly, 104 (1997), 359-360.

Cf. A011251, A011254.

a(n) = my(m = 1); while(sigma(m)+eulerphi(m) != n*m, m++); m; \\ Michel Marcus, Oct 04 2017

Name, offset and data corrected by Farideh Firoozbakht, Jan 22 2008

a(5) confirmed by Martin Ehrenstein, Jan 15 2024

A112729 Numbers of the form 2^(k+2)3(72^k-1) where 72^k-1 is prime.

Original entry on oeis.org

312, 85632, 22013952, 1443107438592, 369435881766912, 24211351590301335552, 103986963299971520879061368832

Offset: 1

Farideh Firoozbakht, Dec 01 2005

This sequence is a subsequence of A011251 and A011774, namely if m is in the sequence then phi(m)+sigma(m)=3*m (see Comments line of A011251).

Number of digits of all the 20 known terms of this sequence are respectively 3, 5, 8, 13, 15, 20, 30, 109, 11069, 13566, 14787, 15722, 20988, 25263, 40594, 42272, 101802, 104453, 107155 and 219110.

			103986963299971520879061368832 is in the sequence because 103986963299971520879061368832=2^(45+2)*3*(7*2^45-1) and 7*2^45-1 is prime.

Cf. A001771. A011251, A011774.

Do[If[PrimeQ[7*2^n-1], Print[3*2^(n+2)*(7*2^n-1)]], {n, 177}]

A328951 Numbers m such that sigma(m) + tau(m) = 3m.

Original entry on oeis.org

60, 5472, 2500704, 24361213461200

Offset: 1

Jaroslav Krizek, Nov 10 2019

Abundant numbers m with abundance A(m) = m - tau(m) = A049820(m), where A049820(n) is the number of non-divisors of n.
Subsequence of A056076.
Corresponding values of A(m) = m - tau(m): 48, 5436, 2500632, ...
4 is the only number m with deficiency D(m) = m - tau(m).
808989640739424 is also a term. - Giovanni Resta, Nov 14 2019

			60 is a term because sigma(60) + tau(60) = 3*60; 168 + 12 = 180 = 3*60.

Cf. A000005, A000203, A033880, A049820, A056076.
Cf. A083874 (numbers m such that sigma(m) + tau(m) = 2m).
Cf. A011251 (numbers m such that sigma(m) + phi(m) = 3m).
Cf. A329104 (numbers m with abundance A(m) = tau(m)).

[m: m in [1..10^7] | SumOfDivisors(m) - 2*m eq m - NumberOfDivisors(m)];

Select[Range[3*10^6], DivisorSigma[0, #] + DivisorSigma[1, #] == 3# &] (* Amiram Eldar, Nov 10 2019 *)

isok(m) = my(f=factor(m)); sigma(f) + numdiv(m) == 3*m; \\ Michel Marcus, Nov 13 2019

a(4) from Martin Ehrenstein, Jul 25 2023

cp's OEIS Frontend

A001771 Numbers k such that 7*2^k - 1 is prime.

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A011774 Nonprimes k that divide sigma(k) + phi(k).

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A011254 Numbers k such that phi(k) + sigma(k) = 4*k.

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A015704 a(n) is the smallest number m such that phi(m) + sigma(m) = n*m.

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A112729 Numbers of the form 2^(k+2)*3*(7*2^k-1) where 7*2^k-1 is prime.

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A328951 Numbers m such that sigma(m) + tau(m) = 3m.

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Magma

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A112729 Numbers of the form 2^(k+2)3(72^k-1) where 72^k-1 is prime.