A057447 -id:A057447 - cp's OEIS frontend

A002827 Unitary perfect numbers: numbers k such that usigma(k) - k = k.

6, 60, 90, 87360, 146361946186458562560000

Offset: 1

d is a unitary divisor of k if gcd(d,k/d)=1; usigma(k) is their sum (A034448).
The prime factors of a unitary perfect number (A002827) are the Higgs primes (A057447). - Paul Muljadi, Oct 10 2005
It is not known if a(6) exists. - N. J. A. Sloane, Jul 27 2015
Frei proved that if there is a unitary perfect number that is not divisible by 3, then it is divisible by 2^m with m >= 144, it has at least 144 distinct odd prime factors, and it is larger than 10^440. - Amiram Eldar, Mar 05 2019
Conjecture: Subsequence of A083207 (Zumkeller numbers). Verified for all present terms. - Ivan N. Ianakiev, Jan 20 2020

			Unitary divisors of 60 are 1,4,3,5,12,20,15,60, with sum 120 = 2*60.
146361946186458562560000 = 2^18 * 3 * 5^4 * 7 * 11 * 13 * 19 * 37 * 79 * 109 * 157 * 313.

R. K. Guy, Unsolved Problems in Number Theory, Sect. B3.
F. Le Lionnais, Les Nombres Remarquables. Paris: Hermann, p. 59, 1983.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section III.45.1.
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).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 147-148.

H. A. M. Frei, Über unitar perfekte Zahlen, Elemente der Mathematik, Vol. 33, No. 4 (1978), pp. 95-96.
Takeshi Goto, Upper Bounds for Unitary Perfect Numbers and Unitary Harmonic Numbers, Rocky Mountain Journal of Mathematics, Vol. 37, No. 5 (2007), pp. 1557-1576.
A. V. Lelechenko, The Quest for the Generalized Perfect Numbers, in Theoretical and Applied Aspects of Cybernetics, TAAC 2014, Kiev.
M. V. Subbarao, Letter to N. J. A. Sloane, Feb 18 1974
M. V. Subbarao, T. J. Cook, R. S. Newberry and J. M. Weber, On unitary perfect numbers, Delta, 3 (No. 1, 1972), 22-26.
G. Villemin's Almanac of Numbers, Nombres Unitairement Parfaits
C. R. Wall, Letter to P. Hagis, Jr., Jan 13 1972
C. R. Wall, The fifth unitary perfect number, Canad. Math. Bull., 18 (1975), 115-122.
C. R. Wall, On the largest odd component of a unitary perfect number, Fib. Quart., 25 (1987), 312-316.
Eric Weisstein's World of Mathematics, Unitary Perfect Number.
Wikipedia, Unitary perfect number

Cf. A034460, A034448, A057447.
Subsequence of the following sequences: A003062, A290466 (seemingly), A293188, A327157, A327158.
Gives the positions of ones in A327159.

usnQ[n_]:=Total[Select[Divisors[n],GCD[#,n/#]==1&]]==2n; Select[Range[ 90000],usnQ] (* This will generate the first four terms of the sequence; it would take a very long time to attempt to generate the fifth term. *) (* Harvey P. Dale, Nov 14 2012 *)

is(n)=sumdivmult(n, d, if(gcd(d, n/d)==1, d))==2*n \\ Charles R Greathouse IV, Aug 01 2016

If m is a term and omega(m) = A001221(m) = k, then m < 2^(2^k) (Goto, 2007). - Amiram Eldar, Jun 06 2020

A007459 Higgs's primes: a(n+1) = smallest prime > a(n) such that a(n+1)-1 divides the product (a(1)...a(n))^2.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 43, 47, 53, 59, 61, 67, 71, 79, 101, 107, 127, 131, 139, 149, 151, 157, 173, 181, 191, 197, 199, 211, 223, 229, 263, 269, 277, 283, 311, 317, 331, 347, 349, 367, 373, 383, 397, 419, 421, 431, 461, 463, 491, 509, 523, 547, 557, 571

Offset: 1

N. J. A. Sloane

Named after the British mathematician Denis A. Higgs (1932-2011). - Amiram Eldar, Jun 05 2021

No prime of the form a*b^k + 1 (those in A089200) with a > 0, b > 1 and k > 2 is a Higgs's prime. - Mauro Fiorentini, Aug 08 2023

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
Stanley Burris and Simon Lee, Tarski's high school identities, Amer. Math. Monthly, Vol. 100, No. 3 (1993), pp. 231-236.
Robert G. Wilson v, Note to N. J. A. Sloane with attachment, (Annotated scanned copy of The Am. Math. Mo. Vol. 100 No. 3 pp. 233, Mar. 1993).
Robert G. Wilson v, Letter to N. J. A. Sloane, Oct. 1993

Cf. A057447, A057448, A057459, A089200, A282027.

a007459 n = a007459_list !! (n-1)
a007459_list = f 1 a000040_list where
  f q (p:ps) = if mod q (p - 1) == 0 then p : f (q * p ^ 2) ps else f q ps
-- Reinhard Zumkeller, Apr 14 2013

a:=[2]; P:=1; j:=1;
for n from 2 to 32 do
P:=P*a[n-1]^2;
  for i from j+1 to 250 do
  if (P mod (ithprime(i)-1)) = 0 then
  a:=[op(a),ithprime(i)]; j:=i; break; fi;
od:
od:
a; # N. J. A. Sloane, Feb 12 2017

f[ n_List ] := (a = n; b = Apply[ Times, a^2 ]; d = NextPrime[ a[ [ -1 ] ] ]; While[ ! IntegerQ[ b/(d - 1) ] || d > b, d = NextPrime[ d ] ]; AppendTo[ a, d ]; Return[ a ]); Nest[ f, {2}, 75 ]
nxt[{p_,a_}]:=Module[{np=NextPrime[a]},While[PowerMod[p,2,np-1] != 0,np = NextPrime[np]];{p*np,np}]; NestList[nxt,{2,2},60][[All,2]] (* Harvey P. Dale, Jul 09 2021 *)

step(v)=my(N=vecprod(v)^2);forprime(p=v[#v]+1,,if(N%(p-1)==0,return(concat(v,p))))
first(n)=my(v=[2]);for(i=2,n,v=step(v));v \\ Charles R Greathouse IV, Jun 11 2015

More terms from David W. Wilson

Definition clarified by N. J. A. Sloane, Feb 12 2017

A095074 Primes in whose binary expansion the number of 0-bits is less than or equal to number of 1-bits.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 71, 79, 83, 89, 101, 103, 107, 109, 113, 127, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 271, 283, 307, 311, 313, 317, 331, 347, 349, 359

Offset: 1

Antti Karttunen, Jun 01 2004

			From _Indranil Ghosh_, Feb 03 2017: (Start)
29 is in the sequence because 29_10 = 11101_2. '11101' has one 0 and three 1's.
37 is in the sequence because 37_10 = 100101_2. '100101' has three 1's and 3 0's. (End)

Indranil Ghosh, Table of n, a(n) for n = 1..25000
A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Complement of A095071 in A000040. Differs from A057447 first time at n=18, where a(n)=71, while A057447(18)=67. Cf. A095054.

Select[Prime[Range[50]], DigitCount[#, 2, 0] <= DigitCount[#, 2, 1] &] (* Alonso del Arte, Jan 11 2011 *)

forprime(p=2,359,v=binary(p);s=0;for(k=1,#v,s+=if(v[k]==0,+1,-1));if(s<=0,print1(p,", "))) \\ Washington Bomfim, Jan 13 2011

from sympy import isprime
i=1
j=1
while j<=25000:
    if isprime(i) and bin(i)[2:].count("0")<=bin(i)[2:].count("1"):
        print(str(j)+" "+str(i))
        j+=1
    i+=1 # Indranil Ghosh, Feb 03 2017

A057448 a(n+1) = next prime such that a(n+1)-1 | (a(1)...a(n))^5.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 263, 269, 271, 277, 281

Offset: 1

Robert G. Wilson v, Sep 25 2000

nonn

Cf. A007459 & A057447.

NextPrime[ n_Integer ] := Module[ {k = n + 1}, While[ ! PrimeQ[ k ], k++ ]; Return[ k ] ]; f[ n_List ] := (a = n; b = Apply[ Times, a^5 ]; d = NextPrime[ a[[ -1 ]] ]; While[ ! IntegerQ[ b/(d - 1) ] && d < b+2, d = NextPrime[ d ] ]; AppendTo[ a, d ]; Return[ a ]); Nest[ f, {2}, 75 ]

A122516 Primes in A046992.

Original entry on oeis.org

3, 5, 11, 19, 23, 37, 43, 61, 83, 107, 181, 271, 283, 349, 467, 499, 547, 563, 743, 821, 863, 947, 991, 1013, 1571, 2341, 2437, 2633, 2803, 2837, 2939, 3299, 3373, 3677, 3833, 4073, 4793, 4973, 5387, 5479, 5573, 6043, 6091, 6737, 7907, 8017, 8693, 8867

Offset: 1

Roger L. Bagula, Sep 16 2006

nonn

A subset of A057447. - Alexander Adamchuk, Sep 17 2006

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A046992, A013918, A057447, A122933.

a122516 n = a122516_list !! (n-1)
a122516_list = filter ((== 1) . a010051) a046992_list
-- Reinhard Zumkeller, Feb 25 2012

Flatten[Table[If[PrimeQ[Sum[ PrimePi[n], {n, 1, m}]], Sum[PrimePi[n], {n, 1, m}], {}], {m, 1, 200}]]

a(n) = Prime[ A122933[n] ]. - Alexander Adamchuk, Sep 20 2006

Edited by N. J. A. Sloane, Sep 17 2006
More terms from Alexander Adamchuk, Sep 17 2006
Definition corrected, Sep 30 2006

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A002827 Unitary perfect numbers: numbers k such that usigma(k) - k = k.

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A007459 Higgs's primes: a(n+1) = smallest prime > a(n) such that a(n+1)-1 divides the product (a(1)...a(n))^2.

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A095074 Primes in whose binary expansion the number of 0-bits is less than or equal to number of 1-bits.

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A057448 a(n+1) = next prime such that a(n+1)-1 | (a(1)...a(n))^5.

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A122516 Primes in A046992.

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