A046655
Numbers whose sum of the squares of divisors is also a square number.
Original entry on oeis.org
1, 42, 246, 287, 728, 1434, 1673, 1880, 4264, 6237, 9799, 9855, 18330, 21352, 21385, 24856, 36531, 39990, 46655, 57270, 66815, 92664, 125255, 156570, 182665, 208182, 212949, 242879, 273265, 380511, 391345, 411558, 539560, 627215, 693160, 730145, 741096
Offset: 1
a(3) = 246 = 2*3*41 with 8 divisors: 1, 2, 3, 6, 41, 82, 123, 246. The sum of squares of the divisors is 84100 = 290^2, also a square.
- Jean-Marie De Koninck, Ces nombres qui nous fascinent, Entry 42, p. 16, Ellipses, Paris, 2008.
-
Select[Range[750000], IntegerQ[Sqrt[DivisorSigma[2, #]]] &] (* Jayanta Basu, Jun 27 2013 *)
-
for( n=1,10^6, issquare(sigma(n,2)) && print1(n", ")) \\ M. F. Hasler, Oct 05 2008
A064028
Sum of the unitary divisors of n!.
Original entry on oeis.org
1, 3, 12, 36, 216, 1020, 8160, 61920, 507744, 4383392, 52600704, 624249600, 8739494400, 109190390400, 1583122968000, 25318378008000, 455730804144000, 8193040840252800, 163860816805056000, 3256371347261760000, 67204676251838361600, 1366492477414792734720
Offset: 1
n=6, 6! = 720, sum of the 8 unitary ones of its 30 divisors is 1020, a(6) = 720+1+16+45+9+80+5+144 = 1020.
- Amiram Eldar, Table of n, a(n) for n = 1..450
- Charles R. Wall, Problem H-374, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 22, No. 3 (1984), p. 280; Bounds of Joy, Solution to Problem H-374 by the proposer, ibid., Vol. 24, No. 2 (1986), p. 188.
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usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); usigma/@ (Range[17]!) (* Amiram Eldar, Jun 23 2019 *)
-
valp(n,p)=my(s); while(n\=p, s+=n); s
a(n)=my(s=1); forprime(p=2,n, s*=p^valp(n,p)+1); s \\ Charles R Greathouse IV, Jan 26 2023
A064138
Sum of non-unitary divisors of n!.
Original entry on oeis.org
0, 0, 0, 24, 144, 1398, 11184, 97200, 973296, 10950696, 131408352, 1593191808, 22304685312, 333297226080, 5103130001760, 81686161277280, 1470350902991040, 26490792085668288, 529815841713365760, 10635027891469974720
Offset: 1
For n = 6, 6! = 720, the sum of its 30 divisors is 2418, the sum of the 8 unitary divisors is 1020, so the remaining 22 divisors give a(6) = 1398.
Cf.
A034448,
A048105,
A046656,
A056657,
A056171,
A056172,
A000203,
A000142,
A062569,
A063955,
A063960.
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f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := p^e + 1; a[n_] := Times @@ f1 @@@ (fct = FactorInteger[n!]) - Times @@ f2 @@@ fct; a[1] = 0; Array[a, 20] (* Amiram Eldar, Apr 01 2024 *)
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usigma(n)= { local(f,s=1); f=factor(n); for(i=1, matsize(f)[1], s*=1 + f[i, 1]^f[i, 2]); return(s) }
{ n=0; f=1; for (n=1, 100, f*=n; write("b064138.txt", n, " ", sigma(f) - usigma(f)); ) } \\ Harry J. Smith, Sep 08 2009
Showing 1-3 of 3 results.