A381742
Numbers k such that k^2 is abundant but d*k is nonabundant for any proper divisor d of k.
Original entry on oeis.org
14, 124, 585, 1016, 16748, 32085, 33892, 37882, 39962, 41925, 46665, 121605, 134589, 181305, 212175, 388455, 495465, 522488, 524224, 544065, 839865, 1061565, 1152921, 1165515, 1243275, 1247103, 1335411, 1676829, 1943638, 2151075, 2290869, 2478075, 2625514, 2673998
Offset: 1
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q[k_] := DivisorSigma[-1, k^2] > 2 && AllTrue[Divisors[k], DivisorSigma[-1, #*k] <= 2 || # == k &]; Select[Range[10^6], q]
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isok(k) = fordiv(k, d, if(d < k && sigma(d*k, -1) > 2, return(0))); sigma(k^2, -1) > 2;
A190213
Integers m such that m divides (2^m-2)^2 and (m-2)^((k-1)*(1+k*(m-1))) == 1 (mod k), where k = 2^m - 1.
Original entry on oeis.org
1, 3, 4, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839
Offset: 1
For n=3, k=2^3-1=7, m=1+6*2=13, x=m*k=13*7=91, 2^(x-1)==(a+1) (mod x) with 2^90 == (63+1)(mod 91), fixes a=63. m^(x-1) == (b+1) (mod x) with 13^90 == (77+1) (mod 91) fixes b=77. The two conditions are satisfied: 63 == 0 (mod 7) and 77 == 0 (mod 7). Therefore n=3 is in the sequence.
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isA190213 := proc(n) local k,m,x,a,b ; k := 2^n-1 ; m := (k-1)*(n-1)+1 ; x := k*m ; a := modp( 2 &^ (x-1),x) -1 ; b := modp( m &^ (x-1),x) -1 ; return ( modp(a,k) = 0 and modp(b,k)=0 ) ; end proc:
for n from 2 do if isA190213(n) then printf("%d,\n",n); end if; end do; # avoids n=1 and undefined 0^0, R. J. Mathar, Jun 11 2011
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okQ[n_] := Module[{k, m, x, a, b}, k = 2^n - 1; m = 1 + (k - 1)(n - 1); x = m k; a = PowerMod[2, x - 1, x] - 1; b = PowerMod[m, x - 1, x] - 1; Mod[a, k] == 0 && Mod[b, k] == 0];
Reap[For[n = 1, n < 10^4, n++, If[okQ[n], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Oct 30 2019 *)
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is(n)=my(k=2^n-1,m=(k-1)*(n-1)+1,e=m*k-1); Mod(2,k)^e==1 && Mod(m,k)^e==1 \\ Charles R Greathouse IV, Sep 16 2022
A237499
Odious Mersenne exponents.
Original entry on oeis.org
2, 7, 13, 19, 31, 61, 107, 127, 521, 607, 1279, 3217, 11213, 21701, 44497, 132049, 216091, 756839, 1257787, 3021377, 6972593, 20996011, 24036583, 30402457, 37156667, 43112609, 82589933
Offset: 1
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Select[MersennePrimeExponent[Range[47]], OddQ @ DigitCount[#, 2][[1]] &] (* Amiram Eldar, Dec 10 2019 *)
a(13) and a(18) inserted and more terms added by
Amiram Eldar, Dec 10 2019
Showing 1-3 of 3 results.
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