A318262
Numbers m such that 2^phi(m) mod m is a prime power (in the sense of A246655).
Original entry on oeis.org
6, 12, 14, 20, 24, 28, 30, 40, 48, 56, 60, 62, 72, 80, 84, 96, 112, 120, 124, 126, 144, 168, 192, 224, 240, 248, 252, 254, 272, 288, 320, 336, 340, 384, 408, 448, 480, 496, 504, 508, 510, 544, 576, 584, 640, 672, 680, 768, 816, 896, 960, 992, 1008, 1016, 1020
Offset: 1
The odd part of the first few terms can be arranged as follows:
3,
3, 7, 5,
3, 7, 15, 5,
3, 7, 15, 31, 9, 5, 21,
3, 7, 15, 31, 63, 9, 21,
3, 7, 15, 31, 63, 127, 17, 9, 5, 21, 85,
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Select[Range[2^10], And[PrimePowerQ@ #, ! PrimeQ@ #] &@ Mod[2^EulerPhi@ #, #] &] (* Michael De Vlieger, Sep 04 2018 *)
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isok(n) = isprimepower(lift(Mod(2, n)^eulerphi(n))); \\ Michel Marcus, Sep 06 2018
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def isA318262(n):
m = power_mod(2, euler_phi(n), n)
return m.is_prime_power()
def A318262_list(search_bound):
return [n for n in range(2,search_bound+1,2) if isA318262(n)]
print(A318262_list(1020))
A364859
Lesser of a pair of S-amicable numbers k < m such that s(k) = m and s(m) = k, where s(k) = A364858(k).
Original entry on oeis.org
186, 1184, 2030, 6232, 10744, 66928, 522405, 643336, 5459176, 7677248, 13223490, 16137628, 25596544, 26090325, 28118032, 31772524, 34364912, 40504324, 133178325
Offset: 1
186 is a term since A364858(186) = 198 > 186, and A364858(198) = 186.
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seq[nmax_] := Module[{s = {1}, sum, sum2, am = {}, ak}, Do[sum = Total[Select[Divisors[n], MemberQ[s, #] &]]; If[sum <= n, AppendTo[s, n]; If[sum < n, sum2 = Total[Select[Most[Divisors[sum]], MemberQ[s, #] &]]; If[sum2 == n, AppendTo[am, sum]]]], {n, 2, nmax}]; am]; seq[10^4]
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lista(nmax) = {my(c = 0, s, s2); for(n=2, nmax, s = sumdiv(n, d, !bittest(c, d)*d) - n; if(s > n, c+=1<M. F. Hasler at A181487
A364860
Greater of a pair of S-amicable numbers k < m such that s(k) = m and s(m) = k, where s(k) = A364858(k).
Original entry on oeis.org
198, 1210, 2220, 6368, 10856, 66992, 525915, 652664, 5495264, 7684672, 13727466, 16150628, 25640096, 26138475, 28128368, 33642028, 34380688, 40803868, 133471275
Offset: 1
198 is a term since A364858(198) = 186 < 198, and A364858(186) = 198.
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seq[nmax_] := Module[{s = {1}, sum, sum2, am = {}, ak}, Do[sum = Total[Select[Divisors[n], MemberQ[s, #] &]]; If[sum <= n, AppendTo[s, n]; If[sum < n, sum2 = Total[Select[Most[Divisors[sum]], MemberQ[s, #] &]]; If[sum2 == n, AppendTo[am, n]]]], {n, 2, nmax}]; am]; seq[10^4]
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lista(nmax) = {my(c = 0, s, s2); for(n=2, nmax, s = sumdiv(n, d, !bittest(c, d)*d) - n; if(s > n, c+=1<M. F. Hasler at A181487
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