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A254748
Numbers without superdivisors: numbers n such that n/k + n fails to divide at least one of (n/k)^(n/k) + n, (n/k)^n + n/k or n^(n/k) + n/k for any divisor k of n.
4
2, 4, 6, 8, 12, 14, 16, 18, 20, 24, 26, 28, 30, 32, 38, 40, 42, 44, 48, 50, 52, 54, 56, 60, 62, 64, 66, 68, 72, 74, 80, 84, 86, 88, 90, 92, 96, 98, 100, 102, 104, 108, 110, 112, 114, 120, 122, 124, 126, 128, 132, 134, 138, 140, 144, 146, 148, 150, 152, 158, 160, 164, 168, 170, 172, 174
OFFSET
1,1
COMMENTS
Zerosuperdivisor numbers. Numbers n such that A247477(n) = 0.
A000027 = zerosuperdivisor numbers U onesuperdivisor numbers U twosuperdivisor numbers U threesuperdivisor numbers U ...
Conjecture: Perfect numbers (A000396) are zerosuperdivisor numbers.
Conjecture: Average of twin prime pairs (A014574) are zerosuperdivisor numbers.
None of these numbers are odd or 10 mod 12 or 36 mod 40 or 78 mod 84 or 136 mod 144 or ... - Charles R Greathouse IV, Feb 19 2015
LINKS
Michael De Vlieger and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 3592 terms from Michael De Vlieger)
EXAMPLE
2 is in this sequence because 2/1 + 2 does not divide (2/1)^(2/1) + 2, (2/1)^2 + 2/1, 2^(2/1) + 2/1 and 2/2 + 2 does not divide (2/2)^(2/2) + 2, (2/2)^2 + 2/2, 2^(2/2) + 2/2: 4 does not divide 6, 6, 6 and 3 does not divide 3, 2, 3.
MATHEMATICA
superdivisors[n_] := Select[Range@ n, And[Mod[(n/#)^(n/#) + n, n/# + n] == 0, Mod[(n/#)^n + n/#, n/# + n] == 0, Mod[n^(n/#) + n/#, n/# + n] == 0] &] /. {} -> 0; Position[Array[superdivisors, 174], 0] // Flatten (* Michael De Vlieger, Feb 09 2015 *)
PROG
(PARI) is(n)=fordiv(n, d, my(m=n/d, k=d+n); if(Mod(d, k)^d==-n && Mod(d, k)^n==-d && Mod(n, k)^d==-d, return(0))); 1 \\ Charles R Greathouse IV, Feb 19 2015
CROSSREFS
Cf. A000027, A000396, A014574, A247477 (definition of superdivisor).
Sequence in context: A164530 A058817 A328593 * A217562 A088879 A316470
KEYWORD
nonn,easy
AUTHOR
STATUS
approved