A228870 -id:A228870 - cp's OEIS frontend

A226872 1 together with even numbers n >= 2 such that 1^n + 2^n + 3^n + ... + n^n == n/2 (mod n).

1, 2, 4, 8, 10, 14, 16, 22, 26, 28, 32, 34, 38, 44, 46, 50, 52, 56, 58, 62, 64, 68, 70, 74, 76, 82, 86, 88, 92, 94, 98, 104, 106, 112, 116, 118, 122, 124, 128, 130, 134, 136, 142, 146, 148, 152, 154, 158, 164, 166, 170, 172, 176, 178, 182, 184, 188, 190, 194, 196

Offset: 1

José María Grau Ribas, Jun 20 2013

nonn

For n>1, a(n) is even. Alternatively, the even terms of this sequence can be characterized in any of the following ways: (i) even integers n such that n*B(n) == n/2 (mod n), where B(n) is the n-th Bernulli number; OR (ii) integers n such that gcd(n,A027642(n)) = 2; OR (iii) even integers n such that (p-1) does not divide n for every odd prime p dividing n (cf. A124240). - Max Alekseyev, Sep 05 2013

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A014117, A031971, A228869, A228870.

Join[{1}, Select[Range[200], Mod[Sum[PowerMod[k, #, #], {k, #}], #] == #/2 &]] (* T. D. Noe, Sep 04 2013 *)

is(n)=if(n%2,return(n==1));my(f=factor(n)[,1]);for(i=2,#f,if(n%(f[i]-1)==0,return(0)));1 \\ Charles R Greathouse IV, Sep 04 2013

A228869 Numbers n such that 2 * (1^n + 2^n + 3^n + ... + n^n) == 0 (mod n).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 81, 82, 83

Offset: 1

T. D. Noe, Sep 06 2013

nonn

See A228870 for the numbers not in this sequence.
Union of A226872 and the positive odd integers (A005408).
Also, positive integers n such that (p-1) does not divide n for every odd prime p dividing n (cf. A124240). - Max Alekseyev, Sep 07 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A228870, A226872.

Select[Range[100], Mod[2*Sum[PowerMod[k, #, #], {k, #}], #] == 0 &]

A280187 Numbers n such that 2 * (1^n + 2^n + 3^n + ... + n^n) is not 0 (mod n), but 2 * (1^d + 2^d + 3^d + ... + d^d) is 0 (mod d) for each other d | n.

Original entry on oeis.org

6, 20, 110, 272, 506, 812, 2162, 2756, 3422, 4970, 6806, 7832, 11342, 12656, 17030, 18632, 22052, 27722, 29756, 31862, 36290, 38612, 51302, 54056, 56882, 62750, 65792, 68906, 72092, 85556, 96410, 100172, 120062, 124256, 128522

Offset: 1

Charles R Greathouse IV, Dec 28 2016

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..4000

Primitive elements of A228870.

Subsequence of A002943. Also a subsequence of A028689, A036689, A053198, A068377, A079143, A128672, A220211 and other sequences ...- Paolo P. Lava, Jan 10 2017

has(n)=my(f=factor(n)[,1]); for(i=1,#f, if(n%(f[i]-1)==0 && f[i]>2, return(1))); 0
is(n)=if(n%2, return(0)); if(n%3==0, return(n==6)); if(n%20==0, return(n==20)); if(!has(n), return(0)); my(f=factor(n)[,1]); for(i=1,#f, if(has(n/f[i]), return(0))); 1 \\ Charles R Greathouse IV, Dec 28 2016

cp's OEIS Frontend

A226872 1 together with even numbers n >= 2 such that 1^n + 2^n + 3^n + ... + n^n == n/2 (mod n).

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A228869 Numbers n such that 2 * (1^n + 2^n + 3^n + ... + n^n) == 0 (mod n).

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A280187 Numbers n such that 2 * (1^n + 2^n + 3^n + ... + n^n) is not 0 (mod n), but 2 * (1^d + 2^d + 3^d + ... + d^d) is 0 (mod d) for each other d | n.

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