A323251 -id:A323251 - cp's OEIS frontend

A323249 Sequence lists numbers k > 1 such that k^2 == d(k) (mod sigma(k)), where d = A000005 and sigma = A000203.

8, 9, 14, 26, 38, 62, 74, 86, 122, 134, 146, 158, 194, 206, 218, 254, 278, 302, 314, 326, 362, 386, 398, 422, 446, 458, 482, 542, 554, 566, 614, 626, 662, 674, 698, 734, 746, 758, 794, 818, 842, 866, 878, 914, 926, 974, 998, 1046, 1082, 1094, 1142, 1154, 1202, 1214

Offset: 1

Paolo P. Lava, Jan 08 2019

A112772 is a subsequence. After 8 and 9, the following numbers not in A112772 are 5438, 5462, 5504, 11096, ...

			sigma(8) = 15 and 8^2 mod 15 = 4 = d(8).

Jinyuan Wang, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A112772, A323250, A323251.

with(numtheory): op(select(n->n^2 mod sigma(n)=tau(n),[$1..1214]));

Select[Range[1225], PowerMod[#1, 2, #3] == #2 & @@ Prepend[DivisorSigma[{0, 1}, #], #] &] (* Michael De Vlieger, Jan 18 2019 *)

for(k=1, 2000, x=sigma(k); if(Mod(k,x)^2==Mod(numdiv(k), x), print1(k, ", "))) \\ Jinyuan Wang, Feb 03 2019

Solutions of k^2 mod sigma(k) = d(k).

A323250 Sequence lists numbers k > 1 such that k^3 == d(k) (mod sigma(k)), where d = A000005 and sigma = A000203.

Original entry on oeis.org

2, 110, 152, 506, 830, 882, 8138, 13826, 15878, 19514, 37634, 93242, 99002, 153216, 218978, 576902, 998978, 2259758, 3041798, 5326106, 6654278, 7709006, 7772762, 31833002, 44564438, 106657202, 279422306, 1702668664, 1774448104, 2132364366, 3932536504, 3966201002, 4954728904

Offset: 1

Paolo P. Lava, Jan 08 2019

nonn

From Jinyuan Wang, Feb 03 2019: (Start)
Conjecture: All terms are even.
a(34) > 5*10^9. (End)

			sigma(2) = 3 and 2^3 mod 3 = 2 = d(2).

Cf. A000005, A000203, A323249, A323251.

with(numtheory): op(select(n->n^3 mod sigma(n)=tau(n),[$1..7772762]));

Select[Range[10^8], PowerMod[#1, 3, #3] == #2 & @@ Prepend[DivisorSigma[{0, 1}, #], #] &] (* Michael De Vlieger, Jan 18 2019 *)

for(k=1,10^8,x=sigma(k);if(Mod(k,x)^3==Mod(numdiv(k),x),print1(k,", "))) \\ Jinyuan Wang, Feb 02 2019

Solutions of k^3 mod sigma(k) = d(k).

a(24)-a(25) from Michael De Vlieger, Jan 18 2019

a(26)-a(33) from Jinyuan Wang, Feb 02 2019

cp's OEIS Frontend

A323249 Sequence lists numbers k > 1 such that k^2 == d(k) (mod sigma(k)), where d = A000005 and sigma = A000203.

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