A378792 -id:A378792 - cp's OEIS frontend

A378789 Numbers k such that tau(k) == 1 (mod(2*(tau(k+prime(k)-1)))), where tau(k) = A000005(k).

1, 81, 225, 256, 441, 625, 1521, 2601, 3025, 3249, 3364, 4761, 5929, 6561, 7225, 8281, 9025, 9216, 12321, 13225, 14161, 17689, 18225, 20449, 21025, 25921, 27556, 31684, 34225, 41209, 43681, 45369, 45796, 46225, 47089, 47961, 48841, 50176, 50625, 57600, 61009, 62001, 67081

Offset: 1

Claude H. R. Dequatre, Dec 07 2024

98 terms < 5*10^5 were found.

All terms are squares because their number of divisors is odd (see formula field in A000005: a(n) is odd iff n is square).

			1 is a term because tau(1) = 1, tau(1 + 2 - 1) = 2 and 1 modulo 4 is 1.
81 is a term because tau(81) = 5, tau(81 + 419 - 1) = 2 and 5 modulo 4 is 1.
85 is not a term because tau(85) = 4, tau(85 + 439 - 1) = 2 and 4 modulo 4 = 0.

Cf. A000005, A000040.

Cf. A378792, A378794.

isok(k)=my(d_1=numdiv(k),d_2=numdiv(k+prime(k)-1));d_1%(2*d_2)==1;
for(k=1,1000,if(isok(k),print1(k", ")))

A378794 Numbers k such that tau(k) == 1 (mod(2*(tau(k^2 + k - 1)))), where tau(k) = A000005(k).

Original entry on oeis.org

1, 16, 36, 100, 196, 225, 256, 441, 484, 625, 676, 1089, 1156, 1225, 1296, 2116, 2601, 3136, 3249, 3364, 3844, 4225, 5929, 6561, 7056, 7225, 7569, 8100, 8281, 8836, 9216, 10000, 11236, 12321, 13924, 14161, 14884, 15129, 16641, 17689, 17956, 19881, 20164, 20449

Offset: 1

Claude H. R. Dequatre, Dec 07 2024

736 terms < 10^7 were found.

All terms are squares because their number of divisors is odd (see formula field in A000005: a(n) is odd iff n is square).

			1 is a term because tau(1) = 1, tau(1 + 1 - 1) = 1 and 1 modulo 2 is 1.
16 is a term because tau(16) = 5, tau(256 + 16 - 1) = 2 and 5 modulo 4 is 1.
50 is not a term because tau(50) = 6, tau(2500 + 50 -1) = 2 and 6 modulo 4 is 2.

Cf. A000005, A000040.

Cf. A378789, A378792.

isok(k)=my(d_1=numdiv(k),d_2=numdiv(k^2+k-1));d_1%(2*d_2)==1;
for(k=1,1000,if(isok(k),print1(k", ")))

cp's OEIS Frontend

A378789 Numbers k such that tau(k) == 1 (mod(2*(tau(k+prime(k)-1)))), where tau(k) = A000005(k).

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