A140480 -id:A140480 - cp's OEIS frontend

A160550 a(n) = A001065(n) mod A000005(n).

0, 1, 1, 0, 1, 2, 1, 3, 1, 0, 1, 4, 1, 2, 1, 0, 1, 3, 1, 4, 3, 2, 1, 4, 0, 0, 1, 4, 1, 2, 1, 1, 3, 0, 1, 1, 1, 2, 1, 2, 1, 6, 1, 4, 3, 2, 1, 6, 2, 1, 1, 4, 1, 2, 1, 0, 3, 0, 1, 0, 1, 2, 5, 0, 3, 6, 1, 4, 3, 2, 1, 3, 1, 0, 1, 4, 3, 2, 1, 6, 0, 0, 1, 8, 3, 2, 1, 4, 1, 0, 1, 4, 3, 2, 1, 0, 1, 1, 3, 0, 1, 2, 1, 2, 7

Offset: 1

Ctibor O. Zizka, May 19 2009

A054022(n) for n > 1 gives the numbers k such that a(k) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000203, A000027, A000005, A054022, A001065, A003601, A140480.

[ (SumOfDivisors(n)-n) mod NumberOfDivisors(n): n in [1..105] ]; // Klaus Brockhaus, May 21 2009

Table[Mod[DivisorSigma[1,n]-n,DivisorSigma[0,n]],{n,110}] (* Harvey P. Dale, Jun 22 2013 *)

A160550(n)=lift(Mod(sigma(n)-n,numdiv(n))); \\ Michael B. Porter, Oct 13 2009

Extended and comment edited by Klaus Brockhaus, May 21 2009

A145191 Numbers m such that Sum_{i=1..m} omega(i)^2 is divisible by m, where omega is A001221.

Original entry on oeis.org

1, 20, 68, 903, 3876, 3890, 19096, 19122, 19127, 110990, 111004, 111007, 111010, 111013, 774276, 774277, 774278, 774279, 774303, 774313, 774314, 774315, 6615593, 70607550, 70607559, 959878582, 959878737, 959878753, 959878836, 959878846, 959878888, 959878902, 959878914

Offset: 1

Ctibor O. Zizka, Oct 03 2008

nonn

If for some m is c square, then we have RootMeanSquare(omega(1),...,omega(n)) = c.

Cf. A001221, A063971, A013939, A140480.

With[{max = 10^5}, Position[Accumulate[PrimeNu[Range[max]]^2]/Range[max], ?IntegerQ] // Flatten] (* _Amiram Eldar, Sep 22 2024 *)

isok(m) = !frac(sum(i=1, m, omega(i)^2)/m); \\ Michel Marcus, Mar 15 2022

lista(nn) = {my(v = vector(nn, k, omega(k)^2)); print1(1, ", "); for (n=2, nn, v[n] += v[n-1]; if (! frac(v[n]/n), print1(n, ", ")););} \\ Michel Marcus, Mar 16 2022

listaa(nn) = {my(v = vector(nn, k, omega(k)^2)); print1(1, ", "); for (n=2, nn, v[n] += v[n-1]; if (! frac(v[n]/n), print1(n, ", "));); for (m=1, 100, last = v[nn]; v = vector(nn, k, omega(k+m*nn)^2); v[1] += last; for (n=2, nn, v[n] += v[n-1]; if (! frac(v[n]/(m*nn+n)), print1(n+m*nn, ", "));););} \\ Michel Marcus, Mar 16 2022

a(7)-a(9) from Michel Marcus, Mar 15 2022
a(10)-a(25) from Michel Marcus, Mar 16 2022
a(26)-a(33) from Amiram Eldar, Sep 22 2024

A164986 Numbers of the form 2p^2 = q^2 + 1, where p and q are primes.

Original entry on oeis.org

50, 1682, 3971273138702695316402, 367680737852094722224630791187352516632102802

Offset: 1

Rick L. Shepherd, Sep 03 2009

nonn

A079704 INTERSECT A002522. Subsequence of A088920 (Solutions k to the Diophantine equation k = 2n^2 = m^2+1): those terms for which associated m in A002315 and n in A001653 are both prime.
Corresponding p are prime Pell numbers (prime denominators of continued fraction convergents to sqrt(2)).
Corresponding q are prime numerators of the continued fraction convergents to sqrt(2).
Corresponding p, q, p^2, q^2, (p,q), (q,p), etc., form subsequences of many other OEIS sequences; see cross-references.
Any further terms are too large to include here.

			a(1) = 50 as 50 = 2*5^2 = 7^2 + 1, where 5 and 7 are prime.

Cf. A088920, A118612, A086397, A086395, A002315 (NSW numbers), A088165 (prime NSW numbers = prime RMS numbers (A140480)), A001653, A000129 (Pell numbers), A086383, A101411, A079704, A002522, A008843, A104683, A163742, etc.

a(n) = 2*(A118612(n+1))^2 = (A086397(n+1))^2 + 1.

cp's OEIS Frontend

A160550 a(n) = A001065(n) mod A000005(n).

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A145191 Numbers m such that Sum_{i=1..m} omega(i)^2 is divisible by m, where omega is A001221.

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A164986 Numbers of the form 2p^2 = q^2 + 1, where p and q are primes.

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