A061285 -id:A061285 - cp's OEIS frontend

A283262 Numbers m such that tau(m^2) is a prime.

2, 3, 4, 5, 7, 8, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227

Offset: 1

Jaroslav Krizek, Mar 08 2017

nonn

tau(m) is the number of positive divisors of m (A000005).
Numbers m such that A000005(A000290(m)) = A048691(m) is a prime.
Union of A000040 (primes) and A051676.
Supersequence of A055638 (sigma(m^2) is prime).
Subsequence of A000961 (powers of primes).
Prime powers p^e with 2e+1 prime (e >= 1).
See A061285(m) = the smallest number k such that tau(k^2) = m-th prime.

			tau(4^2) = tau(16) = 5 (prime).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A000290, A000961, A048691, A051676, A055638, A061285.

[n: n in [2..100000] | IsPrime(NumberOfDivisors(n^2))];

N:= 1000: # to get all terms <= N
es:= select(t -> isprime(2*t+1), [$1..ilog2(N)]):
Ps:= select(isprime, [2,seq(i,i=3..N,2)]):
sort(select(`<=`, [seq(seq(p^e,e=es),p=Ps)],N)): # Robert Israel, Mar 16 2017

Select[Range@ 227, PrimeQ[DivisorSigma[0, #^2]] &] (* Michael De Vlieger, Mar 09 2017 *)

isok(n)=isprime(numdiv(n^2)) \\ Indranil Ghosh, Mar 09 2017

A104137 Number of distinct necklaces with p beads of two possible colors, allowing turning over, p being a prime greater than 2.

Original entry on oeis.org

4, 8, 18, 126, 380, 4112, 14310, 184410, 9272780, 34669602, 1857545300, 26818405352, 102282248574, 1497215711538, 84973644983780, 4885261149611790, 18900353608280300, 1101298162244236182, 16628051030379615882

Offset: 1

Lekraj Beedassy, Mar 07 2005

nonn

For the general necklace problem, see A000029.

Martin Gardner, The Colossal Book of Mathematics, pp. 19, W. W. Norton & Co., NY 2001 (or, New Mathematical Diversions, pp. 243-4 MAA Washington DC 1995).

Cf. A000029.

for p from 2 to 30 do printf(`%d,`,(2^(ithprime(p)-1)-1)/ithprime(p) + 2^((ithprime(p)-1)/2) + 1) od: # James Sellers, Apr 10 2005

a(n) = (2^(p-1) - 1)/p + 2^{(p-1)/2} + 1 = A007663(n) + A061285(n) + 1.

More terms from James Sellers, Apr 10 2005

A133155 Numbers formed by setting bits representing odd primes, where bit_no = (prime - 1)/2. Setting bit number b is the same as OR-ing with 2^b (i.e., bit numbers start at zero).

Original entry on oeis.org

2, 6, 14, 46, 110, 366, 878, 2926, 19310, 52078, 314222, 1362798, 3459950, 11848558, 78957422, 615828334, 1689570158, 10279504750, 44639243118, 113358719854, 663114533742, 2862137789294, 20454323833710, 301929300544366, 1427829207386990, 3679629021072238

Offset: 1

Alan Griffiths, Oct 08 2007

nonn

			a(3) = 14 because 3, 5 and 7 are odd primes so therefore bits 1, 2 and 3 are set and bit zero is not. 1110_2 = 14.

Partial sums of A061285.

import gmpy
a = gmpy.mpz(0)
i = 0
for p in range(3,100,2):
    if gmpy.is_prime(p):
        a = gmpy.setbit(a,(p-1)/2)
        i += 1
        print(i,a)

a(n) = setbit(a(n-1),(p-1)/2) where p is the n-th odd prime.

cp's OEIS Frontend

A283262 Numbers m such that tau(m^2) is a prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

A104137 Number of distinct necklaces with p beads of two possible colors, allowing turning over, p being a prime greater than 2.

Views

Author

Keywords

Comments

References

Crossrefs

Programs

Maple

Formula

Extensions

A133155 Numbers formed by setting bits representing odd primes, where bit_no = (prime - 1)/2. Setting bit number b is the same as OR-ing with 2^b (i.e., bit numbers start at zero).

Views

Author

Keywords

Examples

Crossrefs

Programs

Python

Formula