A240971 -id:A240971 - cp's OEIS frontend

A334813 Arithmetic numbers k (A003601) such that sigma(k)/d(k) is also an arithmetic number, where d(k) is the number of divisors of k (A000005) and sigma(k) is their sum (A000203).

1, 5, 6, 11, 13, 14, 15, 20, 29, 37, 38, 39, 41, 43, 44, 45, 49, 53, 54, 56, 57, 59, 60, 61, 65, 68, 73, 78, 83, 85, 86, 87, 89, 95, 96, 97, 101, 102, 107, 109, 110, 111, 113, 114, 116, 118, 123, 125, 129, 131, 134, 135, 137, 139, 142, 143, 145, 147, 150, 153

Offset: 1

Amiram Eldar, May 12 2020

nonn

The number of terms not exceeding 10^k for k = 1, 2, ... is 3, 36, 426, 4744, 50442, 533806, 5585745, 58013810, 599272790, 6162302702, ... Apparently, this sequence has asymptotic density ~0.6.
Includes all the primes p such that (p+1)/2 is an odd prime, i.e., A005383 without the first term 3.
If p is in A240971 then p^2 is a term.

			5 is a term since sigma(5)/d(5) = 6/2 = 3 is an integer, and so is sigma(3)/d(3) = 4/2 = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A003601, A102187.

rat[n_] := DivisorSigma[1, n]/DivisorSigma[0, n]; Select[Range[200], IntegerQ[(r = rat[#])] && IntegerQ[rat[r]] &]

A338299 Primes of the form (p^2+p+1)/3 where p is prime.

Original entry on oeis.org

19, 61, 127, 331, 631, 1801, 3169, 3571, 5419, 8269, 13267, 16651, 19441, 24571, 55897, 59221, 145861, 151201, 176419, 246247, 260191, 292969, 347821, 360187, 368551, 377011, 398581, 698419, 733591, 863497, 915769, 929077, 990151, 1024921, 1155061, 1177507, 1324681, 1372957, 1618471, 1980469

Offset: 1

J. M. Bergot and Robert Israel, Oct 21 2020

nonn

All terms == 1 (mod 6).

			a(3) = 127 is prime and 127 = (19^2+19+1)/3 where 19 is prime.

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

Cf. A240971.

A240971:= select(t -> isprime(t) and isprime((t^2+t+1)/3), [seq(i,i=1..1000,6)]):
map(t -> (t^2+t+1)/3, A240971);

a(n) = (A240971(n)^2 + A240971(n)+1)/3.

A240971(n) = (sqrt(12*a(n)-3)-1)/2.

A338300 Primes p of the form (q^2+q+1)/3 where q is prime and (p^2+p+1)/3 is prime.

Original entry on oeis.org

19, 127, 3169, 24571, 698419, 863497, 3348577, 5684257, 6156169, 7174987, 7646437, 10790137, 16293691, 18637669, 19271071, 28210267, 30384919, 33156901, 36760501, 45782227, 47533141, 58887991, 62503981, 88210519, 92224441, 100450747, 113559769, 129356767, 138577237, 156233617, 159017041

Offset: 1

J. M. Bergot and Robert Israel, Oct 21 2020

nonn

			a(3) = 3169 is a term because 3169 = (97^2+97+1)/3 and (3169^2+3169+1)/3 = 3348577, and 97, 3169 and 3348577 are all prime.

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

Intersection of A240971 and A338299.

A:= select(t -> isprime(t) and isprime((t^2+t+1)/3), [seq(i,i=1..30000,6)]):
B:= map(t -> (t^2+t+1)/3, A):
select(t -> isprime((t^2+t+1)/3), B);

cp's OEIS Frontend

A334813 Arithmetic numbers k (A003601) such that sigma(k)/d(k) is also an arithmetic number, where d(k) is the number of divisors of k (A000005) and sigma(k) is their sum (A000203).

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A338299 Primes of the form (p^2+p+1)/3 where p is prime.

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A338300 Primes p of the form (q^2+q+1)/3 where q is prime and (p^2+p+1)/3 is prime.

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Author

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Maple