A340871 -id:A340871 - cp's OEIS frontend

A341214 a(n) is the smallest prime p such that p, p - 1, p - 2, ..., p - n + 1 have 2, 4, 6, ..., 2*n divisors respectively.

2, 7, 47, 1019, 55414379

Offset: 1

Jaroslav Krizek, Feb 07 2021

a(n) is the smallest prime p such that tau(p) = tau(p - 1)/2 = tau(p - 2)/3 = ... = tau(p - n + 1)/n = 2, where tau(k) = the number of divisors of k (A000005).

No such prime p exists for n > 5, so a(5) is the final term. - Jon E. Schoenfield, Feb 07 2021

			a(4) = 1019 because 1016, 1017, 1018 and 1019 have 8, 6, 4, and 2 divisors respectively and there is no smaller prime having this property (see A340872).

Jon E. Schoenfield, A proof that a(5) is the final term

Cf. A341213 (similar sequence for natural numbers).

Cf. A000005, A294528, A340871, A340872.

A340872 Primes p such that p, p - 1, p - 2 and p - 3 have 2, 4, 6 and 8 divisors respectively.

Original entry on oeis.org

1019, 226379, 645419, 893099, 967259, 1744859, 1880939, 1996859, 2238779, 2435339, 2999099, 3316619, 3734219, 4092059, 4269899, 4344779, 4511099, 4681739, 4692539, 5857499, 5993579, 6476699, 6744539, 6848939, 6855419, 8160059, 8608619, 8755499, 9254459

Offset: 1

Jaroslav Krizek, Jan 24 2021

nonn

Term 55414379 of this sequence is the smallest prime p such that p, p - 1, p - 2, p - 3 and p - 4 have 2, 4, 6, 8 and 10 divisors respectively.

			tau (1016) = 8, tau(1017) = 6, tau(1018) = 4, tau(1019) = 2.

Cf. A000005 (tau), A340871.

[m: m in [1..10^7] | IsPrime(m) and #Divisors(m - 1) eq 4 and #Divisors(m - 2) eq 6 and #Divisors(m - 3) eq 8]

Select[Range[10^6], DivisorSigma[0, # - {0, 1, 2, 3}] == {2, 4, 6, 8} &] (* Amiram Eldar, Jan 25 2021 *)
Select[Prime[Range[620000]],DivisorSigma[0,#-{0,1,2,3}]=={2,4,6,8}&] (* Harvey P. Dale, Nov 19 2021 *)

cp's OEIS Frontend

A341214 a(n) is the smallest prime p such that p, p - 1, p - 2, ..., p - n + 1 have 2, 4, 6, ..., 2*n divisors respectively.

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