A335398 -id:A335398 - cp's OEIS frontend

A335399 Starts of runs of 5 consecutive numbers that have an equal number of unitary and nonunitary divisors (A048109).

146447622, 2259799749, 2559357269, 2647718871, 3660580374, 4262858871, 4708102374, 5188831623, 5341658373, 5494129749, 5728055749, 5876715750, 6127708374, 6455588247, 6608437623, 6612840374, 6617111750, 6689113623, 6722600373, 7456747623, 7923798375, 8272111445

Offset: 1

Zak Seidov and Amiram Eldar, Jun 06 2020

nonn

Do longer runs of consecutive numbers with an equal number of unitary and nonunitary divisors exist for any length of run?

Starts of runs of 6 consecutive numbers that have an equal number of unitary and nonunitary divisors, from Giovanni Resta's bfile, 80566783622, 117243671750, 390773539750, 573122731621, 636972066374. - Zak Seidov, Jun 07 2020

			146447622 is a term since 146447622, 146447623, 146447624, 146447625 and 146447626 each have an equal number of unitary and nonunitary divisors. 146447622 has 32 unitary divisors and 32 nonunitary divisors, 146447623, 146447625 and 146447626 each have 8 and 8, and 146447624 has 16 and 16.

Giovanni Resta, Table of n, a(n) for n = 1..3000

Subsequence of A048109, A335328, A335397 and A335398.

Cf. A000005, A034444, A048105.

q[n_] := DivisorSigma[0, n] == 2^(PrimeNu[n] + 1); v = q /@ Range[5]; seq = {}; Do[v = Append[Drop[v, 1], q[k]]; If[And @@ v, AppendTo[seq, k - 4]], {k, 6, 3*10^8}]; seq

A348100 Starts of runs of 4 consecutive numbers that have an equal number of unitary and nonunitary prime divisors (A348097).

Original entry on oeis.org

56347, 906596, 906597, 1090250, 1141550, 1243275, 4972372, 5684822, 6288488, 6379748, 6486325, 6907974, 7480447, 8587249, 9129248, 11112173, 12133672, 12133673, 13852924, 14185448, 17519948, 19293208, 19293209, 19751750, 20738672, 21560848, 21721796, 21959350

Offset: 1

Amiram Eldar, Sep 30 2021

nonn

			56347 is a term since 56347 = 29^2 * 67, 56347 + 1 = 56348 = 2^2 * 14087, 56347 + 2 = 56349 = 3^3 * 2087 and 56347 + 3 = 56350 = 2 * 5^2 * 7^2 * 23 all have the same number of unitary and nonunitary prime divisors.

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

Subsequence of A348097, A348098 and A348099.

Cf. A335398.

q[n_] := n == 1 || Count[(e = FactorInteger[n][[;; , 2]]), 1] == Length[e]/2; v = q /@ Range[4]; seq = {}; Do[v = Append[Drop[v, 1], q[k]]; If[And @@ v, AppendTo[seq, k - 3]], {k, 5, 1.3*10^6}]; seq

cp's OEIS Frontend

A335399 Starts of runs of 5 consecutive numbers that have an equal number of unitary and nonunitary divisors (A048109).

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