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A363657 Numbers m where A217854(m) is a record minimum.

1, 4, 9, 16, 36, 100, 144, 324, 400, 576, 900, 1764, 2304, 3600, 7056, 8100, 14400, 28224, 32400, 44100, 57600, 108900, 112896, 129600, 176400, 396900, 435600, 518400, 608400, 705600, 1587600, 2822400, 5336100, 6350400, 14288400, 15681600, 17640000, 21344400

Offset: 1

Simon Jensen, Jun 13 2023

nonn

(-m)^tau(m) < 0 and (-m)^tau(m) < (-k)^tau(k) for all positive k < m, where tau is the number of divisors function.
All terms are squares.
It is conjectured that if m is a term, then abs((-m)^tau(m)) <= abs((-k)^tau(k)) for some k < m. See the link.

			9 is a term since (-9)^tau(9) = (-9)^3 = -729 and -729 < (-k)^tau(k) for k = 1,...,8.
25 is not a term since (-25)^tau(5) = (-25)^3 = -15625 > (-16)^tau(16) = (-16)^5 = -1048576 and 16 < 25.

Simon K. Jensen, On an extended divisor product summatory function

Cf. A363658, A000290, A067128, A000005, A217854, A224914.

isok(m) = my(x=(-m)^numdiv(m)); for (k=1, m-1, if (x >= (-k)^numdiv(k), return(0))); return(1); \\ Michel Marcus, Jun 18 2023

A363658 Positive numbers m where A217854(m) is positive and increases to a record.

Original entry on oeis.org

2, 3, 5, 6, 8, 10, 12, 18, 20, 24, 30, 40, 42, 48, 60, 72, 84, 90, 96, 108, 120, 168, 180, 240, 336, 360, 420, 480, 504, 540, 600, 630, 660, 672, 720, 840, 1080, 1260, 1440, 1680, 2160, 2520, 3360, 3780, 3960, 4200, 4320, 4620, 4680, 5040, 7560, 9240, 10080

Offset: 1

Simon Jensen, Jun 13 2023

nonn

(-m)^tau(m) > 0 and (-m)^tau(m) > (-k)^tau(k) for all positive k < m, where tau is the number of divisors function.
There are no squares in this sequence.
It appears that if n > 13, then a(n) = A067128(n). See the link.
Only a finite number of terms in A002093 can also be terms in this sequence. See the link.

			5 is a term since (-5)^tau(5) = (-5)^2 = 25 and 25 > (-k)^tau(k) for k = 1,...,4.

Simon Jensen, Table of n, a(n) for n = 1..135
Simon Jensen, On an extended divisor product summatory function

Cf. A363657, A067128, A000005, A002093, A217854, A224914.

isok(m) = my(x=(-m)^numdiv(m)); if (x>0, for (k=1, m-1, if (x <= (-k)^numdiv(k), return(0))); return(1)); \\ Michel Marcus, Aug 31 2023

A224914 Partial sums of A217854.

Original entry on oeis.org

-1, 3, 12, -52, -27, 1269, 1318, 5414, 4685, 14685, 14806, 3000790, 3000959, 3039375, 3090000, 2041424, 2041713, 36053937, 36054298, 100054298, 100248779, 100483035, 100483564, 110175797740, 110175782115, 110176239091, 110176770532, 110658660836

Offset: 1

Simon Jensen, Apr 19 2013

sign

If there is some n > 47 such that a(n) < 0, then there is some k^2 > 47 such that a(k^2) < 0.
If n > 1 is a square number, then a(n) = a(n-1) - n^tau(n).
If n > 1 is a nonsquare number, then a(n) = a(n-1) + n^tau(n).
If n > 1 is a prime, then a(n) = a(n-1) + n^2.

			a(4) = a(1) + a(2) + a(3) + (-4)^tau(4) = (-1) + 3 + 12 + (-64) = -52.

Simon Jensen, Table of n, a(n) for n = 1..10000
Simon Jensen, On an extended divisor product summatory function

Accumulate@ Table[(-n)^DivisorSigma[0, n], {n, 28}] (* Michael De Vlieger, Mar 18 2016 *)

a(n) = sum(k=1, n, (-k)^numdiv(k)); \\ Michel Marcus, Mar 18 2016

a(n) = Sum_{i=1..n} (-i)^tau(i) = Sum_{i=1..n} (-i)^A000005(i) = Sum_{i=1..n} A217854(i).

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User: Simon Jensen

A363657 Numbers m where A217854(m) is a record minimum.

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A363658 Positive numbers m where A217854(m) is positive and increases to a record.

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A224914 Partial sums of A217854.

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