A350756 -id:A350756 - cp's OEIS frontend

A046952 Sets record for f(n) = |{(a,b):a*b=n and a|b}|. Also squares of highly composite numbers A002182.

1, 4, 16, 36, 144, 576, 1296, 2304, 3600, 14400, 32400, 57600, 129600, 518400, 705600, 1587600, 2822400, 6350400, 25401600, 57153600, 101606400, 228614400, 406425600, 635040000, 768398400, 2057529600, 2540160000, 3073593600

Offset: 1

Simon Colton (simonco(AT)cs.york.ac.uk)

Invented by the HR automatic theory formation program.
Also, integers whose number of square divisors sets a new record. - Bernard Schott, Jan 14 2022
As a(n) is the square of n-th highly composite number (A002182), the record number of square divisors of a(n) is A046951(a(n)) = tau(A002182(n)) = A002183(n) where tau is the divisor counting function (A000005). - Bernard Schott, Jan 15 2022
Integers m for which number of solutions (A353282) to the Diophantine equation S(x,y) = (x+y) + (x-y) + (x*y) + (x/y) = m sets a new record; these records are respectively 0, 1, 2, 3, 5, 7, ... Example: the 5 solutions for S(x,y) = 144 are (36,1), (32,2), (27,3), (20,5), (11,11). - Bernard Schott, Apr 19 2022

			f(1)=1, (first with 1), f(4)=2 (first with 2), f(16)=3 (first with 3).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Simon Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999, #2.
Simon Colton, HR - Automatic Theory Formation in Pure Mathematics

Cf. A000005, A002182, A002183, A046951.

Cf. A350756 (similar, with triangular divisors).

a(n) = A002182(n)^2. - Bernard Schott, Jan 14 2022

A356020 Positions of records in A356018, i.e., integers whose number of evil divisors sets a new record.

Original entry on oeis.org

1, 3, 6, 12, 18, 30, 60, 90, 120, 180, 360, 540, 720, 1080, 1440, 2160, 3780, 4320, 6120, 7560, 8640, 12240, 15120, 24480, 27720, 30240, 36720, 48960, 50400, 55440, 73440, 83160, 110880, 128520, 138600, 166320, 221760, 257040, 277200, 332640, 471240, 514080, 554400

Offset: 1

Bernard Schott, Jul 24 2022

Corresponding records of number of evil divisors are 0, 1, 2, 3, 4, 6, 9, 10, 12, 15, ...

			60 is in the sequence because A356018(60) = 9 is larger than any earlier value in A356018.

David A. Corneth, Table of n, a(n) for n = 1..189

Cf. A001969, A093688, A093696, A356018, A356019.

Similar sequences: A093036, A093037, A330815, A350756, A355969.

f[n_] := DivisorSum[n, 1 &, EvenQ[DigitCount[#, 2, 1]] &]; fm = -1; s = {}; Do[If[(fn = f[n]) > fm, fm = fn; AppendTo[s, n]], {n, 1, 10^5}]; s (* Amiram Eldar, Jul 24 2022 *)

upto(n) = my(res = List(), r=-1); forfactored(i=1, n, if(numdiv(i[2]) > r, d = divisors(i[2]); c=sum(j=1, #d, isevil(d[j])); if(c>r, r=c; listput(res,i[1])))); res
isevil(n) = bitand(hammingweight(n), 1)==0 \\ David A. Corneth, Jul 24 2022

from sympy import divisors
from itertools import count, islice
def c(n): return bin(n).count("1")&1 == 0
def f(n): return sum(1 for d in divisors(n, generator=True) if c(d))
def agen(record=-1):
    for k in count(1):
        if f(k) > record: record = f(k); yield k
print(list(islice(agen(), 40))) # Michael S. Branicky, Jul 24 2022

More terms from Amiram Eldar, Jul 24 2022

A355304 Integers whose number of normal undulating divisors sets a new record.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 1080, 1260, 1440, 1680, 2160, 2520, 5040, 7560, 10080, 15120, 21840, 28080, 32760, 56160, 65520, 98280, 131040, 196560, 393120, 589680, 786240, 1113840, 1670760, 2227680, 3341520, 6683040, 13366080, 20049120

Offset: 1

Bernard Schott, Jun 30 2022

Normal undulating integers are in A355301.
The first 14 terms are also the first 14 highly composite numbers in A002182, then A002182(15) = 840 while a(15) = 1080. Indeed, 840 is the smallest integer that has 32 divisors of which only 28 are normal undulating integers, while 1080 has also 32 divisors of which 30 are normal undulating integers.
Corresponding records of number of normal undulating divisors are 1, 2, 3, 4, 6, 8, 9, 10, 12, ...

			a(6) = 24 is in the sequence because A355302(24) is larger than any earlier value in A355302.

Cf. A002182, A355301, A355302, A355303.

Similar, but with divisors that are: A046952 (squares), A053624 (odd), A181808 (even), A093036 (palindromes), A340548 (repdigits), A340549 (repunits), A350756 (triangular).

nuQ[n_] := AllTrue[(s = Sign[Differences[IntegerDigits[n]]]), # != 0 &] && AllTrue[Differences[s], # != 0 &]; dm = -1; seq = {}; Do[If[(d = DivisorSum[n, 1 &, nuQ[#] &]) > dm, dm = d; AppendTo[seq, n]], {n, 1, 10^5}]; seq (* Amiram Eldar, Jun 30 2022 *)

More terms from Amiram Eldar, Jun 30 2022

A356179 Positions of records in A279497, i.e., integers whose number of pentagonal divisors sets a new record.

Original entry on oeis.org

1, 5, 35, 70, 210, 420, 2310, 4620, 18480, 32340, 60060, 120120, 240240, 720720, 1141140, 2042040, 4084080, 4564560, 13693680, 19399380, 38798760, 77597520, 232792560, 387987600

Offset: 1

Bernard Schott, Jul 28 2022

The first fourteen terms are the same as A356132; then a(15) = 1141140 while A356132(15) = 1261260.

Corresponding records of number of pentagonal divisors are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, ...

			210 is in the sequence because A279497(210) = 5 is larger than any earlier value in A279497.

Cf. A000326, A279497, A356132.

Similar sequences: A046952, A093036, A350756, A355595.

f[n_] := DivisorSum[n, 1 &, IntegerQ[(1 + Sqrt[1 + 24*#])/6] &]; fm = -1; s = {}; Do[If[(fn = f[n]) > fm, fm = fn; AppendTo[s, n]], {n, 1, 10^5}]; s (* Amiram Eldar, Jul 28 2022 *)

lista(nn) = my(m=0); for (n=1, nn, my(new = sumdiv(n, d, ispolygonal(d, 5))); if (new > m, m = new; print1(n, ", "));); \\ Michel Marcus, Jul 28 2022

a(23)-a(24) from David A. Corneth, Jul 28 2022

cp's OEIS Frontend

A046952 Sets record for f(n) = |{(a,b):a*b=n and a|b}|. Also squares of highly composite numbers A002182.

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A356020 Positions of records in A356018, i.e., integers whose number of evil divisors sets a new record.

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A355304 Integers whose number of normal undulating divisors sets a new record.

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A356179 Positions of records in A279497, i.e., integers whose number of pentagonal divisors sets a new record.

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Mathematica

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