cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A340548 Integers whose number of repdigit divisors sets a new record.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 66, 132, 264, 792, 3960, 14652, 26664, 29304, 79992, 146520, 399960, 1025640, 2666664, 7999992, 13333320, 39999960, 269333064, 807999192, 1346665320, 4039995960, 28279971720, 7999999999992, 8080799919192, 13333333333320, 13467999865320, 39999999999960, 40403999595960
Offset: 1

Views

Author

Bernard Schott, Jan 11 2021

Keywords

Comments

The first 10 terms are the same as A093036, then A093036(11) = 1848 while a(11) = 3960, because from a(1) to a(10), all palindromic divisors are also repdigits, and then 616 is a non-repdigit palindromic divisor of 1848.
Number of repdigit divisors: 1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 16, 17, 18, ...
Indices of repdigits: 1, 2, 3, 4, 7, ...

Examples

			132 has 12 divisors: {1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132} of which 10 are repdigits: {1, 2, 3, 4, 6, 11, 22, 33, 44, 66}. No positive integer smaller than 132 has as many as ten repdigit divisors; hence 132 is a term.

Links

Crossrefs

Cf. A010785, A190217.
Similar for: A053624 (odd), A181808 (even), A093036 (palindromes), A340549 (repunits).

Programs

  • Mathematica
    repQ[n_] := Length @ Union @ IntegerDigits[n] == 1; s[n_] := DivisorSum[n, 1 &, repQ[#] &]; smax =  0; seq = {}; Do[s1 = s[n]; If[s1 > smax, smax = s1; AppendTo[seq, n]], {n, 1, 10^5}]; seq (* Amiram Eldar, Jan 11 2021 *)
  • PARI
    isrd(n) = {1 == #Set(digits(n))}; \\ A010785
f(n) = sumdiv(n, d, isrd(d));
lista(nn) = {my(m = 0); for (n=1, nn, my(x = f(n)); if (x > m, print1(n, ", "); m = x););} \\ Michel Marcus, Jan 11 2021

Extensions

a(16)-a(20) from Michel Marcus, Jan 11 2021
a(21)-a(26) from Amiram Eldar, Jan 12 2021
a(27) from Chai Wah Wu, Jan 14 2021
More terms from David A. Corneth, Jan 15 2021

A181803 Triangle read by rows: T(n,k) is the k-th smallest divisor d of n such that n sets a record for the number of its divisors that are multiples of d.

Original entry on oeis.org

1, 1, 2, 3, 1, 2, 4, 5, 1, 3, 6, 7, 2, 4, 8, 9, 5, 10, 11, 1, 2, 3, 6, 12, 13, 7, 14, 15, 4, 8, 16, 17, 3, 9, 18, 19, 5, 10, 20, 21, 11, 22, 23, 1, 2, 4, 6, 12, 24, 25, 13, 26, 27, 7, 14, 28, 29, 5, 15, 30, 31, 8, 16, 32, 33, 17, 34, 35, 1, 3, 6, 9, 18, 36, 37, 19, 38, 39, 10, 20, 40, 41, 7, 21, 42
Offset: 1

Views

Author

Matthew Vandermast, Nov 27 2010

Keywords

Comments

In other words, row n contains a particular divisor d of n iff more multiples of d appear among the divisors of n than appear among the divisors of any smaller positive integer. Cf. A181808.
Row n contains A181801(n) numbers, the largest of which is n. T(n,k) * A180802(n, A181801(n)-k+1) = n.
For all positive integer values (j,k) such that jk = n, the number of divisors of n that are multiples of j equals A000005(k). Therefore, j appears in row n iff k=n/j is a member of A002182.

Examples

			First rows read: 1; 1,2; 3; 1,2,4; 5; 1,3,6; 7; 2,4,8; 9; 5,10; 11; 1,2,3,6,12;...
6 has four divisors (1, 2, 3 and 6). Of those divisors, 1, 3 and 6 appear in row 6.
a. The divisors of 6 include four multiples of 1 (1, 2, 3 and 6); two multiples of 3 (3 and 6), and one multiple of 6 (6). No positive integer smaller than 6 has more than three multiples of 1 among its divisors; hence, 1 appears in row 6. Also, no positive integer smaller than 6 has more than one multiple of 3 among its divisors, or has any multiple of 6 among its divisors. Hence, 3 and 6 both appear in row 6.
b. On the other hand, although 6 includes two multiples of 2 among its divisors (2 and 6), so does a smaller positive integer (4, whose even divisors are 2 and 4). Accordingly, 2 is not included in row 6.
The divisors of 6 that appear in row 6 are therefore 1, 3 and 6. Note that 1, 3 and 6 equal 6/6, 6/2 and 6/1 respectively, and all of the denominators in those fractions are highly composite numbers (A002182).

Links

Crossrefs

For the highly composite divisors of n, see A181802. See also A181808, A181809, A181810.

Formula

T(n,k) = n/(A180802(n, A181801(n)-k+1)).

A181809 Numbers n such that both n and n/2 are highly composite (A002182).

Original entry on oeis.org

2, 4, 12, 24, 48, 120, 240, 360, 720, 1680, 2520, 5040, 10080, 15120, 20160, 50400, 55440, 110880, 166320, 221760, 332640, 554400, 665280, 1441440, 2162160, 2882880, 4324320, 7207200, 8648640, 14414400, 17297280, 21621600, 43243200, 73513440
Offset: 1

Views

Author

Matthew Vandermast, Nov 27 2010

Keywords

Comments

These are the numbers that set records both for total number of divisors and for number of even divisors; intersection of A002182 and A181808.
For all positive integer values (j,k) such that jk = n, the number of divisors of n that are multiples of j equals A000005(k). Therefore, n sets a record for the number of its divisors that are multiples of j iff k=n/j is highly composite (A002182).

Examples

			The number 12 is both highly composite (A002182(5) = 12) and twice another highly composite number (12 = 2*6 = 2*A002182(4)).  It therefore has more divisors (A002183(5)=6) than any smaller positive integer, and more even divisors (A002183(4)=4) than any smaller positive integer. Since 12 is the third positive integer with the properties that define this sequence, a(3)=12.

Links

Crossrefs

Numbers n such that 1 and 2 both appear in row n of A181803. See also A181808, A181810.
A053624 gives numbers that set records for number of odd divisors. No number sets records both for its number of odd divisors and its number of even divisors. Only the number 1 sets a record for its number of odd divisors and its number of total divisors.
Subsequence of A025487.

A181810 a(n) = largest number k such that A002182(n)/j is highly composite for each integer j from 1 to k.

Original entry on oeis.org

1, 2, 2, 1, 3, 2, 1, 2, 1, 2, 1, 2, 3, 4, 1, 1, 2, 3, 4, 1, 2, 3, 2, 1, 1, 1, 2, 2, 1, 2, 3, 2, 1, 4, 1, 2, 4, 1, 1, 2, 3, 2, 1, 4, 1, 2, 4, 1, 2, 2, 3, 1, 1, 6, 1, 2, 1, 2, 2, 1, 2, 2, 3, 1, 1, 6, 3, 2, 1, 4, 1, 2, 1, 2, 2, 3, 1, 6, 3, 2, 4, 1, 1, 1, 1, 2, 2
Offset: 1

Views

Author

Matthew Vandermast, Nov 27 2010

Keywords

Comments

Also, largest number k such that, for each integer j from 1 to k, more multiples of j appear among the divisors of A002182(n) than appear among the divisors of any smaller positive integer.
For all positive integer values (j,k) such that jk = n, the number of divisors of n that are multiples of j equals A000005(k). Therefore, n sets a record for the number of its divisors that are multiples of j iff k = n/j is highly composite (A002182).

Examples

			360 is a member of A002182, twice a member of A002182 (360/2 = 180), and three times a member of A002182 (360/3 = 120), but is not four times a member of A002182 (360/4 = 90 is not a member of A002182). Since A002182(13) = 360, a(13) = 3.
360 also sets records for the number of its divisors, the number of its divisors that are multiples of 2 (cf. A181808), and the number of its divisors that are multiples of 3, but not the number of its divisors that are multiples of 4.

Links

Crossrefs

a(n) equals the largest number k such that each number from 1 to k appears in row A002182(n) of A181803. a(n) also equals the largest number k such that each of the first k members of row A002182(n) of A056538 is highly composite.
See also A181801, A181808, A181809.

Programs

  • Mathematica
    f[hc_, n_] := Module[{k=1}, While[MemberQ[hc, n/k], k++]; k-1]; s={}; hc={}; dm = 0; Do[d = DivisorSigma[0, n]; If[d > dm, dm = d; AppendTo[hc, n]]; AppendTo[s, f[hc, n]], {n, 1, 10^5}]; s (* Amiram Eldar, Jul 08 2019 *)

Extensions

a(5) corrected and more terms added by Amiram Eldar, Jul 08 2019

A350756 Integers whose number of divisors that are triangular numbers sets a new record.

Original entry on oeis.org

1, 3, 6, 30, 90, 180, 210, 420, 630, 1260, 2520, 6930, 13860, 27720, 41580, 83160, 138600, 180180, 360360, 540540, 1081080, 1413720, 2162160, 3063060, 6126120, 12252240, 18378360, 36756720, 73513440, 91891800, 116396280, 183783600, 232792560, 349188840
Offset: 1

Views

Author

Bernard Schott, Jan 13 2022

Keywords

Comments

Terms that are triangular: 1, 3, 6, 210, 630, 2162160, ...
The number of triangular divisors of a(n) is A007862(a(n)): 1, 2, 3, 5, 6, 7, 8, 9, 10, 12, ...

Examples

			1260 has 36 divisors of which 12 are triangular numbers {1, 3, 6, 10, 15, 21, 28, 36, 45, 105, 210, 630}. No positive integer smaller than 1260 has as many as twelve triangular divisors; hence 1260 is a term.

Crossrefs

Cf. A000217, A007862, A130317.
Similar for A046952 (squares), A053624 (odd), A093036 (palindromes), A181808 (even), A340548 (repdigits), A340549 (repunits) divisors.

Programs

  • Mathematica
    max=0;Do[If[(d=Length@Select[Divisors@k,IntegerQ[(Sqrt[8#+1]-1)/2]&])>max,Print@k;max=d],{k,10^10}] (* Giorgos Kalogeropoulos, Jan 13 2022 *)
  • PARI
    lista(nn) = {my(r=0); for (n=1, nn, my(m = sumdiv(n, d, ispolygonal(d,3))); if (m>r, r=m; print1(n", ")));} \\ Michel Marcus, Jan 14 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

Views

Author

Bernard Schott, Jun 30 2022

Keywords

Comments

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, ...

Examples

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

Crossrefs

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).

Programs

  • Mathematica
    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 *)

Extensions

More terms from Amiram Eldar, Jun 30 2022

A369151 Numbers with a record high excess of even over odd divisors; so indices of record lows in A048272.

Original entry on oeis.org

1, 2, 4, 8, 16, 24, 48, 96, 144, 192, 240, 480, 720, 960, 1440, 2880, 3360, 5040, 6720, 10080, 20160, 30240, 40320, 60480, 80640, 100800, 110880, 181440, 201600, 221760, 332640, 443520, 665280, 887040, 1108800, 1330560, 1995840, 2217600, 2661120, 2882880, 4324320, 5765760, 8648640, 11531520, 14414400
Offset: 1

Views

Author

Keith F. Lynch, Jan 14 2024

Keywords

Comments

Every term is the product of primorials, i.e., this is a subsequence of A025487, i.e., no prime factor of any term has a lower exponent than the following prime has.

Examples

			24 is a term because 24 has 6 even divisors, {2,4,6,8,12,24}, and 2 odd divisors, {1,3}, giving a difference of 4, more than that of any number less than 24.

Links

Crossrefs

Cf. A048272.

Formula

If n > 2, a(n) = 2*A181808(n-2) = 4*A002182(n-2).
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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