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.

Previous Showing 11-14 of 14 results.

A065303 Neither n nor sigma(n) is squarefree.

Original entry on oeis.org

12, 24, 27, 28, 32, 40, 44, 48, 52, 54, 56, 60, 63, 68, 75, 76, 81, 84, 88, 90, 92, 96, 98, 99, 108, 112, 120, 124, 125, 126, 132, 135, 136, 140, 147, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 184, 188, 189, 192, 198, 204, 207, 212, 216, 220
Offset: 1

Views

Author

Labos Elemer, Oct 29 2001

Keywords

Examples

			n = 147 = 3*7*7, sigma(147) = 2*2*3*19 = 228.

Links

Crossrefs

Cf. A000203, A008683, A065299, A065200, A065201, A065302.

Programs

  • Mathematica
    Select[Range@ 220, Nor[SquareFreeQ@ #, SquareFreeQ@ DivisorSigma[1, #]] &] (* Michael De Vlieger, Mar 18 2017 *)
Select[Range[250],NoneTrue[{#,DivisorSigma[1,#]},SquareFreeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 22 2019 *)
  • PARI
    n=0; for (m = 1, 10^9, if (!moebius(m) && !moebius(sigma(m)), write("b065303.txt", n++, " ", m); if (n==1000, return)) ) \\ Harry J. Smith, Oct 16 2009
  • PARI
    sigmaSquarefree(f)=my(v=vector(#f~,i, (f[i,1]^(f[i,2]+1)-1) / (f[i,1]-1))); for(i=2,#v, for(j=1,i-1, if(gcd(v[i],v[j])>1, return(0)))); for(i=1,#v, if(!issquarefree(v[i]), return(0))); 1
list(lim)=my(v=List()); forfactored(k=12,lim\1, if(!issquarefree(k) && !sigmaSquarefree(k[2]), listput(v,k[1]))); Vec(v) \\ Charles R Greathouse IV, Jan 08 2018
  • Python
    from sympy import divisor_sigma
from sympy.ntheory.factor_ import core
def is_squarefree(n): return core(n) == n
print([i for i in range(1, 251) if not is_squarefree(i) and not is_squarefree(divisor_sigma(i,1))]) # Indranil Ghosh, Mar 18 2017

A375397 Numbers divisible by the square of some prime factor other than the least. Non-hooklike numbers.

Original entry on oeis.org

18, 36, 50, 54, 72, 75, 90, 98, 100, 108, 126, 144, 147, 150, 162, 180, 196, 198, 200, 216, 225, 234, 242, 245, 250, 252, 270, 288, 294, 300, 306, 324, 338, 342, 350, 360, 363, 375, 378, 392, 396, 400, 414, 432, 441, 450, 468, 484, 486, 490, 500, 504, 507, 522
Offset: 1

Views

Author

Gus Wiseman, Aug 16 2024

Keywords

Comments

Contains no squarefree numbers A005117 or prime powers A000961, but some perfect powers A131605.
Also numbers k such that the minima of the maximal anti-runs in the weakly increasing sequence of prime factors of k (with multiplicity) are not identical. Here, an anti-run is a sequence with no adjacent equal parts, and the minima of the maximal anti-runs in a sequence are obtained by splitting it into maximal anti-run subsequences and taking the least term of each. Note the prime factors can alternatively be taken in weakly decreasing order.
Includes all terms of A036785 = non-products of a squarefree number and a prime power.
The asymptotic density of this sequence is 1 - (1/zeta(2)) * (1 + Sum_{p prime} (1/(p^2-p)) / Product_{primes q <= p} (1 + 1/q)) = 0.11514433883... . - Amiram Eldar, Oct 26 2024

Examples

			The prime factors of 300 are {2,2,3,5,5}, with maximal anti-runs ((2),(2,3,5),(5)), with minima (2,2,5), so 300 is in the sequence.
The terms together with their prime indices begin:
    18: {1,2,2}
    36: {1,1,2,2}
    50: {1,3,3}
    54: {1,2,2,2}
    72: {1,1,1,2,2}
    75: {2,3,3}
    90: {1,2,2,3}
    98: {1,4,4}
   100: {1,1,3,3}
   108: {1,1,2,2,2}
   126: {1,2,2,4}
   144: {1,1,1,1,2,2}

Links

Crossrefs

A superset of A036785.
The complement for maxima is A065200, counted by A034296.
For maxima instead of minima we have A065201, counted by A239955.
A version for compositions is A374520, counted by A374640.
Also positions of non-constant rows in A375128, sums A374706, ranks A375400.
The complement is A375396, counted by A115029.
The complement for distinct minima is A375398, counted by A375134.
For distinct instead of identical minima we have A375399, counts A375404.
Partitions of this type are counted by A375405.
A000041 counts integer partitions, strict A000009.
A003242 counts anti-run compositions, ranks A333489.
A number's prime factors (A027746, reverse A238689) have sum A001414, min A020639, max A006530.
A number's prime indices (A112798, reverse A296150) have sum A056239, min A055396, max A061395.
Both have length A001222, distinct A001221.
Cf. A000005, A013661, A046660, A272919, A319066, A358905, A374686, A374704, A374742, A375133, A375136, A375401.

Programs

  • Mathematica
    Select[Range[100],!SameQ@@Min /@ Split[Flatten[ConstantArray@@@FactorInteger[#]],UnsameQ]&]
  • PARI
    is(k) = if(k > 1, my(e = factor(k)[, 2]); vecprod(e) > e[1], 0); \\ Amiram Eldar, Oct 26 2024

Extensions

Name edited by Peter Munn, May 08 2025

A375403 Numbers whose maximal anti-runs of weakly increasing prime factors (with multiplicity) do not have distinct maxima.

Original entry on oeis.org

4, 8, 9, 16, 18, 24, 25, 27, 32, 36, 40, 48, 49, 50, 54, 56, 64, 72, 75, 80, 81, 88, 96, 98, 100, 104, 108, 112, 120, 121, 125, 128, 135, 136, 144, 147, 150, 152, 160, 162, 168, 169, 176, 184, 189, 192, 196, 200, 208, 216, 224, 225, 232, 240, 242, 243, 245
Offset: 1

Views

Author

Gus Wiseman, Aug 15 2024

Keywords

Comments

First differs from A299117 in having 150.
An anti-run is a sequence with no adjacent equal terms. The maxima of maximal anti-runs in a sequence are obtained by splitting it into maximal anti-run subsequences and taking the greatest term of each.
The partitions with these Heinz numbers are those with (1) some part appearing more than twice or (2) the greatest part appearing more than once.
Note the prime factors can alternatively be written in weakly decreasing order.

Examples

			The maximal anti-runs of prime factors of 150 are ((2,3,5),(5)), with maxima (5,5), so 150 is in the sequence.
The maximal anti-runs of prime factors of 180 are ((2),(2,3),(3,5)), with maxima (2,3,5), so 180 is not in the sequence.
The maximal anti-runs of prime factors of 300 are ((2),(2,3,5),(5)), with maxima (2,5,5), so 300 is in the sequence.
The terms together with their prime indices begin:
    4: {1,1}
    8: {1,1,1}
    9: {2,2}
   16: {1,1,1,1}
   18: {1,2,2}
   24: {1,1,1,2}
   25: {3,3}
   27: {2,2,2}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   48: {1,1,1,1,2}

Crossrefs

For identical instead of distinct we have A065201, complement A065200.
The complement for minima is A375398, counted by A375134.
For minima instead of maxima we have A375399, counted by A375404.
Partitions of this type are counted by A375401.
The complement is A375402, counted by A375133.
A000041 counts integer partitions, strict A000009.
A003242 counts anti-run compositions, ranks A333489.
A number's prime factors (A027746, reverse A238689) have sum A001414, min A020639, max A006530.
A number's prime indices (A112798, reverse A296150) have sum A056239, min A055396, max A061395.
Both have length A001222, distinct A001221.
Cf. A046660, A066328, A358836, A374632, A374706, A374768, A374767, A375128, A375136, A375396, A375400.

Programs

  • Mathematica
    Select[Range[150],!UnsameQ@@Max /@ Split[Flatten[ConstantArray@@@FactorInteger[#]],UnsameQ]&]

A375405 Number of integer partitions of n with a repeated part other than the least.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 3, 5, 8, 13, 20, 29, 42, 62, 83, 117, 158, 214, 283, 377, 488, 641, 823, 1058, 1345, 1714, 2154, 2713, 3387, 4222, 5230, 6474, 7959, 9782, 11956, 14591, 17737, 21529, 26026, 31422, 37811, 45425, 54418, 65097, 77652, 92510, 109943, 130468
Offset: 0

Views

Author

Gus Wiseman, Aug 17 2024

Keywords

Comments

Also partitions whose minima of maximal anti-runs are not identical. An anti-run is a sequence with no adjacent equal terms. The minima of maximal anti-runs in a sequence are obtained by splitting it into maximal anti-run subsequences and taking the least term of each.

Examples

			The a(0) = 0 through a(10) = 13 partitions:
  .  .  .  .  .  (221)  (2211)  (331)    (332)     (441)      (442)
                                (2221)   (3221)    (3321)     (3322)
                                (22111)  (3311)    (4221)     (3331)
                                         (22211)   (22221)    (4411)
                                         (221111)  (32211)    (5221)
                                                   (33111)    (32221)
                                                   (222111)   (33211)
                                                   (2211111)  (42211)
                                                              (222211)
                                                              (322111)
                                                              (331111)
                                                              (2221111)
                                                              (22111111)

Links

Crossrefs

The complement for maxima instead of minima is A034296.
The complement is counted by A115029, ranks A375396.
For maxima instead of minima we have A239955, ranks A073492.
These partitions have ranks A375397.
For distinct instead of identical we have A375404, ranks A375399.
A000041 counts integer partitions, strict A000009.
A003242 counts anti-run compositions, ranks A333489.
A055887 counts sequences of partitions with total sum n.
A375128 lists minima of maximal anti-runs of prime indices, sums A374706.
Cf. A046660, A065200, A065201, A141199, A358836, A374704, A375133, A375134, A375136, A375401.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], !SameQ@@Min/@Split[#,UnsameQ]&]],{n,0,30}]
- or -
Table[Length[Select[IntegerPartitions[n], !UnsameQ@@DeleteCases[#,Min@@#]&]],{n,0,30}]
  • PARI
    A_x(N) = {my(x='x+O('x^N), f=sum(i=1,N,sum(j=i+1,N-i, ((x^(i+(2*j)))/(1-x^i))*prod(k=i+1,N-i-(2*j), if(kJohn Tyler Rascoe, Aug 21 2024

Formula

G.f.: Sum_{i>0} (Sum_{j>i} ( (x^(i+(2*j)))/(1-x^i) * Product_{k>=i} (1-[kJohn Tyler Rascoe, Aug 21 2024
Previous Showing 11-14 of 14 results.

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

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