A027383 -id:A027383 - cp's OEIS frontend

A247905 Start with a single hexagon; at n-th generation add a hexagon at each expandable vertex (this is the "vertex to side" version); a(n) is the sum of all label values at n-th generation. (See comment for construction rules.)

1, 7, 19, 43, 79, 139, 223, 355, 535, 811, 1183, 1747, 2503, 3643, 5167, 7459, 10519, 15115, 21247, 30451, 42727, 61147, 85711, 122563, 171703, 245419, 343711, 491155, 687751, 982651, 1375855, 1965667, 2752087, 3931723, 5504575, 7863859, 11009575, 15728155, 22019599, 31456771, 44039671, 62914027, 88079839, 125828563, 176160199, 251657659

Offset: 0

Kival Ngaokrajang, Sep 26 2014

Refer to A247620, which is the "vertex to vertex" expansion version. For this case, the expandable vertices of the existing generation will contact the sides of the new ones i.e. "vertex to side" expansion version. Let us assign the label "1" the hexagon at the origin; at n-th generation add a hexagon at each expandable vertex, i.e. each vertex where the added generations will not overlap the existing ones, although overlaps among new generations are allowed. The non-overlapping hexagons will have the same label value as a predecessor; for the overlapping ones, the label value will be sum of label values of predecessors. a(n) is the sum of all label values at n-th generation. The hexagons count is A003215. See illustration. For n >= 1, (a(n) - a(n-1))/6 is A027383.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Kival Ngaokrajang, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,2).

Cf. Vertex to vertex version: A061777, A247618, A247619, A247620.
Cf. Vertex to side version: A101946, A247903, A247904.
Cf. A003215, A027383.

[3*2^(n/2)*((7+5*Sqrt(2)) + (-1)^n*(7-5*Sqrt(2))) -(12*n+41): n in [0..50]]; // G. C. Greubel, Feb 17 2022

LinearRecurrence[{2,1,-4,2}, {1,7,19,43}, 50] (* G. C. Greubel, Feb 17 2022 *)

{
b=0; a=1; print1(1, ", ");
for (n=0, 50,
     b=b+2^floor(n/2);
     a=a+6*b;
     print1(a, ", ")
    )
}

Vec(-(2*x^3+4*x^2+5*x+1)/((x-1)^2*(2*x^2-1)) + O(x^100)) \\ Colin Barker, Sep 26 2014

[3*2^(n/2)*((7+5*sqrt(2)) + (-1)^n*(7-5*sqrt(2))) -(12*n+41) for n in (0..50)] # G. C. Greubel, Feb 17 2022

a(0) = 1, for n >= 1, a(n) = 6*A027383(n) + a(n-1).
a(n) = 2*a(n-1) +a(n-2) -4*a(n-3) +2*a(n-4). - Colin Barker, Sep 26 2014
G.f.: (1+5*x+4*x^2+2*x^3)/((1-x)^2*(1-2*x^2)). - Colin Barker, Sep 26 2014
a(n) = 3*2^(n/2)*((1+sqrt(2))^3 + (-1)^n*(1-sqrt(2))^3) -12*n - 41. - G. C. Greubel, Feb 18 2022

A351010 Numbers k such that the k-th composition in standard order is a concatenation of twins (x,x).

Original entry on oeis.org

0, 3, 10, 15, 36, 43, 58, 63, 136, 147, 170, 175, 228, 235, 250, 255, 528, 547, 586, 591, 676, 683, 698, 703, 904, 915, 938, 943, 996, 1003, 1018, 1023, 2080, 2115, 2186, 2191, 2340, 2347, 2362, 2367, 2696, 2707, 2730, 2735, 2788, 2795, 2810, 2815, 3600, 3619

Offset: 1

Gus Wiseman, Feb 01 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms together with their binary expansions and the corresponding compositions begin:
    0:         0  ()
    3:        11  (1,1)
   10:      1010  (2,2)
   15:      1111  (1,1,1,1)
   36:    100100  (3,3)
   43:    101011  (2,2,1,1)
   58:    111010  (1,1,2,2)
   63:    111111  (1,1,1,1,1,1)
  136:  10001000  (4,4)
  147:  10010011  (3,3,1,1)
  170:  10101010  (2,2,2,2)
  175:  10101111  (2,2,1,1,1,1)
  228:  11100100  (1,1,3,3)
  235:  11101011  (1,1,2,2,1,1)
  250:  11111010  (1,1,1,1,2,2)
  255:  11111111  (1,1,1,1,1,1,1,1)

The case of twins (binary weight 2) is A000120.
The Heinz numbers of these compositions are given by A000290.
All terms are evil numbers A001969.
Partitions of this type are counted by A035363, any length A351004.
These compositions are counted by A077957(n-2), see also A016116.
The strict case (distinct twins) is A351009, counted by A032020 with 0's.
The anti-run case is A351011, counted by A003242 interspersed with 0's.
A011782 counts integer compositions.
A085207/A085208 represent concatenation of standard compositions.
A333489 ranks anti-runs, complement A348612.
A345167/A350355/A350356 rank alternating compositions.
A351014 counts distinct runs in standard compositions.
Cf. A018819, A025047, A027383, A035457, A053738, A088218, A106356, A238279, A344604, A351012, A351015.
Selected statistics of standard compositions:
- Length is A000120.
- Sum is A070939.
- Heinz number is A333219.
- Number of distinct parts is A334028.
Selected classes of standard compositions:
- Partitions are A114994, strict A333256.
- Multisets are A225620, strict A333255.
- Strict compositions are A233564.
- Constant compositions are A272919.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],And@@EvenQ/@Length/@Split[stc[#]]&]

A363493 Number T(n,k) of partitions of [n] having exactly k parity changes within their blocks, n>=0, 0<=k<=max(0,n-1), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 4, 6, 4, 1, 10, 18, 17, 6, 1, 25, 61, 68, 38, 10, 1, 75, 210, 292, 202, 83, 14, 1, 225, 778, 1252, 1116, 576, 170, 22, 1, 780, 3008, 5670, 5928, 3899, 1490, 341, 30, 1, 2704, 12219, 26114, 32382, 25320, 12655, 3856, 678, 46, 1, 10556, 52268, 126073, 177666, 163695, 98282, 39230, 9418, 1319, 62, 1

Offset: 0

Alois P. Heinz, Jun 05 2023

			T(4,0) = 4: 13|24, 13|2|4, 1|24|3, 1|2|3|4.
T(4,1) = 6: 124|3, 12|3|4, 134|2, 1|23|4, 14|2|3, 1|2|34.
T(4,2) = 4: 123|4, 12|34, 14|23, 1|234.
T(4,3) = 1: 1234.
T(5,2) = 17: 1235|4, 123|4|5, 1245|3, 12|34|5, 125|3|4, 12|3|45, 1345|2, 134|25, 14|235, 14|23|5, 15|234, 1|234|5, 1|23|45, 145|2|3, 14|25|3, 1|25|34, 1|2|345.
Triangle T(n,k) begins:
     1;
     1;
     1,     1;
     2,     2,     1;
     4,     6,     4,     1;
    10,    18,    17,     6,     1;
    25,    61,    68,    38,    10,     1;
    75,   210,   292,   202,    83,    14,    1;
   225,   778,  1252,  1116,   576,   170,   22,   1;
   780,  3008,  5670,  5928,  3899,  1490,  341,  30,  1;
  2704, 12219, 26114, 32382, 25320, 12655, 3856, 678, 46, 1;
  ...

Alois P. Heinz, Rows n = 0..150, flattened
Wikipedia, Partition of a set

Columns k=0-2 give: A124419, A363511, A363588.
Row sums give A000110.
T(n+1,n) gives A000012.
T(n+2,n) gives A027383.
T(2n+1,n) gives A363495.
Cf. A152874, A363496, A363519.

b:= proc(n, x, y) option remember; `if`(n=0, 1,
     `if`(y=0, 0, expand(b(n-1, y-1, x+1)*y*z))+
        b(n-1, y, x)*x + b(n-1, y, x+1))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..12);

b[n_, x_, y_] := b[n, x, y] = If[n == 0, 1,
  If[y == 0, 0, Expand[b[n - 1, y - 1, x + 1]*y*z]] +
  b[n - 1, y, x]*x + b[n - 1, y, x + 1]];
T[n_] := CoefficientList[b[n, 0, 0], z];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Sep 05 2023, after Alois P. Heinz *)

Sum_{k=0..max(0,n-1)} k * T(n,k) = A363496(n).

A035491 Relevant part of deck in Guy's shuffling problem (A035485): row n of the table lists the first 2n "cards" (numbers) after the n-th shuffle.

Original entry on oeis.org

2, 1, 3, 2, 4, 1, 1, 3, 5, 2, 6, 4, 6, 1, 4, 3, 7, 5, 8, 2, 5, 6, 8, 1, 2, 4, 9, 3, 10, 7, 9, 5, 3, 6, 10, 8, 7, 1, 11, 2, 12, 4, 1, 9, 11, 5, 2, 3, 12, 6, 4, 10, 13, 8, 14, 7, 4, 1, 10, 9, 13, 11, 8, 5, 14, 2, 7, 3, 15, 12, 16, 6, 2, 4, 7, 1, 3, 10, 15, 9, 12, 13, 16, 11, 6, 8, 17, 5, 18, 14

Offset: 1

Wouter Meeussen

			{}, {2, 1}, {3, 2, 4, 1}, {1, 3, 5, 2, 6, 4}, {6, 1, 4, 3, 7, 5, 8, 2}, ...
From _M. F. Hasler_, Aug 11 2022: (Start)
The first rows of the table are: (sequence = right part of the following table)
  row | first 2n cards (followed in the deck by 2n+1, 2n+2, ...)
------+---------------------------------------------------------
   0  |  -     (followed by 1, 2, 3, ...)
   1  |  2 1       (followed by 3, 4, 5, ...)
   2  |  3 2 4 1     (followed by 5, 6, 7, ...)
   3  |  1 3 5 2 6 4   (followed by 7, 8, 9, ...)
   4  |  6 1 4 3 7 5 8 2 (followed by 9, 10, 11, ...)
   5  |  5 6 8 1 2 4 9 3 10 7 (followed by 11, 12, 13, ...)
   6  |  9 5 3 6 10 8 7 1 11 2 12 4 (followed by 13, 14, 15, ...)
   7  |  1 9 11 5 2 3 12 6 4 10 13 8 14 7 (followed by 15, 16, 17, ...)
   8  |  4 1 10 9 13 11 8 5 14 2 7 3 15 12 16 6 (followed by 17, 18, 19, ...)
   (...)
The largest numbers in row n are 2n - k, located at column 2n + 1 - d(k) with d(k) = 2*A027383(k) = A347789(k+2) = 2, 4, 8, 12, 20, 28, ..., for k >= 0, d(k) <= 2n. (End)

D. Gale, Mathematical Entertainments: "Careful Card-Shuffling and Cutting Can Create Chaos," The Mathematical Intelligencer, vol. 14, no. 1, 1992, pages 54-56.
D. Gale, Tracking the Automatic Ant and Other Mathematical Explorations, A Collection of Mathematical Entertainments Columns from The Mathematical Intelligencer, Springer, 1998.

Cf. A035485, A035490 - A035494.

Flatten[NestList[riguy, {}, 12]] (* See A035490. *)

A35491=Map(); d=[]; A035491_row(n)={while(#dM. F. Hasler, Aug 11 2022

from itertools import count, islice
def agen(): # generator of terms
    deck = []
    for n in count(1):
        deck += [2*n-1, 2*n]
        first, next = deck[:n], deck[n:2*n]
        deck[0:2*n:2], deck[1:2*n:2] = next, first
        yield from deck
print(list(islice(agen(), 90))) # Michael S. Branicky, Aug 11 2022

T[n, 2*n + 1 - 2*A027383(k)] = 2n - k for all n and k >= 0, A027383(k) <= n. - M. F. Hasler, Aug 13 2022

A133629 a(1)=1, a(n) = a(n-1) + (p-1)*p^(n/2-1) if n is even, otherwise a(n) = a(n-1) + p^((n-1)/2), where p=5.

Original entry on oeis.org

1, 5, 10, 30, 55, 155, 280, 780, 1405, 3905, 7030, 19530, 35155, 97655, 175780, 488280, 878905, 2441405, 4394530, 12207030, 21972655, 61035155, 109863280, 305175780, 549316405, 1525878905, 2746582030, 7629394530, 13732910155, 38146972655, 68664550780

Offset: 1

Hieronymus Fischer, Sep 19 2007

Partial sums of A133632.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,5,-5).

Sequences with similar recurrence rules: A027383 (p=2), A087503 (p=3), A133628 (p=4).
Related sequences: A132666, A132667, A132668, A132669.
Other related sequences for different p: A016116 (p=2), A038754 (p=3), A084221 (p=4), A133632 (p=5).

a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=5*a[n-2]+5 od: seq(a[n], n=1..29); # Zerinvary Lajos, Mar 17 2008

Vec(x*(1 + 4*x) / ((1 - x) * (1 - 5*x^2)) + O(x^40)) \\ Colin Barker, Nov 25 2016

def A133629(n): return (5+((n&1)<<2))*5**(n>>1)-5>>2 # Chai Wah Wu, Sep 02 2025

a(n) = Sum_{k=1..n} A133632(k).
The following formulas are given for a general natural parameter p > 1 (p=5 for this sequence).
G.f.: x(1+(p-1)x)/((1-px^2)(1-x)).
a(n) = (p/(p-1))*(p^(n/2)-1) if n is even, otherwise a(n)=(p/(p-1))*((2p-1)*p^((n-3)/2)-1).
a(n) = (p/(p-1))*(p^floor(n/2) + p^floor((n-1)/2) - p^floor((n-2)/2)-1).
a(n) = p^floor(n/2) + (p^floor((n+1)/2)-p)/(p-1).
a(n) = A132669(a(n+1)) - 1.
a(n) = A132669(a(n-1)+1) for n > 0.
A132669(a(n)) = a(n-1)+1 for n > 0.
From Colin Barker, Nov 25 2016: (Start)
a(n) = 5*(5^(n/2) - 1)/4 for n even.
a(n) = (9*5^(n/2-1/2) - 5)/4 for n odd.
a(n) = a(n-1) + 5*a(n-2) - 5*a(n-3) for n > 3.
G.f.: x*(1 + 4*x) / ((1 - x) * (1 - 5*x^2)).
(End)

A350352 Products of three or more distinct prime numbers.

Original entry on oeis.org

30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 210, 222, 230, 231, 238, 246, 255, 258, 266, 273, 282, 285, 286, 290, 310, 318, 322, 330, 345, 354, 357, 366, 370, 374, 385, 390, 399, 402, 406, 410, 418, 426, 429, 430

Offset: 1

Gus Wiseman, Jan 11 2022

nonn

First differs from A336568 in lacking 420.

			The terms and their prime indices begin:
     30: {1,2,3}     182: {1,4,6}      285: {2,3,8}
     42: {1,2,4}     186: {1,2,11}     286: {1,5,6}
     66: {1,2,5}     190: {1,3,8}      290: {1,3,10}
     70: {1,3,4}     195: {2,3,6}      310: {1,3,11}
     78: {1,2,6}     210: {1,2,3,4}    318: {1,2,16}
    102: {1,2,7}     222: {1,2,12}     322: {1,4,9}
    105: {2,3,4}     230: {1,3,9}      330: {1,2,3,5}
    110: {1,3,5}     231: {2,4,5}      345: {2,3,9}
    114: {1,2,8}     238: {1,4,7}      354: {1,2,17}
    130: {1,3,6}     246: {1,2,13}     357: {2,4,7}
    138: {1,2,9}     255: {2,3,7}      366: {1,2,18}
    154: {1,4,5}     258: {1,2,14}     370: {1,3,12}
    165: {2,3,5}     266: {1,4,8}      374: {1,5,7}
    170: {1,3,7}     273: {2,4,6}      385: {3,4,5}
    174: {1,2,10}    282: {1,2,15}     390: {1,2,3,6}

Michael De Vlieger, Table of n, a(n) for n = 1..10000

This is the squarefree case of A033942.
Including squarefree semiprimes gives A120944.
The squarefree complement consists of 1 and A167171.
These are the Heinz numbers of the partitions counted by A347548.
A000040 lists prime numbers (exactly 1 prime factor).
A005117 lists squarefree numbers.
A006881 lists squarefree numbers with exactly 2 prime factors.
A007304 lists squarefree numbers with exactly 3 prime factors.
A046386 lists squarefree numbers with exactly 4 prime factors.
Cf. A000009, A000111, A001250, A002808, A027383, A349796.

Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]>=3&]

is(n,f=factor(n))=my(e=f[,2]); #e>2 && vecmax(e)==1 \\ Charles R Greathouse IV, Jul 08 2022

list(lim)=my(v=List()); forsquarefree(n=30,lim\1, if(#n[2][,2]>2, listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Jul 08 2022

from sympy import factorint
def ok(n): f = factorint(n, multiple=True); return len(f) == len(set(f)) > 2
print([k for k in range(431) if ok(k)]) # Michael S. Branicky, Jan 14 2022

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A350352(n):
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n+x-sum(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,i)) for i in range(3,x.bit_length())))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f,n,n) # Chai Wah Wu, Sep 11 2024

A053599 Number of nonempty subsequences {s(k)} of 1..n such that the difference sequence is palindromic.

Original entry on oeis.org

1, 3, 7, 13, 23, 37, 59, 89, 135, 197, 291, 417, 607, 861, 1243, 1753, 2519, 3541, 5075, 7121, 10191, 14285, 20427, 28617, 40903, 57285, 81859, 114625, 163775, 229309, 327611, 458681, 655287, 917429, 1310643, 1834929, 2621359, 3669933, 5242795, 7339945

Offset: 1

John W. Layman, Jan 19 2000

Equals (n-1)-th row sums of triangle A152202. - Gary W. Adamson, Nov 29 2008
a(n) is the number of positive integers < 2^n such that the binary representation of the odd part is palindromic; i.e., palindromic without the final 0's. - Andrew Woods, May 19 2012
a(n) is the number of ideals of the quotient ring Z_{2^n}[u]/ for indeterminate u. - Fatih Temiz, Oct 11 2017
Conjecture:  let b(n) be the number of subsets S of {1,2,...,n} having more than one element such that (sum of least two elements of S) > max(S).  Then b(0) = b(1) = 0 and b(n+2) = a(n+1) for n >= 0. - Clark Kimberling Sep 27 2022

			For n=4 the 13 sequences are 1,2,3,4,12,13,14,23,24,34,123,234,1234.

Andrew Woods, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,2).

Cf. A027383 (first differences), A016116 (second differences).

Cf. A152202.

t={1,1};Do[AppendTo[t,t[[-2]]+t[[-1]]];AppendTo[t,2*t[[-2]]],{n,40}];Nest[Accumulate,t,2] (* Vladimir Joseph Stephan Orlovsky, Jan 29 2012 *)

a(1)=1, a(2)=3 and, for n > 2, a(n) = 2*a(n-2) + 2*n - 1.
G.f.: x*(1+x)/((1-x)^2*(1-2*x^2)). - Colin Barker, Mar 28 2012
a(n) = 5*2^((n+1)/2) - 2*n - 7 for odd n, 7*2^(n/2) - 2*n - 7 for even n. - Andrew Woods, May 19 2012

Corrected by T. D. Noe, Nov 08 2006

A180918 'DPE(n,k)' triangle read by rows. DPE(n,k) is the number of k-double-palindromes of n up to cyclic equivalence.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 2, 2, 1, 1, 0, 3, 2, 3, 1, 1, 0, 3, 3, 3, 3, 1, 1, 0, 4, 3, 6, 3, 4, 1, 1, 0, 4, 4, 6, 6, 4, 4, 1, 1, 0, 5, 4, 10, 6, 10, 4, 5, 1, 1, 0, 5, 5, 10, 10, 10, 10, 5, 5, 1, 1, 0, 6, 5, 15, 10, 20, 10, 15, 5, 6, 1, 1, 0, 6, 6, 15, 15, 20, 20, 15, 15, 6, 6, 1, 1

Offset: 1

John P. McSorley, Sep 23 2010

A k-composition of n is an ordered collection of k positive integers (parts) which sum to n. Two k-compositions of n are cyclically equivalent if one can be obtained from the other by a cyclic permutation of its parts.
A palindrome is a word which is the same when written backwards.
A k-double-palindrome of n is a k-composition of n which is the concatenation of two palindromes, PP'=P|P', where both |P|, |P'|>=1.
See sequence A180653. For example 1123532=11|23532 is a 7-double-palindrome of 17 since both 11 and 23532 are palindromes.
Let DPE(n,k) denote the number of k-double-palindromes of n up to cyclic equivalence.
This sequence is the 'DPE(n,k)' triangle read by rows.

			The triangle begins:
  0
  0 1
  0 1 1
  0 2 1 1
  0 2 2 1 1
  0 3 2 3 1 1
  0 3 3 3 3 1 1
  0 4 3 6 3 4 1 1
  0 4 4 6 6 4 4 1 1
  0 5 4 10 6 10 4 5 1 1
  ...
For example, row 8 is: 0 4 3 6 3 4 1 1.
We have DPE(8,3)=3 because there are 3 3-double-palindromes of 8 up to cyclic equivalence: {116, 611}, {224, 422}, and {233, 332}.
We have DPE(8,4)=6 because there are 6 4-double-palindromes of 8: up to cyclic equivalence: {1115, 5111, 1511, 1151}, {1214, 4121, 1412, 2141}, {1133, 3311}, {1313, 3131}, {1232, 2123, 3212, 2321}, and {2222}.

John P. McSorley: Counting k-compositions of n with palindromic and related structures. Preprint, 2010.

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Row sums are A027383(n-1).
If we remove the cyclic equivalence requirement, and just count k-double-palindromes of n, then we get sequence A180653.
If we replace the left hand column of 0's by 1's in the triangle above, we get the triangle 'RE(n, k)' where RE(n, k) is the number of k-reverses of n up to cyclic equivalence, see the McSorley reference above for more details and also sequence A119963.
See sequence A179181 for the triangle whose (n, k) term gives the number of k-palindromes (single-palindromes) of n up to cyclic equivalence.

T(n, k) = {if(k<=1, 0, binomial((n-k%2)\2, k\2))} \\ Andrew Howroyd, Sep 27 2019

T(n, 1) = 0; T(n, k) = A119963(n,k) for k > 1.

Terms a(56) and beyond from Andrew Howroyd, Sep 27 2019

A350140 Nonsquarefree numbers whose prime signature has at least one odd part other the first or last.

Original entry on oeis.org

60, 84, 120, 132, 140, 150, 156, 168, 204, 220, 228, 240, 260, 264, 270, 276, 280, 294, 300, 308, 312, 315, 336, 340, 348, 364, 372, 378, 380, 408, 420, 440, 444, 456, 460, 476, 480, 490, 492, 495, 516, 520, 528, 532, 540, 552, 560, 564, 572, 580, 585, 588

Offset: 1

Gus Wiseman, Dec 25 2021

nonn

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

Also Heinz numbers of non-weakly alternating non-strict integer partitions, where we define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. These partitions are counted by A349796. This sequence involves the somewhat degenerate case where no strict increases are allowed.

			The terms together with their Heinz partitions begin (A-E = 10-14):
     60: (3211)      276: (9211)      420: (43211)
     84: (4211)      280: (43111)     440: (53111)
    120: (32111)     294: (4421)      444: (C211)
    132: (5211)      300: (33211)     456: (82111)
    140: (4311)      308: (5411)      460: (9311)
    150: (3321)      312: (62111)     476: (7411)
    156: (6211)      315: (4322)      480: (3211111)
    168: (42111)     336: (421111)    490: (4431)
    204: (7211)      340: (7311)      492: (D211)
    220: (5311)      348: (A211)      495: (5322)
    228: (8211)      364: (6411)      516: (E211)
    240: (321111)    372: (B211)      520: (63111)
    260: (6311)      378: (42221)     528: (521111)
    264: (52111)     380: (8311)      532: (8411)
    270: (32221)     408: (72111)     540: (322211)

Including all nonsquarefree numbers gives A013929, complement A005117.
Subsets include A088860 and A110286.
Signatures of this type are counted by A274230, complement A027383.
The strict instead of non-strict version is A336568, counted by A347548.
A version for compositions allowing strict is A349057, counted by A349053.
Allowing strict partitions gives A349794, counted by A349061.
These partitions are counted by A349796.
The complement in nonsquarefree partitions is A350137, counted by A349795.
A000041 = integer partitions, strict A000009.
A001250 = alternating permutations, ranked by A349051, complement A348615.
A003242 = Carlitz (anti-run) compositions.
A025047/A025048/A025049 = alternating compositions, ranked by A345167.
A056239 adds up prime indices, row sums of A112798, row lengths A001222.
A096441 = weakly alternating 0-appended partitions.
A124010 = prime signature, sorted A118914.
A345164 = alternating permutations of prime indices, complement A350251.
A345170 = partitions w/ an alternating permutation, ranked by A345172.
A349052/A129852/A129853 = weakly alternating compositions.
A349056 = weakly alternating permutations of prime indices.
A349058 = weakly alternating patterns, complement A350138.
A349060 = weakly alternating partitions, strong A349801.
A349798 = weakly but not strongly alternating perms of prime indices.
Cf. A000111, A047967, A333213, A335448, A344615, A344653, A345173, A349054, A349059, A349797, A349799.

Select[Range[300],!SquareFreeQ[#]&&PrimeNu[#]>1&& !And@@EvenQ/@Take[Last/@FactorInteger[#],{2,-2}]&]

Complement of A005117 in A349794.

A351011 Numbers k such that the k-th composition in standard order has even length and alternately equal and unequal parts, i.e., all run-lengths equal to 2.

Original entry on oeis.org

0, 3, 10, 36, 43, 58, 136, 147, 228, 235, 528, 547, 586, 676, 698, 904, 915, 2080, 2115, 2186, 2347, 2362, 2696, 2707, 2788, 2795, 3600, 3619, 3658, 3748, 3770, 8256, 8323, 8458, 8740, 8747, 8762, 9352, 9444, 9451, 10768, 10787, 10826, 11144, 11155, 14368

Offset: 1

Gus Wiseman, Feb 03 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms together with their binary expansions and standard compositions begin:
    0:           0  ()
    3:          11  (1,1)
   10:        1010  (2,2)
   36:      100100  (3,3)
   43:      101011  (2,2,1,1)
   58:      111010  (1,1,2,2)
  136:    10001000  (4,4)
  147:    10010011  (3,3,1,1)
  228:    11100100  (1,1,3,3)
  235:    11101011  (1,1,2,2,1,1)
  528:  1000010000  (5,5)
  547:  1000100011  (4,4,1,1)
  586:  1001001010  (3,3,2,2)
  676:  1010100100  (2,2,3,3)
  698:  1010111010  (2,2,1,1,2,2)
  904:  1110001000  (1,1,4,4)
  915:  1110010011  (1,1,3,3,1,1)

The case of twins (binary weight 2) is A000120.
All terms are evil numbers A001969.
These compositions are counted by A003242 interspersed with 0's.
Partitions of this type are counted by A035457, any length A351005.
The Heinz numbers of these compositions are A062503.
Taking singles instead of twins gives A333489, complement A348612.
This is the anti-run case of A351010.
The strict case (distinct twins) is A351009, counted by A077957(n-2).
A011782 counts compositions.
A085207/A085208 represent concatenation of standard compositions.
A345167 ranks alternating compositions, counted by A025047.
A350355 ranks up/down compositions, counted by A025048.
A350356 ranks down/up compositions, counted by A025049.
A351014 counts distinct runs in standard compositions.
Cf. A008965, A018819, A027383, A032020, A035363, A088218, A106356, A122129, A122134, A238279, A351007.
Selected statistics of standard compositions:
- Length is A000120.
- Sum is A070939.
- Heinz number is A333219.
- Number of distinct parts is A334028.
Selected classes of standard compositions:
- Partitions are A114994, strict A333256.
- Multisets are A225620, strict A333255.
- Strict compositions are A233564.
- Constant compositions are A272919.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],And@@(#==2&)/@Length/@Split[stc[#]]&]

cp's OEIS Frontend

A247905 Start with a single hexagon; at n-th generation add a hexagon at each expandable vertex (this is the "vertex to side" version); a(n) is the sum of all label values at n-th generation. (See comment for construction rules.)

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A351010 Numbers k such that the k-th composition in standard order is a concatenation of twins (x,x).

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A363493 Number T(n,k) of partitions of [n] having exactly k parity changes within their blocks, n>=0, 0<=k<=max(0,n-1), read by rows.

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A035491 Relevant part of deck in Guy's shuffling problem (A035485): row n of the table lists the first 2n "cards" (numbers) after the n-th shuffle.

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A133629 a(1)=1, a(n) = a(n-1) + (p-1)*p^(n/2-1) if n is even, otherwise a(n) = a(n-1) + p^((n-1)/2), where p=5.

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A350352 Products of three or more distinct prime numbers.

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A053599 Number of nonempty subsequences {s(k)} of 1..n such that the difference sequence is palindromic.

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