Displaying 1-10 of 58 results found.
Number of odd parts in the partition having Heinz number n.
+10
75
0, 1, 0, 2, 1, 1, 0, 3, 0, 2, 1, 2, 0, 1, 1, 4, 1, 1, 0, 3, 0, 2, 1, 3, 2, 1, 0, 2, 0, 2, 1, 5, 1, 2, 1, 2, 0, 1, 0, 4, 1, 1, 0, 3, 1, 2, 1, 4, 0, 3, 1, 2, 0, 1, 2, 3, 0, 1, 1, 3, 0, 2, 0, 6, 1, 2, 1, 3, 1, 2, 0, 3, 1, 1, 2, 2, 1, 1, 0, 5, 0, 2, 1, 2, 2, 1, 0, 4
COMMENTS
We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436. In the Maple program the subprogram B yields the partition with Heinz number n.
REFERENCES
George E. Andrews and Kimmo Eriksson, Integer Partitions, Cambridge Univ. Press, Cambridge, 2004.
Miklós Bóna, A Walk Through Combinatorics, World Scientific Publishing Co., 2002.
FORMULA
Totally additive with a(p) = 1 if primepi(p) is odd, and 0 otherwise.
EXAMPLE
a(12) = 2 because the partition having Heinz number 12 = 2*2*3 is [1,1,2], having 2 odd parts.
MAPLE
with(numtheory): a := proc (n) local B, ct, q: B := proc (n) local nn, j, m: nn := op(2, ifactors(n)): for j to nops(nn) do m[j] := op(j, nn) end do: [seq(seq(pi(op(1, m[i])), q = 1 .. op(2, m[i])), i = 1 .. nops(nn))] end proc: ct := 0: for q to nops(B(n)) do if `mod`(B(n)[q], 2) = 1 then ct := ct+1 else end if end do: ct end proc: seq(a(n), n = 1 .. 135);
# second Maple program:
a:= n-> add(`if`(numtheory[pi](i[1])::odd, i[2], 0), i=ifactors(n)[2]):
Numbers with as many even as odd prime indices, counted with multiplicity.
+10
64
1, 6, 14, 15, 26, 33, 35, 36, 38, 51, 58, 65, 69, 74, 77, 84, 86, 90, 93, 95, 106, 119, 122, 123, 141, 142, 143, 145, 156, 158, 161, 177, 178, 185, 196, 198, 201, 202, 209, 210, 214, 215, 216, 217, 219, 221, 225, 226, 228, 249, 262, 265, 278, 287, 291, 299
COMMENTS
These are Heinz numbers of the integer partitions counted by A045931. A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The integers in the multiplicative subgroup of positive rational numbers generated by the products of two consecutive primes ( A006094). The sequence is closed under multiplication, prime shift ( A003961), and - where the result is an integer - under division. Using these closures, all the terms can be derived from the presence of 6. For example, A003961(6) = 15, A003961(15) = 35, 6 * 35 = 210, 210/15 = 14. Closed also under A297845, since A297845 can be defined using squaring, prime shift and multiplication. - Peter Munn, Oct 05 2020
EXAMPLE
The sequence of terms together with their prime indices begins:
1: {}
6: {1,2}
14: {1,4}
15: {2,3}
26: {1,6}
33: {2,5}
35: {3,4}
36: {1,1,2,2}
38: {1,8}
51: {2,7}
58: {1,10}
65: {3,6}
69: {2,9}
74: {1,12}
77: {4,5}
84: {1,1,2,4}
86: {1,14}
90: {1,2,2,3}
93: {2,11}
95: {3,8}
MATHEMATICA
Select[Range[100], Total[Cases[If[#==1, {}, FactorInteger[#]], {p_, k_}:>k*(-1)^PrimePi[p]]]==0&]
PROG
(PARI) is(n) = {my(v = vector(2), f = factor(n)); for(i = 1, #f~, v[1 + primepi(f[i, 1])%2]+=f[i, 2]); v[1] == v[2]} \\ David A. Corneth, Oct 06 2020 (Python)
from sympy import factorint, primepi
def ok(n):
v = [0, 0]
for p, e in factorint(n).items(): v[primepi(p)%2] += e
return v[0] == v[1]
CROSSREFS
Cf. A000712, A001222, A001405, A006094, A026010, A045931, A063886, A097613, A112798, A130780, A171966, A239241, A241638, A325700.
Numbers n such that n = product (p_k)^(c_k) and set of its (c_k k's)'s is a self-conjugate partition, where p_k is k-th prime and c_k > 0.
+10
61
1, 2, 6, 9, 20, 30, 56, 75, 84, 125, 176, 210, 264, 350, 416, 441, 624, 660, 735, 1088, 1100, 1386, 1560, 1632, 1715, 2310, 2401, 2432, 2600, 3267, 3276, 3648, 4080, 5390, 5445, 5460, 5888, 6800, 7546, 7722, 8568, 8832, 9120, 12705, 12740, 12870, 13689
COMMENTS
The Heinz numbers of the self-conjugate partitions. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] to be Product(p_j-th prime, j=1..r) (a concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 1, 4] we get 2*2*2*7 = 56. It is in the sequence since [1,1,1,4] is self-conjugate. - Emeric Deutsch, Jun 05 2015
EXAMPLE
20 is in the sequence because 20 = 2^2 * 5^1 = (p_1)^2 *(p_3)^1, (two 1's, one 3's) = (1,1,3) is a self-conjugate partition of 5.
The terms together with their prime indices begin:
1: ()
2: (1)
6: (2,1)
9: (2,2)
20: (3,1,1)
30: (3,2,1)
56: (4,1,1,1)
75: (3,3,2)
84: (4,2,1,1)
125: (3,3,3)
176: (5,1,1,1,1)
210: (4,3,2,1)
264: (5,2,1,1,1)
(End)
MAPLE
with(numtheory): c := proc (n) local B, C: B := proc (n) local pf: pf := op(2, ifactors(n)): [seq(seq(pi(op(1, op(i, pf))), j = 1 .. op(2, op(i, pf))), i = 1 .. nops(pf))] end proc: C := proc (P) local a: a := proc (j) local c, i: c := 0: for i to nops(P) do if j <= P[i] then c := c+1 else end if end do: c end proc: [seq(a(k), k = 1 .. max(P))] end proc: mul(ithprime(C(B(n))[q]), q = 1 .. nops(C(B(n)))) end proc: SC := {}: for i to 14000 do if c(i) = i then SC := `union`(SC, {i}) else end if end do: SC; # Emeric Deutsch, May 09 2015
MATHEMATICA
Select[Range[14000], Function[n, n == If[n == 1, 1, Module[{l = #, m = 0}, Times @@ Power @@@ Table[l -= m; l = DeleteCases[l, 0]; {Prime@ Length@ l, m = Min@ l}, Length@ Union@ l]] &@ Catenate[ConstantArray[PrimePi@ #1, #2] & @@@ FactorInteger@ n]]]] (* Michael De Vlieger, Aug 27 2016, after JungHwan Min at A122111 *)
CROSSREFS
A002110 (primorial numbers) is a subsequence.
After a(1) and a(2), a subsequence of A241913. These partitions are counted by A000700. These are the positions of zeros in A352491. A325039 counts partitions w/ product = conjugate product, ranked by A325040.
Heinz number (rank) and partition:
Number of even parts in the partition having Heinz number n.
+10
61
0, 0, 1, 0, 0, 1, 1, 0, 2, 0, 0, 1, 1, 1, 1, 0, 0, 2, 1, 0, 2, 0, 0, 1, 0, 1, 3, 1, 1, 1, 0, 0, 1, 0, 1, 2, 1, 1, 2, 0, 0, 2, 1, 0, 2, 0, 0, 1, 2, 0, 1, 1, 1, 3, 0, 1, 2, 1, 0, 1, 1, 0, 3, 0, 1, 1, 0, 0, 1, 1, 1, 2, 0, 1, 1, 1, 1, 2, 1, 0, 4, 0, 0, 2, 0, 1, 2
COMMENTS
We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436. In the Maple program the subprogram B yields the partition with Heinz number n.
REFERENCES
George E. Andrews and Kimmo Eriksson, Integer Partitions, Cambridge Univ. Press, Cambridge, 2004.
Miklós Bóna, A Walk Through Combinatorics, World Scientific Publishing Co., 2002.
FORMULA
Totally additive with a(p) = 1 if primepi(p) is even, and 0 otherwise.
EXAMPLE
a(18) = 2 because the partition having Heinz number 18 = 2*3*3 is [1,2,2], having 2 even parts.
MAPLE
with(numtheory): a := proc (n) local B, ct, q: B := proc (n) local nn, j, m: nn := op(2, ifactors(n)): for j to nops(nn) do m[j] := op(j, nn) end do: [seq(seq(pi(op(1, m[i])), q = 1 .. op(2, m[i])), i = 1 .. nops(nn))] end proc: ct := 0: for q to nops(B(n)) do if `mod`(B(n)[q], 2) = 0 then ct := ct+1 else end if end do: ct end proc: seq(a(n), n = 1 .. 135);
# second Maple program:
a:= n-> add(`if`(numtheory[pi](i[1])::even, i[2], 0), i=ifactors(n)[2]):
MATHEMATICA
a[n_] := Sum[If[PrimePi[i[[1]]] // EvenQ, i[[2]], 0], {i, FactorInteger[n]} ]; a[1] = 0; Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Dec 10 2016 after Alois P. Heinz *)
Number of partitions of n with equal number of even and odd parts.
+10
60
1, 0, 0, 1, 0, 2, 1, 3, 2, 5, 5, 7, 9, 11, 16, 18, 25, 28, 41, 44, 62, 70, 94, 107, 140, 163, 207, 245, 302, 361, 440, 527, 632, 763, 904, 1090, 1285, 1544, 1812, 2173, 2539, 3031, 3538, 4202, 4896, 5793, 6736, 7934, 9221, 10811, 12549, 14661, 16994, 19780
COMMENTS
The trivariate g.f. with x marking weight (i.e., sum of the parts), t marking number of odd parts and s marking number of even parts, is 1/product((1-tx^(2j-1))(1-sx^(2j)), j=1..infinity). - Emeric Deutsch, Mar 30 2006
FORMULA
G.f.: Sum_{k>=0} x^(3*k)/Product_{i=1..k} (1-x^(2*i))^2. - Vladeta Jovovic, Aug 18 2007
EXAMPLE
a(9) = 5 because we have [8,1], [7,2], [6,3], [5,4] and [2,2,2,1,1,1].
The a(0) = 1 through a(12) = 9 partitions (A = 10, empty columns indicated by dots):
() . . 21 . 32 2211 43 3221 54 3322 65 4332
41 52 4211 63 4321 74 4431
61 72 4411 83 5322
81 5221 92 5421
222111 6211 A1 6321
322211 6411
422111 7221
8211
22221111
(End)
MAPLE
g:=1/product((1-t*x^(2*j-1))*(1-s*x^(2*j)), j=1..30): gser:=simplify(series(g, x=0, 56)): P[0]:=1: for n from 1 to 53 do P[n]:=subs(s=1/t, coeff(gser, x^n)) od: seq(coeff(t*P[n], t), n=0..53); # Emeric Deutsch, Mar 30 2006
MATHEMATICA
p[n_] := p[n] = Select[IntegerPartitions[n], Count[#, _?OddQ] == Count[#, _?EvenQ] &]; t = Table[p[n], {n, 0, 10}] (* partitions of n with # odd parts = # even parts *)
TableForm[t] (* partitions, vertical format *)
Table[Length[p[n]], {n, 0, 30}] (* A045931 *)
CROSSREFS
The version for subsets of {1..n} is A001405. Dominated by A027187 (partitions of even length). This is column k = 0 of the triangle A240009. A half-conjugate version is A277579. These partitions are ranked by A325698.
Alternating bit sum for n: replace 2^k with (-1)^k in binary expansion of n.
+10
44
0, 1, -1, 0, 1, 2, 0, 1, -1, 0, -2, -1, 0, 1, -1, 0, 1, 2, 0, 1, 2, 3, 1, 2, 0, 1, -1, 0, 1, 2, 0, 1, -1, 0, -2, -1, 0, 1, -1, 0, -2, -1, -3, -2, -1, 0, -2, -1, 0, 1, -1, 0, 1, 2, 0, 1, -1, 0, -2, -1, 0, 1, -1, 0, 1, 2, 0, 1, 2, 3, 1, 2, 0, 1, -1, 0, 1, 2, 0, 1, 2, 3, 1, 2, 3, 4, 2, 3, 1, 2, 0, 1, 2, 3, 1, 2, 0, 1, -1, 0, 1, 2, 0, 1, -1, 0, -2
COMMENTS
Notation: (2)[n](-1)
a(n) == 0 (mod 3) iff n == 0 (mod 3).
a(n) == 0 (mod 6) iff (n == 0 (mod 3) and n/3 not in A036556). a(n) == 3 (mod 6) iff (n == 0 (mod 3) and n/3 in A036556). (End) First occurrence of k and -k: 0, 1, 2, 5, 10, 21, 42, 85, ..., ( A000975); i.e., first 0 occurs for 0, first 1 occurs for 1, first -1 occurs at 2, first 2 occurs for 5, etc.; a(n)=-3 only if n mod 3 = 0,
a(n)=-2 only if n mod 3 = 1,
a(n)=-1 only if n mod 3 = 2,
a(n)= 0 only if n mod 3 = 0,
a(n)= 1 only if n mod 3 = 1,
a(n)= 2 only if n mod 3 = 2,
a(n)= 3 only if n mod 3 = 0, ..., . (End)
In the Koch curve, number the segments starting with n=0 for the first segment. The net direction (i.e., the sum of the preceding turns) of segment n is a(n)*60 degrees. This is since in the curve each base-4 digit 0,1,2,3 of n is a sub-curve directed respectively 0, +60, -60, 0 degrees, which is the net 0,+1,-1,0 of two bits in the sum here. - Kevin Ryde, Jan 24 2020
FORMULA
G.f.: (1/(1-x)) * Sum_{k>=0} (-1)^k*x^2^k/(1+x^2^k). - Ralf Stephan, Mar 07 2003 a(0) = 0, a(2n) = -a(n), a(2n+1) = 1-a(n). - Ralf Stephan, Mar 07 2003 G.f. A(x) satisfies: A(x) = x / (1 - x^2) - (1 + x) * A(x^2). - Ilya Gutkovskiy, Jul 28 2021
EXAMPLE
Alternating bit sum for 11 = 1011 in binary is 1 - 1 + 0 - 1 = -1, so a(11) = -1.
MAPLE
A065359 := proc(n) local dgs ; dgs := convert(n, base, 2) ; add( -op(i, dgs)*(-1)^i, i=1..nops(dgs)) ; end proc: # R. J. Mathar, Feb 04 2011
MATHEMATICA
f[0]=0; f[n_] := Plus @@ (-(-1)^Range[ Floor[ Log2@ n + 1]] Reverse@ IntegerDigits[n, 2]); Array[ f, 107, 0]
PROG
(PARI)
SumAD(x)= { local(a=1, s=0); while (x>9, s+=a*(x-10*(x\10)); x\=10; a=-a); return(s + a*x) }
baseE(x, b)= { local(d, e=0, f=1); while (x>0, d=x-b*(x\b); x\=b; e+=d*f; f*=10); return(e) }
{ for (n=0, 1000, b=baseE(n, 2); write("b065359.txt", n, " ", SumAD(b)) ) } \\ Harry J. Smith, Oct 17 2009 (PARI) for(n=0, 106, s=0; u=1; for(k=0, #binary(n)-1, s+=bittest(n, k)*u; u=-u); print1(s, ", ")) /* Washington Bomfim, Jan 18 2011 */ (PARI) a(n) = my(b=binary(n)); b*[(-1)^k|k<-[-#b+1..0]]~; \\ Ruud H.G. van Tol, Oct 16 2023 (Haskell)
a065359 0 = 0
a065359 n = - a065359 n' + m where (n', m) = divMod n 2
(Python)
def a(n):
return sum((-1)**k for k, bi in enumerate(bin(n)[2:][::-1]) if bi=='1')
(Python)
from sympy.ntheory import digits
def A065359(n): return sum((0, 1, -1, 0)[i] for i in digits(n, 4)[1:]) # Chai Wah Wu, Jul 19 2024
Integers in the multiplicative subgroup of positive rationals generated by the products of two consecutive primes and the cubes of primes. Numbers k for which A048675(k) is a multiple of three.
+10
25
1, 6, 8, 14, 15, 20, 26, 27, 33, 35, 36, 38, 44, 48, 50, 51, 58, 63, 64, 65, 68, 69, 74, 77, 84, 86, 90, 92, 93, 95, 106, 110, 112, 117, 119, 120, 122, 123, 124, 125, 141, 142, 143, 145, 147, 156, 158, 160, 161, 162, 164, 170, 171, 177, 178, 185, 188, 196, 198, 201, 202, 208, 209, 210, 214, 215, 216, 217, 219, 221, 225
COMMENTS
The positive integers are partitioned between this sequence, A332821 and A332822, which list the integers in respective cosets of the subgroup. As the sequence lists the integers in a multiplicative subgroup of the positive rationals, the sequence is closed under multiplication and, provided the result is an integer, under division.
It follows that for any n in this sequence, all powers n^k are present (k >= 0), as are all cubes.
If we take each odd term of this sequence and replace each prime in its factorization by the next smaller prime, the resulting numbers are a permutation of the full sequence; and if we take the square root of each square term we get the full sequence.
There are no primes in the sequence, therefore if k is present and p is a prime, k*p and k/p are absent (noting that k/p might not be an integer). This property extends from primes to all terms of A050376 (often called Fermi-Dirac primes), therefore to squares of primes, 4th powers of primes etc. The terms are the even numbers in A332821 halved. The terms are also the numbers m such that 5m is in A332821, and so on for alternate primes: 11, 17, 23 etc. Likewise, the terms are the numbers m such that 3m is in A332822, and so on for alternate primes: 7, 13, 19 etc. The numbers that are half of the even terms of this sequence are in A332822, which consists exactly of those numbers. The numbers that are one third of the terms that are multiples of 3 are in A332821, which consists exactly of those numbers. These properties extend in a pattern of alternating primes as described in the previous paragraph. If k is an even number, exactly one of {k/2, k, 2k} is in the sequence (cf. A191257 / A067368 / A213258); and generally if k is a multiple of a prime p, exactly one of {k/p, k, k*p} is in the sequence. If m and n are in this sequence then so is m*n (the definition of "multiplicative semigroup"), while if n is in this sequence, and x is in the complement A359830, then n*x is in A359830. This essentially follows from the fact that A048675 is totally additive sequence. Compare to A329609. - Antti Karttunen, Jan 17 2023
FORMULA
{a(n) : n >= 1} = {1} U {2 * A332822(k) : k >= 1} U { A003961(a(k)) : k >= 1}. {a(n) : n >= 1} = {1} U {a(k)^2 : k >= 1} U { A331590(2, A332822(k)) : k >= 1}. {a(n) : n >= 1} = {k : k >= 1, 3| A048675(k)}. {a(n) : n >= 1} = {k : k >= 1, 3| A195017(k)}. (End)
MATHEMATICA
Select[Range@ 225, Or[Mod[Total@ #, 3] == 0 &@ Map[#[[-1]]*2^(PrimePi@ #[[1]] - 1) &, FactorInteger[#]], # == 1] &] (* Michael De Vlieger, Mar 15 2020 *)
PROG
(PARI) isA332820(n) = { my(f = factor(n)); !((sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2)%3); };
CROSSREFS
Positions of zeros in A332823; equivalently, numbers in row 3k of A277905 for some k >= 0.
Encoded multiplication table for polynomials in one indeterminate with nonnegative integer coefficients. Symmetric square array T(n, k) read by antidiagonals, n > 0 and k > 0. See comment for details.
+10
24
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 4, 1, 1, 5, 9, 9, 5, 1, 1, 6, 7, 16, 7, 6, 1, 1, 7, 15, 25, 25, 15, 7, 1, 1, 8, 11, 36, 11, 36, 11, 8, 1, 1, 9, 27, 49, 35, 35, 49, 27, 9, 1, 1, 10, 25, 64, 13, 90, 13, 64, 25, 10, 1, 1, 11, 21, 81, 125, 77, 77, 125, 81
COMMENTS
For any number n > 0, let f(n) be the polynomial in a single indeterminate x where the coefficient of x^e is the prime(1+e)-adic valuation of n (where prime(k) denotes the k-th prime); f establishes a bijection between the positive numbers and the polynomials in a single indeterminate x with nonnegative integer coefficients; let g be the inverse of f; T(n, k) = g(f(n) * f(k)).
This table has many similarities with A248601. For any n > 0 and m > 0, f(n * m) = f(n) + f(m).
Also, f(1) = 0 and f(2) = 1.
The function f can be naturally extended to the set of positive rational numbers: if r = u/v (not necessarily in reduced form), then f(r) = f(u) - f(v); as such, f is a homomorphism from the multiplicative group of positive rational numbers to the additive group of polynomials of a single indeterminate x with integer coefficients.
See A297473 for the main diagonal of T. As a binary operation, T(.,.) is related to A306697(.,.) and A329329(.,.). When their operands are terms of A050376 (sometimes called Fermi-Dirac primes) the three operations give the same result. However the rest of the multiplication table for T(.,.) can be derived from these results because T(.,.) distributes over integer multiplication ( A003991), whereas for A306697 and A329329, the equivalent derivation uses distribution over A059896(.,.) and A059897(.,.) respectively. - Peter Munn, Mar 25 2020 The operation defined by this sequence can be extended to be the multiplicative operator of a ring over the positive rationals that is isomorphic to the polynomial ring Z[x]. The extended function f (described in the author's original comments) is the isomorphism we use, and it has the same relationship with the extended operation that exists between their unextended equivalents.
Denoting this extension of T(.,.) as t_Q(.,.), we get t_Q(n, 1/k) = t_Q(1/n, k) = 1/T(n, k) and t_Q(1/n, 1/k) = T(n, k) for positive integers n and k. The result for other rationals is derived from the distributive property: t_Q(q, r*s) = t_Q(q, r) * t_Q(q, s); t_Q(q*r, s) = t_Q(q, s) * t_Q(r, s). This may look unusual because standard multiplication of rational numbers takes on the role of the ring's additive group.
There are many OEIS sequences that can be shown to be a list of the integers in an ideal of this ring. See the cross-references.
There are some completely additive sequences that similarly define by extension completely additive functions on the positive rationals that can be shown to be homomorphisms from this ring onto the integer ring Z, and these functions relate to some of the ideals. For example, the extended function of A048675, denoted A048675_Q, maps i/j to A048675(i) - A048675(j) for positive integers i and j. For any positive integer k, the set {r rational > 0 : k divides A048675_Q(r)} forms an ideal of the ring; for k=2 and k=3 the integers in this ideal are listed in A003159 and A332820 respectively. (End)
LINKS
Eric Weisstein's World of Mathematics, Ring.
FORMULA
T is completely multiplicative in both parameters:
- for any n > 0
- and k > 0 with prime factorization Prod_{i > 0} prime(i)^e_i:
- T(prime(n), k) = T(k, prime(n)) = Prod_{i > 0} prime(n + i - 1)^e_i.
For any m > 0, n > 0 and k > 0:
- T(n, k) = T(k, n) (T is commutative),
- T(m, T(n, k)) = T(T(m, n), k) (T is associative),
- T(n, 1) = 1 (1 is an absorbing element for T),
- T(n, 2) = n (2 is an identity element for T),
- T(n, 2^i) = n^i for any i >= 0,
- T(n, 3^i) = A003961(n)^i for any i >= 0, From Peter Munn, Mar 13 2020 and Apr 20 2021: (Start) T(n, m*k) = T(n, m) * T(n, k); T(n*m, k) = T(n, k) * T(m, k) (T distributes over multiplication).
(End)
EXAMPLE
Array T(n, k) begins:
n\k| 1 2 3 4 5 6 7 8 9 10
---+------------------------------------------------
3| 1 3 5 9 7 15 11 27 25 21 -> A003961 4| 1 4 9 16 25 36 49 64 81 100 -> A000290 5| 1 5 7 25 11 35 13 125 49 55 -> A357852 6| 1 6 15 36 35 90 77 216 225 210 -> A191002 7| 1 7 11 49 13 77 17 343 121 91
8| 1 8 27 64 125 216 343 512 729 1000 -> A000578 9| 1 9 25 81 49 225 121 729 625 441
10| 1 10 21 100 55 210 91 1000 441 550
The encoding, n, of polynomials, f(n), that is used for the table is further described in A206284. Examples of encoded polynomials: n f(n) n f(n)
1 0 16 4
2 1 17 x^6
3 x 21 x^3 + x
4 2 25 2x^2
5 x^2 27 3x
6 x + 1 35 x^3 + x^2
7 x^3 36 2x + 2
8 3 49 2x^3
9 2x 55 x^4 + x^2
10 x^2 + 1 64 6
11 x^4 77 x^4 + x^3
12 x + 2 81 4x
13 x^5 90 x^2 + 2x + 1
15 x^2 + x 91 x^5 + x^3
(End)
PROG
(PARI) T(n, k) = my (f=factor(n), p=apply(primepi, f[, 1]~), g=factor(k), q=apply(primepi, g[, 1]~)); prod (i=1, #p, prod(j=1, #q, prime(p[i]+q[j]-1)^(f[i, 2]*g[j, 2])))
CROSSREFS
Integers in the ideal of the related ring (see Jun 2021 comment) generated by S: S={3}: A005408, S={4}: A000290\{0}, S={4,3}: A003159, S={5}: A007310, S={5,4}: A339690, S={6}: A325698, S={6,4}: A028260, S={7}: A007775, S={8}: A000578\{0}, S={8,3}: A191257, S={8,6}: A332820, S={9}: A016754, S={10,4}: A340784, S={11}: A008364, S={12,8}: A145784, S={13}: A008365, S={15,4}: A345452, S={15,9}: A046337, S={16}: A000583\{0}, S={17}: A008366. Equivalent sequence for polynomial composition: A326376.
Heinz numbers of non-self-conjugate integer partitions.
+10
24
3, 4, 5, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73
COMMENTS
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions. The sequence lists all Heinz numbers of partitions whose Heinz number is different from that of their conjugate.
EXAMPLE
The terms together with their prime indices begin:
3: (2)
4: (1,1)
5: (3)
7: (4)
8: (1,1,1)
10: (3,1)
11: (5)
12: (2,1,1)
13: (6)
14: (4,1)
15: (3,2)
16: (1,1,1,1)
17: (7)
18: (2,2,1)
For example, the self-conjugate partition (4,3,3,1) has Heinz number 350, so 350 is not in the sequence.
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
conj[y_]:=If[Length[y0]==0, y, Table[Length[Select[y, #>=k&]], {k, 1, Max[y]}]];
Select[Range[100], #!=Times@@Prime/@conj[primeMS[#]]&]
CROSSREFS
These partitions are counted by A330644. These are the positions of nonzero terms in A352491. A098825 counts permutations by unfixed points.
A325039 counts partitions w/ same product as conjugate, ranked by A325040.
A352523 counts compositions by unfixed points, rank statistic A352513.
Heinz number (rank) and partition:
- A122111 = rank of conjugate partition Cf. A000720, A026424, A120383, A175508, A195017, A238745, A301987, A304360, A316524, A324846, A350841.
Number of even parts in the conjugate of the integer partition with Heinz number n.
+10
23
0, 0, 0, 1, 0, 1, 0, 0, 2, 1, 0, 0, 0, 1, 2, 1, 0, 1, 0, 0, 2, 1, 0, 1, 3, 1, 0, 0, 0, 1, 0, 0, 2, 1, 3, 2, 0, 1, 2, 1, 0, 1, 0, 0, 0, 1, 0, 0, 4, 2, 2, 0, 0, 1, 3, 1, 2, 1, 0, 2, 0, 1, 0, 1, 3, 1, 0, 0, 2, 2, 0, 1, 0, 1, 1, 0, 4, 1, 0, 0, 2, 1, 0, 2, 3, 1, 2
COMMENTS
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so a(n) counts even prime indices of n.
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
conj[y_]:=If[Length[y]==0, y, Table[Length[Select[y, #>=k&]], {k, 1, Max[y]}]];
Table[Count[conj[primeMS[n]], _?EvenQ], {n, 100}]
CROSSREFS
Positions of first appearances are A001248. Subtracting from the number of odd conjugate parts gives A350941. Subtracting from the number of odd parts gives A350942. Subtracting from the number of even parts gives A350950. There are four statistics:
There are six possible pairings of statistics:
- A349157: # of even parts = # of odd conjugate parts, counted by A277579. - A350848: # of even conj parts = # of odd conj parts, counted by A045931. - A350943: # of even conjugate parts = # of odd parts, counted by A277579. - A350944: # of odd parts = # of odd conjugate parts, counted by A277103. - A350945: # of even parts = # of even conjugate parts, counted by A350948. There are three possible double-pairings of statistics:
A122111 represents partition conjugation using Heinz numbers.
Cf. A028260, A130780, A171966, A195017, A236559, A239241, A241638, A316524, A325700, A350849, A350951.
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