A000469 -id:A000469 - cp's OEIS frontend

A361321 Lexicographically earliest infinite sequence of distinct elements of A000469 such that, for n > 2, a(n) has a common factor with a(n-1) but not with a(n-2).

1, 6, 10, 35, 21, 33, 22, 14, 91, 39, 15, 55, 77, 42, 26, 65, 85, 34, 38, 57, 51, 119, 70, 30, 69, 161, 133, 95, 110, 46, 299, 143, 66, 58, 145, 105, 78, 62, 155, 115, 138, 74, 185, 165, 87, 203, 154, 82, 123, 93, 217, 182, 86, 129, 111, 259, 238, 94, 141, 159

Offset: 1

Scott R. Shannon, Rémy Sigrist and N. J. A. Sloane, Mar 09 2023

nonn

This sequence is a variant of A360519 where we only consider nonprime squarefree numbers (A000469).
Theorem: a(1) = 1, a(2) = 6; thereafter, a(n) is the smallest nonprime squarefree number m not yet in the sequence such that
(i) gcd(m, a(n-1)) > 1,
(ii) gcd(m, a(n-2)) = 1, and
(iii) m does not divide a(n-1).
Conjecture: The sequence is a permutation of A000469.

Scott R. Shannon, Table of n, a(n) for n = 1..100000
Scott R. Shannon, White on black graph of first 50000 terms [The green line is x = y]
Scott R. Shannon, Image of the first 500000 terms in color. The terms with a lowest prime factor of 2,3,5,7,9,11,13,17,19,>=23 are colored white, red, orange, yellow, green, blue, indigo, violet, gray respectively.
Rémy Sigrist, PARI program

Cf. A000469, A336957, A360519.

PARI
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See Links section.
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A239508 Number of partitions of n into nonprime squarefree numbers, cf. A000469.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 5, 6, 7, 7, 8, 8, 10, 12, 14, 14, 16, 17, 20, 22, 25, 26, 31, 33, 37, 40, 45, 49, 57, 60, 66, 71, 80, 86, 98, 104, 115, 125, 138, 147, 164, 175, 193, 209, 230, 244, 269, 289, 318, 343, 374, 398, 437, 468, 510, 548

Offset: 0

Reinhard Zumkeller, Mar 21 2014

nonn

			a(10) = #{10, 6+1+1+1+1, 10x1} = 3;
a(11) = #{10+1, 6+1+1+1+1+1, 11x1} = 3;
a(12) = #{10+1+1, 6+6, 6+6x1, 12x1} = 4;
a(13) = #{10+1+1+1, 6+6+1, 6+7x1, 13x1} = 4;
a(14) = #{14, 10+1+1+1+1, 6+6+1+1, 6+8x1, 14x1} = 5;
a(15) = #{15, 14+1, 10+5x1, 6+6+1+1+1, 6+9x1, 15x1} = 6;
a(16) = #{15+1, 14+1+1, 10+6, 10+6x1, 6+6+4x1, 6+10x1, 16x1} = 7.

Cf. A239509, A073576.

a239508 = p a000469_list where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m

A239509 Number of partitions of n into distinct nonprime squarefree numbers, cf. A000469.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 3, 3, 1, 1, 2, 2, 2, 2, 2, 3, 4, 4, 3, 2, 4, 7, 6, 4, 5, 6, 6, 7, 7, 6, 8, 10, 9, 9, 10, 10, 12, 13, 12, 13, 15, 16, 18, 18, 16, 17, 21, 23, 23, 23, 25, 28, 29, 29, 31, 34, 37, 41, 40, 38, 42, 46

Offset: 0

Reinhard Zumkeller, Mar 21 2014

nonn

			a(30) = #{30, 15+14+1, 14+10+6} = 3;
a(31) = #{30+1, 21+10, 15+10+6, 14+10+6+1} = 4;
a(32) = #{26+6, 22+10, 21+10+1, 15+10+6+1} = 4;
a(33) = #{33, 26+6+1, 22+10+1} = 3;
a(34) = #{34, 33+1} = 2;
a(35) = #{35, 34+1, 21+14, 15+14+6} = 4;
a(36) = #{35+1, 30+6, 26+10, 22+14, 21+15, 21+14+1, 15+14+6+1} = 7.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A239508, A087188.

a239509 = p a000469_list where
   p _      0 = 1
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m

b:= proc(n, i) option remember; `if`(i*(i+1)/2 b(n$2):
seq(a(n), n=0..100);  # Alois P. Heinz, Jun 02 2015

b[n_, i_] := b[n, i] = If[i*(i+1)/2Jean-François Alcover, Jan 15 2016, after Alois P. Heinz *)

A361323 a(n) = k such that A000469(k) = A361321(n).

Original entry on oeis.org

1, 2, 3, 12, 6, 10, 7, 4, 33, 14, 5, 18, 27, 15, 8, 22, 30, 11, 13, 19, 17, 45, 25, 9, 24, 62, 50, 36, 40, 16, 121, 55, 23, 20, 56, 38, 28, 21, 59, 43, 52, 26, 71, 63, 32, 79, 58, 29, 47, 34, 87, 69, 31, 48, 41, 105, 97, 35, 53, 61, 39, 44, 67, 70, 46, 49, 76, 77, 51, 54, 84, 89, 57, 60, 96

Offset: 1

Scott R. Shannon, Rémy Sigrist, and N. J. A. Sloane, Mar 11 2023

nonn

Conjectured to be a permutation of the natural numbers (if not then there exists a k such that A000469(k) does not appear in A361321).

Scott R. Shannon, Table of n, a(n) for n = 1..10000.

Cf. A361321, A000469, A361324, A336957, A360519, A361102.

A361324 a(n) = k such that A361321(k) = A000469(n), or -1 if A000469(n) never appears in A361321.

Original entry on oeis.org

1, 2, 3, 8, 11, 5, 7, 15, 24, 6, 18, 4, 19, 10, 14, 30, 21, 12, 20, 34, 38, 16, 33, 25, 23, 42, 13, 37, 48, 17, 53, 45, 9, 50, 58, 28, 83, 36, 61, 29, 55, 79, 40, 62, 22, 65, 49, 54, 66, 27, 69, 41, 59, 70, 32, 35, 73, 47, 39, 74, 60, 26, 44, 87, 78, 93, 63, 98, 52, 64, 43, 101, 77, 86, 102

Offset: 1

Scott R. Shannon, Rémy Sigrist, and N. J. A. Sloane, Mar 11 2023

nonn

Conjectured to be a permutation of the natural numbers (and if so, -1 will never appear).

Scott R. Shannon, Table of n, a(n) for n = 1..10000.

Cf. A361321, A000469, A361323, A336957, A360519, A361102.

A287299 Number of ways of writing n as a sum of a proper prime power (A246547) and a nonprime squarefree number (A000469).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 0, 2, 1, 0, 1, 2, 2, 0, 0, 2, 2, 1, 1, 3, 0, 1, 1, 4, 3, 0, 2, 2, 2, 0, 3, 4, 3, 1, 2, 6, 3, 1, 0, 5, 4, 2, 2, 4, 3, 0, 2, 3, 5, 0, 1, 3, 4, 3, 2, 4, 3, 3, 4, 5, 4, 0, 2, 5, 5, 0, 4, 6, 2, 1, 1, 7, 3, 1, 2, 7, 4, 2, 4, 5, 5, 1, 3, 6, 5, 1, 3, 6, 6, 3, 4, 4, 4, 2, 4, 7, 6, 3, 1, 4, 4, 0, 4, 6, 5, 2, 2, 7, 5, 2, 1, 7, 8, 4

Offset: 0

Ilya Gutkovskiy, May 23 2017

nonn

Conjecture: a(n) > 0 for all n > 108.

			a(26) = 3 because we have [25, 1], [22, 4] and [16, 10].

Ilya Gutkovskiy, Extended graphical example
Eric Weisstein's World of Mathematics, Prime Power
Eric Weisstein's World of Mathematics, Squarefree

Cf. A000469, A098983, A246547, A282192, A282290, A282318, A282947.

nmax = 120; CoefficientList[Series[(Sum[Boole[SquareFreeQ[k] && ! PrimeQ[k]] x^k, {k, 1, nmax}]) (Sum[Boole[PrimePowerQ[k] && ! PrimeQ[k]] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

x='x+O('x^120); concat([0, 0, 0, 0, 0], Vec(sum(k=1, 120, (issquarefree(k) && !isprime(k))*x^k) * sum(k=1, 120, (isprimepower(k) && !isprime(k))*x^k))) \\ Indranil Ghosh, May 23 2017

G.f.: (Sum_{k>=1} x^A246547(k))*(Sum_{k>=1} x^A000469(k)).

A290136 Positive numbers that are not the sum of two nonprime squarefree numbers (A000469).

Original entry on oeis.org

1, 3, 4, 5, 6, 8, 9, 10, 13, 14, 17, 18, 19, 26, 33, 38, 46, 62, 82

Offset: 1

Ilya Gutkovskiy, Jul 20 2017

The sequence is conjectured to be complete.

Cf. A000469, A014092, A071068, A071331, A098235, A239508, A239509.

nmax = 82; f[x_] := Sum[Boole[SquareFreeQ[k] && PrimeNu[k] != 1] x^k, {k, 1, nmax}]^2; b = Exponent[#, x] & /@ List @@ Normal[Series[f[x], {x, 0, nmax}]]; c = Complement[Range[nmax], b][[1 ;; 19]]

A290137 Number of compositions (ordered partitions) of n into nonprime squarefree parts (A000469).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 7, 9, 12, 16, 22, 30, 41, 55, 73, 96, 128, 173, 235, 317, 426, 570, 763, 1023, 1375, 1848, 2484, 3337, 4482, 6017, 8077, 10843, 14562, 19560, 26276, 35292, 47392, 63632, 85443, 114741, 154098, 206957, 277941, 373254, 501244, 673121, 903945, 1213935, 1630246, 2189330

Offset: 0

Ilya Gutkovskiy, Jul 20 2017

nonn

			a(8) = 4 because we have [6, 1, 1], [1, 6, 1], [1, 1, 6] and [1, 1, 1, 1, 1, 1, 1, 1].

Index entries for sequences related to compositions

Cf. A000469, A239508, A239509, A280194.

nmax = 53; CoefficientList[Series[1/(1 - Sum[Boole[SquareFreeQ[k] && PrimeNu[k] != 1] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 - Sum_{k>=1} x^A000469(k)).

A290397 Least number of nonprime squarefree numbers (A000469) that add up to n.

Original entry on oeis.org

1, 2, 3, 4, 5, 1, 2, 3, 4, 1, 2, 2, 3, 1, 1, 2, 3, 3, 4, 2, 1, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 1, 2

Offset: 1

Ilya Gutkovskiy, Jul 29 2017

nonn

It is conjectured that a(n) <= 5.

			a(6) = 1 because 6 is already nonprime squarefree number.
a(7) = 2 because 7 = 6 + 1 is a partition of 7 into 2 nonprime squarefree parts and there is no such partition with fewer terms.

Cf. A000469, A051034, A225926, A239508, A290136.

A120944 Composite squarefree numbers.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 38, 39, 42, 46, 51, 55, 57, 58, 62, 65, 66, 69, 70, 74, 77, 78, 82, 85, 86, 87, 91, 93, 94, 95, 102, 105, 106, 110, 111, 114, 115, 118, 119, 122, 123, 129, 130, 133, 134, 138, 141, 142, 143, 145, 146, 154, 155, 158, 159, 161

Offset: 1

Zak Seidov, Aug 19 2006

nonn

Intersection of A002808 and A005117: n > 1 such that A008966(n) * (1-A010051(n)) = 1. - Reinhard Zumkeller, Dec 19 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000469 (Nonprime squarefree numbers).
Cf. A001221, A001222, A002808, A005117, A008966, A010051.
Set of powers: A182853.

a120944 n = a120944_list !! (n-1)
a120944_list = filter ((== 1) . a008966) a002808_list
-- Reinhard Zumkeller, Dec 19 2011

[n: n in [6..161] | IsSquarefree(n) and not IsPrime(n)]; // Bruno Berselli, Mar 03 2011

select(not(isprime) and numtheory:-issqrfree, [$2..1000]); # Robert Israel, Jul 07 2015

lst={};Do[If[SquareFreeQ[n],If[ !PrimeQ[n],AppendTo[lst,n]]],{n,2,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 20 2009; updated by Jean-François Alcover, Jun 19 2013 *)
Select[Range[200], PrimeNu[#] > 1 && SquareFreeQ[#] &] (* Carlos Eduardo Olivieri, Jul 07 2015 *)

is(n)=issquarefree(n)&&!isprime(n)&&n>1 \\ Charles R Greathouse IV, Apr 11 2012

from sympy import factorint
def ok(n): f = factorint(n); return len(f) > 1 and all(f[p] < 2 for p in f)
print(list(filter(ok, range(1, 162)))) # Michael S. Branicky, Jun 10 2021

from math import isqrt
from sympy import primepi, mobius
def A120944(n):
    def f(x): return n+1+primepi(x)+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n+1, f(n+1)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 02 2024

From Enrique Pérez Herrero, Apr 01 2012: (Start)
Solutions to floor(omega(x)/bigomega(x))*(1-floor(1/bigomega(x))) = 1, where bigomega is A001222 and omega is A001221.
Sum_{n>=1} 1/a(n)^s  = zeta(s)/zeta(2s) - 1 - PrimeZeta(s). (End)
a(n) = kn + O(n/log n) where k = Pi^2/6. - Charles R Greathouse IV, Aug 02 2024

cp's OEIS Frontend

A361321 Lexicographically earliest infinite sequence of distinct elements of A000469 such that, for n > 2, a(n) has a common factor with a(n-1) but not with a(n-2).

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PARI

A239508 Number of partitions of n into nonprime squarefree numbers, cf. A000469.

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Haskell

A239509 Number of partitions of n into distinct nonprime squarefree numbers, cf. A000469.

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A361323 a(n) = k such that A000469(k) = A361321(n).

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A361324 a(n) = k such that A361321(k) = A000469(n), or -1 if A000469(n) never appears in A361321.

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A287299 Number of ways of writing n as a sum of a proper prime power (A246547) and a nonprime squarefree number (A000469).

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Mathematica

PARI

Formula

A290136 Positive numbers that are not the sum of two nonprime squarefree numbers (A000469).

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Mathematica

A290137 Number of compositions (ordered partitions) of n into nonprime squarefree parts (A000469).

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Mathematica

Formula

A290397 Least number of nonprime squarefree numbers (A000469) that add up to n.

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