A318720 -id:A318720 - cp's OEIS frontend

A318715 Number of strict integer partitions of n with relatively prime parts in which no two parts are relatively prime.

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 1, 0, 4, 0, 3, 0, 1, 0, 5, 0, 8, 0, 2, 0, 5, 0, 10, 0, 4, 0, 13, 0, 15, 0, 3, 1, 13, 0, 19, 0, 9, 1, 24, 0, 20

Offset: 1

Gus Wiseman, Sep 02 2018

nonn

			The a(67) = 10 strict integer partitions are
  (45,12,10) (42,15,10) (40,15,12) (33,22,12) (28,21,18)
  (36,15,10,6) (30,15,12,10) (28,21,12,6) (24,18,15,10)
  (24,15,12,10,6).

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..500

Cf. A007359, A051185, A078374, A303140, A303282, A303283, A305713, A305843, A305854, A318716, A318717, A318720.

Table[Length[Select[IntegerPartitions[n],And[UnsameQ@@#,GCD@@#==1,And@@(GCD[##]>1&)@@@Select[Tuples[#,2],Less@@#&]]&]],{n,50}]

a(71)-a(85) from Robert Price, Sep 08 2018

A036785 Numbers divisible by the squares of two distinct primes.

Original entry on oeis.org

36, 72, 100, 108, 144, 180, 196, 200, 216, 225, 252, 288, 300, 324, 360, 392, 396, 400, 432, 441, 450, 468, 484, 500, 504, 540, 576, 588, 600, 612, 648, 675, 676, 684, 700, 720, 756, 784, 792, 800, 828, 864, 882, 900, 936, 968, 972, 980, 1000, 1008, 1044

Offset: 1

Alford Arnold

Not squarefree, not a nontrivial prime power and not in {squarefree} times {nontrivial prime powers}.

Numbers k such that A056170(k) > 1. The asymptotic density of this sequence is 1 - (6/Pi^2) * (1 + A154945) = 0.05668359058... - Amiram Eldar, Nov 01 2020

CRC Standard Mathematical Tables and Formulae, 30th ed., (1996) page 102-105.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A005117, A025475, A056170, A154945.
Equivalent sequence for 3 distinct primes: A318720.
Cf. A085986, A338539, A339245 (subsequences).
Subsequence of A038838.

Select[Range@ 1050, And[Length@ # > 1, Total@ Boole@ Map[# > 1 &, #[[All, -1]]] > 1] &@ FactorInteger@ # &] (* Michael De Vlieger, Apr 25 2017 *)
dstdpQ[n_]:=Length[Select[Sqrt[#]&/@Divisors[n],PrimeQ]]>1; Select[ Range[ 1100],dstdpQ] (* Harvey P. Dale, Jan 15 2020 *)

is(n)=my(f=vecsort(factor(n)[,2],,4));#f>1&&f[2]>1 \\ Charles R Greathouse IV, Nov 15 2012

More terms from Larry Reeves (larryr(AT)acm.org), Apr 03 2000

New name from Charles R Greathouse IV, Nov 15 2012

A318721 Number of strict relatively prime factorizations of n.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 0, 1, 0, 4, 0, 0, 1, 1, 1, 2, 0, 1, 1, 2, 0, 4, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 1, 2, 1, 1, 0, 6, 0, 1, 1, 0, 1, 4, 0, 1, 1, 4, 0, 4, 0, 1, 1, 1, 1, 4, 0, 2, 0, 1, 0, 6, 1, 1, 1

Offset: 1

Gus Wiseman, Sep 02 2018

nonn

a(1) could be either 0 or 1.

			The a(60) = 6 factorizations are (3*20), (4*15), (5*12), (2*3*10), (2*5*6), (3*4*5).

Cf. A001055, A007716, A045778, A078374, A281116, A317748, A318720.

strfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[strfacs[n/d],Min@@#1>d&],{d,Rest[Divisors[n]]}]];
Table[Length[Select[strfacs[n],GCD@@#==1&]],{n,50}]

A338540 Numbers having exactly three non-unitary prime factors.

Original entry on oeis.org

900, 1764, 1800, 2700, 3528, 3600, 4356, 4500, 4900, 5292, 5400, 6084, 6300, 7056, 7200, 8100, 8712, 8820, 9000, 9800, 9900, 10404, 10584, 10800, 11025, 11700, 12100, 12168, 12348, 12600, 12996, 13068, 13500, 14112, 14400, 14700, 15300, 15876, 16200, 16900, 17100

Offset: 1

Amiram Eldar, Nov 01 2020

nonn

Numbers k such that A056170(k) = A001221(A057521(k)) = 3.
Numbers divisible by the squares of exactly three distinct primes.
Subsequence of A318720 and first differs from it at n = 123.
The asymptotic density of this sequence is (eta_1^3 - 3*eta_1*eta_2 + 2*eta_3)/Pi^2 = 0.0032920755..., where eta_j = Sum_{p prime} 1/(p^2-1)^j (Pomerance and Schinzel, 2011).

			900 = 2^2 * 3^2 * 5^2 is a term since it has exactly 3 prime factors, 2, 3 and 5, that are non-unitary.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Carl Pomerance and Andrzej Schinzel, Multiplicative Properties of Sets of Residues, Moscow Journal of Combinatorics and Number Theory, Vol. 1, No. 1 (2011), pp. 52-66. See pp. 61-62.

Subsequence of A013929, A318720 and A327877.
Cf. A001221, A056170, A057521, A190641, A338539, A338541, A338542.
Cf. A154945 (eta_1), A324833 (eta_2), A324834 (eta_3).

Select[Range[17000], Count[FactorInteger[#][[;;,2]], _?(#1 > 1 &)] == 3 &]

A338541 Numbers having exactly four non-unitary prime factors.

Original entry on oeis.org

44100, 88200, 108900, 132300, 152100, 176400, 213444, 217800, 220500, 260100, 264600, 298116, 304200, 308700, 324900, 326700, 352800, 396900, 426888, 435600, 441000, 456300, 476100, 485100, 509796, 520200, 529200, 544500, 573300, 592900, 596232, 608400, 617400

Offset: 1

Amiram Eldar, Nov 01 2020

nonn

Numbers k such that A056170(k) = A001221(A057521(k)) = 4.
Numbers divisible by the squares of exactly four distinct primes.
The asymptotic density of this sequence is (eta_1^4 - 6*eta_1^2*eta_2 + 3*eta_2^2 + 8*eta_1*eta_3 - 6*eta_4)/(4*Pi^2) = 0.0000970457..., where eta_j = Sum_{p prime} 1/(p^2-1)^j (Pomerance and Schinzel, 2011).

			44100 = 2^2 * 3^2 * 5^2 * 7^2 is a term since it has exactly 4 prime factors, 2, 3, 5 and 7, that are non-unitary.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Carl Pomerance and Andrzej Schinzel, Multiplicative Properties of Sets of Residues, Moscow Journal of Combinatorics and Number Theory, Vol. 1, No. 1 (2011), pp. 52-66. See pp. 61-62.

Subsequence of A013929 and A318720.
Cf. A001221, A056170, A057521, A190641, A338539, A338540, A338542.
Cf. A154945 (eta_1), A324833 (eta_2), A324834 (eta_3), A324835 (eta_4).

Select[Range[620000], Count[FactorInteger[#][[;;,2]], _?(#1 > 1 &)] == 4 &]

A338542 Numbers having exactly five non-unitary prime factors.

Original entry on oeis.org

5336100, 7452900, 10672200, 12744900, 14905800, 15920100, 16008300, 18404100, 21344400, 22358700, 23328900, 25489800, 26680500, 29811600, 31472100, 31840200, 32016600, 36072036, 36808200, 37088100, 37264500, 37352700, 38234700, 39312900, 42380100, 42688800, 43956900

Offset: 1

Amiram Eldar, Nov 01 2020

nonn

Numbers k such that A056170(k) = A001221(A057521(k)) = 5.
Numbers divisible by the squares of exactly five distinct primes.
The asymptotic density of this sequence is (eta_1^5 - 10*eta_1^3*eta_2 + 15*eta_1*eta_2^2 + 20*eta_1^2*eta_3 - 20*eta_2*eta_3 - 30*eta_1*eta_4 + 24*eta_5)/(20*Pi^2) = 0.0000015673..., where eta_j = Sum_{p prime} 1/(p^2-1)^j (Pomerance and Schinzel, 2011).

			5336100 = 2^2 * 3^2 * 5^2 * 7^2 * 11^2 is a term since it has exactly 5 prime factors, 2, 3, 5, 7 and 11, that are non-unitary.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Carl Pomerance and Andrzej Schinzel, Multiplicative Properties of Sets of Residues, Moscow Journal of Combinatorics and Number Theory, Vol. 1, No. 1 (2011), pp. 52-66. See pp. 61-62.

Subsequence of A013929, A318720 and A327877.
Cf. A001221, A056170, A057521, A190641, A338539, A338540, A338541.
Cf. A154945 (eta_1), A324833 (eta_2), A324834 (eta_3), A324835 (eta_4), A324836 (eta_5).

Select[Range[2*10^7], Count[FactorInteger[#][[;;,2]], _?(#1 > 1 &)] == 5 &]

A324912 Binary weight of A324911(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 5, 1, 2, 2, 2, 1, 1, 1, 3, 1

Offset: 1

Antti Karttunen, Apr 14 2019

nonn

After 1 differs from A051903 at n = 900, 1764, 1800, 2700, 3528, 4356, 4500, 4900, 5292, ..., which seems to be a subsequence of A318720.

			For n = 900 = 2^2 * 3^2 * 5^2, A324911(900) = A156552(4) * A156552(9) * A156552(25) = 3*6*12 = 216, which in base-2 is written as "11011000", thus a(900) = 4.

Cf. A000120, A051903, A156552, A318720, A324911.

Differs from A072411 for the first time at n=72, where a(72) = 3, while A072411(72) = 6.

A156552(n) = { my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res }; \\ From A156552
A324911(n) = { my(f=factor(n)); prod(i=1, #f~, A156552(f[i,1]^f[i,2])); };
A324912(n) = hammingweight(A324911(n));

a(n) = A000120(A324911(n)).

cp's OEIS Frontend

A318715 Number of strict integer partitions of n with relatively prime parts in which no two parts are relatively prime.

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A036785 Numbers divisible by the squares of two distinct primes.

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A318721 Number of strict relatively prime factorizations of n.

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A338540 Numbers having exactly three non-unitary prime factors.

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A338541 Numbers having exactly four non-unitary prime factors.

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A338542 Numbers having exactly five non-unitary prime factors.

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A324912 Binary weight of A324911(n).

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