A023023 -id:A023023 - cp's OEIS frontend

A337600 Number of unordered triples of positive integers summing to n whose set of distinct parts is pairwise coprime, where a singleton is always considered coprime.

0, 0, 0, 1, 1, 2, 3, 3, 4, 5, 5, 6, 9, 7, 10, 8, 11, 11, 18, 12, 19, 13, 19, 17, 30, 16, 28, 20, 31, 23, 47, 23, 42, 26, 45, 27, 60, 31, 57, 35, 61, 37, 85, 38, 75, 43, 74, 47, 108, 45, 98, 52, 96, 56, 136, 54, 115, 64, 117, 67, 175, 65, 139, 76, 144, 75, 195

Offset: 0

Gus Wiseman, Sep 20 2020

nonn

First differs from A337601 at a(9) = 5, A337601(9) = 4.

			The a(3) = 1 through a(14) = 10 partitions (A = 10, B = 11, C = 12):
  111  211  221  222  322  332  333  433  443  444  544  554
            311  321  331  431  441  532  533  543  553  743
                 411  511  521  522  541  551  552  661  752
                           611  531  721  722  651  733  761
                                711  811  731  732  751  833
                                          911  741  922  851
                                               831  B11  941
                                               921       A31
                                               A11       B21
                                                         C11

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10000

A220377 is the strict case.
A304712 counts these partitions of any length.
A307719 is the strict case except for any number of 1's.
A337601 does not consider a singleton to be coprime unless it is (1).
A337602 is the ordered version.
A337664 counts compositions of this type and any length.
A000217 counts 3-part compositions.
A000837 counts relatively prime partitions.
A001399/A069905/A211540 count 3-part partitions.
A023023 counts relatively prime 3-part partitions.
A051424 counts pairwise coprime or singleton partitions.
A101268 counts pairwise coprime or singleton compositions.
A304709 counts partitions whose distinct parts are pairwise coprime.
A305713 counts pairwise coprime strict partitions.
A327516 counts pairwise coprime partitions.
A333227 ranks pairwise coprime compositions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
A337461 counts pairwise coprime length-3 compositions.
A337563 counts pairwise coprime length-3 partitions with no 1's.
Cf. A001840, A007359, A007360, A014612, A087087, A284825, A302569, A302696, A328673, A335235, A337603, A337695.

Table[Length[Select[IntegerPartitions[n,{3}],SameQ@@#||CoprimeQ@@Union[#]&]],{n,0,100}]

For n > 0, a(n) = A337601(n) + A079978(n).

A048240 Number of new colors that can be mixed with n units of yellow, blue, red.

Original entry on oeis.org

1, 3, 3, 7, 9, 18, 15, 33, 30, 45, 42, 75, 54, 102, 81, 108, 108, 168, 117, 207, 156, 210, 195, 297, 204, 330, 270, 351, 306, 462, 300, 525, 408, 510, 456, 612, 450, 738, 567, 708, 600, 900, 594, 987, 750, 900, 825, 1173, 792, 1239, 930, 1200

Offset: 0

Jurjen N.E. Bos, N. J. A. Sloane, Robin Trew (trew(AT)hcs.harvard.edu)

T. D. Noe, Table of n, a(n) for n = 0..1000
Vadym Kurylenko, Thin Simplices via Modular Arithmetic, arXiv:2404.03975 [math.CO], 2024. See p. 33.

Cf. A048134, A048241.
A032125(n) = a(2^n).
Cf. A000217, A008683, A000741, A023023.

A048240 := proc(n) local ans, i, j, k; ans := 0; for i from n by -1 to 0 do for j from n by -1 to 0 do k := n - i - j; if 0 <= k and k <= n and gcd(gcd(i, j), k) = 1 then ans := ans + 1; fi; od; od; RETURN(ans); end;

a[n_] := Sum[ MoebiusMu[n/d]*(d+1)*(d+2)/2, {d, Divisors[n]}]; a[0] = 1; Table[a[n], {n, 0, 51}] (* Jean-François Alcover, Jun 14 2012, after Vladeta Jovovic *)

a(n) = number of triples (i, j, k) with i+j+k = n and gcd(i, j, k) = 1.

a(n) = Sum_{d|n} mu(n/d)*(d+1)*(d+2)/2. G.f.: Sum_{k>0} mu(k)/(1-x^k)^3. - Vladeta Jovovic, Dec 22 2002

A338556 Products of three prime numbers of even index.

Original entry on oeis.org

27, 63, 117, 147, 171, 261, 273, 333, 343, 387, 399, 477, 507, 549, 609, 637, 639, 711, 741, 777, 801, 903, 909, 931, 963, 1017, 1083, 1113, 1131, 1179, 1183, 1251, 1281, 1359, 1421, 1443, 1467, 1491, 1557, 1629, 1653, 1659, 1677, 1729, 1737, 1791, 1813, 1869

Offset: 1

Gus Wiseman, Nov 08 2020

nonn

All terms are odd.

Also Heinz numbers of integer partitions with 3 parts, all of which are even. These partitions are counted by A001399.

			The sequence of terms together with their prime indices begins:
      27: {2,2,2}      637: {4,4,6}     1183: {4,6,6}
      63: {2,2,4}      639: {2,2,20}    1251: {2,2,34}
     117: {2,2,6}      711: {2,2,22}    1281: {2,4,18}
     147: {2,4,4}      741: {2,6,8}     1359: {2,2,36}
     171: {2,2,8}      777: {2,4,12}    1421: {4,4,10}
     261: {2,2,10}     801: {2,2,24}    1443: {2,6,12}
     273: {2,4,6}      903: {2,4,14}    1467: {2,2,38}
     333: {2,2,12}     909: {2,2,26}    1491: {2,4,20}
     343: {4,4,4}      931: {4,4,8}     1557: {2,2,40}
     387: {2,2,14}     963: {2,2,28}    1629: {2,2,42}
     399: {2,4,8}     1017: {2,2,30}    1653: {2,8,10}
     477: {2,2,16}    1083: {2,8,8}     1659: {2,4,22}
     507: {2,6,6}     1113: {2,4,16}    1677: {2,6,14}
     549: {2,2,18}    1131: {2,6,10}    1729: {4,6,8}
     609: {2,4,10}    1179: {2,2,32}    1737: {2,2,44}

A014612 allows all prime indices (not just even) (strict: A007304).
A066207 allows products of any length (strict: A258117).
A338471 is the version for odds instead of evens (strict: A307534).
A338557 is the strict case.
A014311 is a ranking of ordered triples (strict: A337453).
A001399(n-3) counts 3-part partitions (strict: A001399(n-6)).
A005117 lists squarefree numbers, with even case A039956.
A008284 counts partitions by sum and length (strict: A008289).
A023023 counts 3-part relatively prime partitions (strict: A101271).
A046316 lists products of exactly three odd primes (strict: A046389).
A066208 lists numbers with all odd prime indices (strict: A258116).
A075818 lists even Heinz numbers of 3-part partitions (strict: A075819).
A307719 counts 3-part pairwise coprime partitions (strict: A220377).
A285508 lists Heinz numbers of non-strict triples.
Cf. A000217, A001221, A001222, A037144, A056239, A112798, A337599, A337600.
Subsequence of A332820.

Select[Range[1000],PrimeOmega[#]==3&&OddQ[Times@@(1+PrimePi/@First/@FactorInteger[#])]&]

isok(m) = my(f=factor(m)); (bigomega(f)==3) && (#select(x->(x%2), apply(primepi, f[,1]~)) == 0); \\ Michel Marcus, Nov 10 2020

from itertools import filterfalse
from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A338556(n):
    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
    def f(x): return int(n+x-sum((primepi(x//(k*m))>>1)-(b>>1)+1 for a,k in filterfalse(lambda x:x[0]&1,enumerate(primerange(3,integer_nthroot(x,3)[0]+1),2)) for b,m in filterfalse(lambda x:x[0]&1,enumerate(primerange(k,isqrt(x//k)+1),a))))
    return bisection(f,n,n) # Chai Wah Wu, Oct 18 2024

A338471 Products of three prime numbers of odd index.

Original entry on oeis.org

8, 20, 44, 50, 68, 92, 110, 124, 125, 164, 170, 188, 230, 236, 242, 268, 275, 292, 310, 332, 374, 388, 410, 412, 425, 436, 470, 506, 508, 548, 575, 578, 590, 596, 605, 628, 668, 670, 682, 716, 730, 764, 775, 782, 788, 830, 844, 902, 908, 932, 935, 964, 970

Offset: 1

Gus Wiseman, Nov 08 2020

nonn

Also Heinz numbers of integer partitions with 3 parts, all of which are odd. These partitions are counted by A001399.

			The sequence of terms together with their prime indices begins:
       8: {1,1,1}      268: {1,1,19}     575: {3,3,9}
      20: {1,1,3}      275: {3,3,5}      578: {1,7,7}
      44: {1,1,5}      292: {1,1,21}     590: {1,3,17}
      50: {1,3,3}      310: {1,3,11}     596: {1,1,35}
      68: {1,1,7}      332: {1,1,23}     605: {3,5,5}
      92: {1,1,9}      374: {1,5,7}      628: {1,1,37}
     110: {1,3,5}      388: {1,1,25}     668: {1,1,39}
     124: {1,1,11}     410: {1,3,13}     670: {1,3,19}
     125: {3,3,3}      412: {1,1,27}     682: {1,5,11}
     164: {1,1,13}     425: {3,3,7}      716: {1,1,41}
     170: {1,3,7}      436: {1,1,29}     730: {1,3,21}
     188: {1,1,15}     470: {1,3,15}     764: {1,1,43}
     230: {1,3,9}      506: {1,5,9}      775: {3,3,11}
     236: {1,1,17}     508: {1,1,31}     782: {1,7,9}
     242: {1,5,5}      548: {1,1,33}     788: {1,1,45}

Robert Israel, Table of n, a(n) for n = 1..10000

A066208 allows products of any length (strict: A258116).
A307534 is the squarefree case.
A338469 is the restriction to odds.
A338556 is the version for evens (strict: A338557).
A000009 counts partitions into odd parts (strict: A000700).
A001399(n-3) counts 3-part partitions (strict: A001399(n-6)).
A008284 counts partitions by sum and length.
A014311 is a ranking of ordered triples (strict: A337453).
A014612 lists Heinz numbers of all triples (strict: A007304).
A023023 counts 3-part relatively prime partitions (strict: A101271).
A023023 counts 3-part relatively prime partitions (strict: A078374).
A046316 lists products of exactly three odd primes (strict: A046389).
A066207 lists numbers with all even prime indices (strict: A258117).
A075818 lists even Heinz numbers of 3-part partitions (strict: A075819).
A285508 lists Heinz numbers of non-strict triples.
A307719 counts 3-part pairwise coprime partitions (strict: A220377).
Cf. A000217, A001221, A001222, A005117, A037144, A056239, A112798.
Subsequence of A332820.

N:= 1000: # for terms <= N
R:= NULL:
for i from 1 by 2 do
  p:= ithprime(i);
  if p^3 >= N then break fi;
  for j from i by 2 do
    q:= ithprime(j);
    if p*q^2 >= N then break fi;
    for k from j by 2 do
      x:= p*q*ithprime(k);
      if x > N then break fi;
      R:= R,x;
od od od:
sort([R]); # Robert Israel, Jun 11 2025

Select[Range[100],PrimeOmega[#]==3&&OddQ[Times@@PrimePi/@First/@FactorInteger[#]]&]

isok(m) = my(f=factor(m)); (bigomega(f)==3) && (#select(x->!(x%2), apply(primepi, f[,1]~)) == 0); \\ Michel Marcus, Nov 10 2020

from sympy import primerange
from itertools import combinations_with_replacement as mc
def aupto(limit):
    pois = [p for i, p in enumerate(primerange(2, limit//4+1)) if i%2 == 0]
    return sorted(set(a*b*c for a, b, c in mc(pois, 3) if a*b*c <= limit))
print(aupto(971)) # Michael S. Branicky, Aug 20 2021

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A338471(n):
    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
    def f(x): return int(n+x-sum((primepi(x//(k*m))+1>>1)-(b+1>>1)+1 for a,k in filter(lambda x:x[0]&1,enumerate(primerange(integer_nthroot(x,3)[0]+1),1)) for b,m in filter(lambda x:x[0]&1,enumerate(primerange(k,isqrt(x//k)+1),a))))
    return bisection(f,n,n) # Chai Wah Wu, Oct 18 2024

A338557 Products of three distinct prime numbers of even index.

Original entry on oeis.org

273, 399, 609, 741, 777, 903, 1113, 1131, 1281, 1443, 1491, 1653, 1659, 1677, 1729, 1869, 2067, 2109, 2121, 2247, 2373, 2379, 2451, 2639, 2751, 2769, 2919, 3021, 3081, 3171, 3219, 3367, 3423, 3471, 3477, 3633, 3741, 3801, 3857, 3913, 3939, 4047, 4053, 4173

Offset: 1

Gus Wiseman, Nov 08 2020

nonn

All terms are odd.
Also sphenic numbers (A007304) with all even prime indices (A031215).
Also Heinz numbers of strict integer partitions with 3 parts, all of which are even. These partitions are counted by A001399.

			The sequence of terms together with their prime indices begins:
     273: {2,4,6}     1869: {2,4,24}    3219: {2,10,12}
     399: {2,4,8}     2067: {2,6,16}    3367: {4,6,12}
     609: {2,4,10}    2109: {2,8,12}    3423: {2,4,38}
     741: {2,6,8}     2121: {2,4,26}    3471: {2,6,24}
     777: {2,4,12}    2247: {2,4,28}    3477: {2,8,18}
     903: {2,4,14}    2373: {2,4,30}    3633: {2,4,40}
    1113: {2,4,16}    2379: {2,6,18}    3741: {2,10,14}
    1131: {2,6,10}    2451: {2,8,14}    3801: {2,4,42}
    1281: {2,4,18}    2639: {4,6,10}    3857: {4,8,10}
    1443: {2,6,12}    2751: {2,4,32}    3913: {4,6,14}
    1491: {2,4,20}    2769: {2,6,20}    3939: {2,6,26}
    1653: {2,8,10}    2919: {2,4,34}    4047: {2,8,20}
    1659: {2,4,22}    3021: {2,8,16}    4053: {2,4,44}
    1677: {2,6,14}    3081: {2,6,22}    4173: {2,6,28}
    1729: {4,6,8}     3171: {2,4,36}    4179: {2,4,46}

For the following, NNS means "not necessarily strict".
A007304 allows all prime indices (not just even) (NNS: A014612).
A046389 allows all odd primes (NNS: A046316).
A258117 allows products of any length (NNS: A066207).
A307534 is the version for odds instead of evens (NNS: A338471).
A337453 is a different ranking of ordered triples (NNS: A014311).
A338556 is the NNS version.
A001399(n-6) counts strict 3-part partitions (NNS: A001399(n-3)).
A005117 lists squarefree numbers, with even case A039956.
A078374 counts 3-part relatively prime strict partitions (NNS: A023023).
A075819 lists even Heinz numbers of strict triples (NNS: A075818).
A220377 counts 3-part pairwise coprime strict partitions (NNS: A307719).
A258116 lists squarefree numbers with all odd prime indices (NNS: A066208).
A285508 lists Heinz numbers of non-strict triples.
Cf. A000217, A001221, A001222, A037144, A056239, A112798, A337605.

Select[Range[1000],SquareFreeQ[#]&&PrimeOmega[#]==3&&OddQ[Times@@(1+PrimePi/@First/@FactorInteger[#])]&]

isok(m) = my(f=factor(m)); (bigomega(f)==3) && (omega(f)==3) && (#select(x->(x%2), apply(primepi, f[,1]~)) == 0); \\ Michel Marcus, Nov 10 2020

from itertools import filterfalse
from math import isqrt
from sympy import primepi, primerange, nextprime, integer_nthroot
def A338557(n):
    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
    def f(x): return int(n+x-sum((primepi(x//(k*m))>>1)-(b>>1) for a,k in filterfalse(lambda x:x[0]&1,enumerate(primerange(3,integer_nthroot(x,3)[0]+1),2)) for b,m in filterfalse(lambda x:x[0]&1,enumerate(primerange(nextprime(k)+1,isqrt(x//k)+1),a+2))))
    return bisection(f,n,n) # Chai Wah Wu, Oct 18 2024

A338469 Products of three odd prime numbers of odd index.

Original entry on oeis.org

125, 275, 425, 575, 605, 775, 935, 1025, 1175, 1265, 1331, 1445, 1475, 1675, 1705, 1825, 1955, 2057, 2075, 2255, 2425, 2575, 2585, 2635, 2645, 2725, 2783, 3175, 3179, 3245, 3425, 3485, 3565, 3685, 3725, 3751, 3925, 3995, 4015, 4175, 4301, 4475, 4565, 4715

Offset: 1

Gus Wiseman, Nov 08 2020

nonn

Also Heinz numbers of integer partitions with 3 parts, all of which are odd and > 1. These partitions are counted by A001399.

			The sequence of terms together with their prime indices begins:
     125: {3,3,3}     1825: {3,3,21}    3425: {3,3,33}
     275: {3,3,5}     1955: {3,7,9}     3485: {3,7,13}
     425: {3,3,7}     2057: {5,5,7}     3565: {3,9,11}
     575: {3,3,9}     2075: {3,3,23}    3685: {3,5,19}
     605: {3,5,5}     2255: {3,5,13}    3725: {3,3,35}
     775: {3,3,11}    2425: {3,3,25}    3751: {5,5,11}
     935: {3,5,7}     2575: {3,3,27}    3925: {3,3,37}
    1025: {3,3,13}    2585: {3,5,15}    3995: {3,7,15}
    1175: {3,3,15}    2635: {3,7,11}    4015: {3,5,21}
    1265: {3,5,9}     2645: {3,9,9}     4175: {3,3,39}
    1331: {5,5,5}     2725: {3,3,29}    4301: {5,7,9}
    1445: {3,7,7}     2783: {5,5,9}     4475: {3,3,41}
    1475: {3,3,17}    3175: {3,3,31}    4565: {3,5,23}
    1675: {3,3,19}    3179: {5,7,7}     4715: {3,9,13}
    1705: {3,5,11}    3245: {3,5,17}    4775: {3,3,43}

Robert Israel, Table of n, a(n) for n = 1..10000

A046316 allows all primes (strict: A046389).
A338471 allows all odd primes (strict: A307534).
A338556 is the version for evens (strict: A338557).
A000009 counts partitions into odd parts (strict: A000700).
A001399(n-3) counts 3-part partitions (strict: A001399(n-6)).
A005408 lists odds (strict: A056911).
A008284 counts partitions by sum and length.
A014311 is a ranking of 3-part compositions (strict: A337453).
A014612 lists Heinz numbers of 3-part partitions (strict: A007304).
A023023 counts 3-part relatively prime partitions (strict: A101271).
A066207 lists numbers with all even prime indices (strict: A258117).
A066208 lists numbers with all odd prime indices (strict: A258116).
A075818 lists even Heinz numbers of 3-part partitions (strict: A075819).
A285508 lists Heinz numbers of non-strict 3-part partitions.
Cf. A001221, A001222, A002620, A005117, A037144, A056239, A112798.

N:= 10000: # for terms <= N
P0:= [seq(ithprime(i),i=3..numtheory:-pi(floor(N/25)),2)]:
sort(select(`<=`,[seq(seq(seq(P0[i]*P0[j]*P0[k],k=1..j),j=1..i),i=1..nops(P0))], N)); # Robert Israel, Nov 12 2020

Select[Range[1,1000,2],PrimeOmega[#]==3&&OddQ[Times@@PrimePi/@First/@FactorInteger[#]]&]

isok(m) = my(f=factor(m)); (m%2) && (bigomega(f)==3) && (#select(x->!(x%2), apply(primepi, f[,1]~)) == 0); \\ Michel Marcus, Nov 10 2020

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A338469(n):
    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
    def f(x): return int(n+x-sum((primepi(x//(k*m))+1>>1)-(b+1>>1)+1 for a,k in filter(lambda x:x[0]&1,enumerate(primerange(5,integer_nthroot(x,3)[0]+1),3)) for b,m in filter(lambda x:x[0]&1,enumerate(primerange(k,isqrt(x//k)+1),a))))
    return bisection(f,n,n) # Chai Wah Wu, Oct 18 2024

A338553 Number of integer partitions of n that are either constant or relatively prime.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 15, 20, 29, 37, 56, 68, 101, 122, 170, 213, 297, 352, 490, 587, 778, 948, 1255, 1488, 1953, 2337, 2983, 3585, 4565, 5393, 6842, 8123, 10088, 12015, 14865, 17534, 21637, 25527, 31085, 36701, 44583, 52262, 63261, 74175, 88936, 104305, 124754

Offset: 0

Gus Wiseman, Nov 03 2020

nonn

The Heinz numbers of these partitions are given by A338555 = A000961 \/ A289509. The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

			The a(1) = 1 through a(7) = 15 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (31)    (41)     (51)      (52)
                    (211)   (221)    (222)     (61)
                    (1111)  (311)    (321)     (322)
                            (2111)   (411)     (331)
                            (11111)  (2211)    (421)
                                     (3111)    (511)
                                     (21111)   (2221)
                                     (111111)  (3211)
                                               (4111)
                                               (22111)
                                               (31111)
                                               (211111)
                                               (1111111)

A023022(n) + A059841(n) is the 2-part version.
A078374(n) + 1 is the strict case (n > 1).
A338554 counts the complement, with Heinz numbers A338552.
A338555 gives the Heinz numbers of these partitions.
A000005 counts constant partitions, with Heinz numbers A000961.
A000837 counts relatively prime partitions, with Heinz numbers A289509.
A282750 counts relatively prime partitions by sum and length.
Cf. A000010, A007360, A008284, A023023, A051424, A101271, A101391, A302698, A304712, A327516, A337664.

Table[Length[Select[IntegerPartitions[n],SameQ@@#||GCD@@#==1&]],{n,0,30}]

For n > 0, a(n) = A000005(n) + A000837(n) - 1.

A340278 Number of partitions of n into 3 parts whose smallest and largest parts are relatively prime.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 4, 4, 5, 6, 9, 8, 11, 11, 13, 15, 19, 18, 23, 22, 24, 26, 32, 31, 37, 38, 42, 43, 50, 47, 55, 55, 59, 62, 68, 68, 78, 78, 83, 85, 96, 93, 105, 104, 108, 112, 124, 121, 132, 131, 138, 141, 154, 151, 162, 164, 172, 176, 191, 187, 205, 205, 212, 217, 228, 226

Offset: 1

Wesley Ivan Hurt, Jan 02 2021

nonn

Index entries for sequences related to partitions

Cf. A023023, A340275, A340276.

Table[Sum[Sum[Floor[1/GCD[k, n - i - k]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 80}]

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} floor(1/gcd(k,n-i-k)).

A340275 Number of partitions of n into 3 parts whose smallest two parts are relatively prime.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 4, 5, 5, 7, 8, 10, 11, 12, 13, 16, 17, 20, 22, 24, 25, 29, 30, 33, 35, 38, 40, 45, 47, 52, 55, 58, 60, 64, 66, 72, 75, 79, 81, 88, 90, 97, 101, 105, 108, 116, 119, 126, 130, 135, 139, 148, 151, 157, 161, 167, 171, 181, 183, 193, 198, 204, 209, 217, 221

Offset: 1

Wesley Ivan Hurt, Jan 02 2021

nonn

Index entries for sequences related to partitions

Cf. A023023, A340276, A340278.

Table[Sum[Sum[Floor[1/GCD[i, k]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} floor(1/gcd(i,k)).

A340276 Number of partitions of n into 3 parts whose largest two parts are relatively prime.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 2, 3, 4, 5, 5, 7, 8, 10, 11, 12, 13, 16, 17, 20, 22, 24, 25, 29, 30, 33, 35, 38, 40, 45, 47, 52, 55, 58, 60, 64, 66, 72, 75, 79, 81, 88, 90, 97, 101, 105, 108, 116, 119, 126, 130, 135, 139, 148, 151, 157, 161, 167, 171, 181, 183, 193, 198, 204, 209, 217

Offset: 1

Wesley Ivan Hurt, Jan 02 2021

nonn

Index entries for sequences related to partitions

Cf. A023023, A340275, A340278.

Table[Sum[Sum[Floor[1/GCD[i, n - i - k]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 80}]

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} floor(1/gcd(i,n-i-k)).

cp's OEIS Frontend

A337600 Number of unordered triples of positive integers summing to n whose set of distinct parts is pairwise coprime, where a singleton is always considered coprime.

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A048240 Number of new colors that can be mixed with n units of yellow, blue, red.

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A338556 Products of three prime numbers of even index.

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A338471 Products of three prime numbers of odd index.

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A338557 Products of three distinct prime numbers of even index.

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A338469 Products of three odd prime numbers of odd index.

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A338553 Number of integer partitions of n that are either constant or relatively prime.

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