A078374 -id:A078374 - cp's OEIS frontend

A072774 Powers of squarefree numbers.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97

Offset: 1

Reinhard Zumkeller, Jul 10 2002

nonn

Essentially the same as A062770. - R. J. Mathar, Sep 25 2008
Numbers m such that in canonical prime factorization all prime exponents are identical: A124010(m,k) = A124010(m,1) for k = 2..A000005(m). - Reinhard Zumkeller, Apr 06 2014
Heinz numbers of uniform partitions. An integer partition is uniform if all parts appear with the same multiplicity. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). - Gus Wiseman, Apr 16 2018

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

Complement of A059404.
Cf. A072775, A072776, A072777 (subsequence), A005117, A072778, A124010, A329332 (tabular arrangement), A384667 (characteristic function).
A subsequence of A242414.
Cf. A000005, A000009, A000837, A007916, A013661, A047966, A052409, A052410, A072774, A078374, A289023, A289509, A300486, A302491, A302796, A302979.

import Data.Map (empty, findMin, deleteMin, insert)
import qualified Data.Map.Lazy as Map (null)
a072774 n = a072774_list !! (n-1)
(a072774_list, a072775_list, a072776_list) = unzip3 $
   (1, 1, 1) : f (tail a005117_list) empty where
   f vs'@(v:vs) m
    | Map.null m || xx > v = (v, v, 1) :
                             f vs (insert (v^2) (v, 2) m)
    | otherwise = (xx, bx, ex) :
                  f vs' (insert (bx*xx) (bx, ex+1) $ deleteMin m)
    where (xx, (bx, ex)) = findMin m
-- Reinhard Zumkeller, Apr 06 2014

isA := n -> n=1 or is(1 = nops({seq(p[2], p in ifactors(n)[2])})):
select(isA, [seq(1..97)]);  # Peter Luschny, Jun 10 2025

Select[Range[100], Length[Union[FactorInteger[#][[All, 2]]]] == 1 &] (* Geoffrey Critzer, Mar 30 2015 *)

is(n)=ispower(n,,&n); issquarefree(n) \\ Charles R Greathouse IV, Oct 16 2015

from math import isqrt
from sympy import mobius, integer_nthroot
def A072774(n):
    def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))-1
    def f(x): return n-2+x-sum(g(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length()))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 19 2024

a(n) = A072775(n)^A072776(n).
Sum_{n>=1} 1/a(n)^s = 1 + Sum_{k>=1} (zeta(k*s)/zeta(2*k*s)-1) for s > 1. - Amiram Eldar, Mar 20 2025
a(n)/n ~ Pi^2/6 (A013661). - Friedjof Tellkamp, Jun 09 2025

A007360 Number of partitions of n into distinct and pairwise relatively prime parts.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 8, 9, 10, 11, 10, 13, 17, 19, 21, 22, 21, 24, 32, 37, 37, 38, 40, 45, 55, 65, 69, 66, 64, 75, 86, 100, 113, 107, 106, 122, 145, 165, 174, 167, 162, 179, 222, 253, 255, 255, 255, 273, 328, 373, 376, 369, 377, 406, 476, 553, 569, 537, 529

Offset: 1

N. J. A. Sloane and Mira Bernstein, following a suggestion from Marc LeBrun

			From _Gus Wiseman_, Sep 23 2019: (Start)
The a(1) = 1 through a(10) = 6 partitions (A = 10):
  (1)  (2)  (3)   (4)   (5)   (6)    (7)   (8)    (9)    (A)
            (21)  (31)  (32)  (51)   (43)  (53)   (54)   (73)
                        (41)  (321)  (52)  (71)   (72)   (91)
                                     (61)  (431)  (81)   (532)
                                           (521)  (531)  (541)
                                                         (721)
(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..750 (terms 1..350 from Alois P. Heinz)
M. LeBrun & D. Hoey, Emails

Number of partitions of n into relatively prime parts = A000837.
The non-strict case is A051424.
Strict relatively prime partitions are A078374.
Cf. A007359, A038348, A084422, A186974, A187106, A303140, A302569, A303362, A304714, A320426, A320436.

$RecursionLimit = 1000; b[n_, i_, s_] := b[n, i, s] = Module[{f}, If[n == 0 || i == 1, 1, If[i<2, 0, f = FactorInteger[i][[All, 1]]; b[n, i-1, Select[s, #Jean-François Alcover, Mar 20 2014, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],Length[#]==1||UnsameQ@@#&&CoprimeQ@@Union[#]&]],{n,0,30}] (* Gus Wiseman, Sep 23 2019 *)

a(n) = A051424(n)-A051424(n-2). - Vladeta Jovovic, Dec 11 2004

More precise definition from Vladeta Jovovic, Dec 11 2004

More terms from Pab Ter (pabrlos2(AT)yahoo.com), Nov 13 2005

A305713 Number of strict integer partitions of n into pairwise coprime parts.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 3, 4, 4, 5, 7, 8, 9, 10, 9, 12, 16, 18, 20, 21, 20, 23, 31, 36, 36, 37, 39, 44, 54, 64, 68, 65, 63, 74, 85, 99, 112, 106, 105, 121, 144, 164, 173, 166, 161, 178, 221, 252, 254, 254, 254, 272, 327, 372, 375, 368, 376, 405, 475, 552, 568, 536

Offset: 1

Gus Wiseman, Sep 01 2018

nonn

			The a(13) = 9 strict partitions are (7,6), (8,5), (9,4), (10,3), (11,2), (12,1), (7,5,1), (5,4,3,1), (7,3,2,1).

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..700

Cf. A000009, A051424, A078374, A302569, A302696, A302797, A303282, A303283, A318717, A318719.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&CoprimeQ@@#&]],{n,30}]

A220377 Number of partitions of n into three distinct and mutually relatively prime parts.

Original entry on oeis.org

1, 0, 2, 1, 3, 1, 6, 1, 7, 3, 7, 3, 14, 3, 15, 6, 14, 6, 25, 6, 22, 10, 25, 9, 42, 8, 34, 15, 37, 15, 53, 13, 48, 22, 53, 17, 78, 17, 65, 30, 63, 24, 99, 24, 88, 35, 84, 30, 126, 34, 103, 45, 103, 38, 166, 35, 124, 57, 128, 51, 184, 44, 150, 67, 172, 52, 218

Offset: 6

Carl Najafi, Dec 13 2012

nonn

The Heinz numbers of these partitions are the intersection of A005117 (strict), A014612 (triples), and A302696 (coprime). - Gus Wiseman, Oct 14 2020

			For n=10 we have three such partitions: 1+2+7, 1+4+5 and 2+3+5.
From _Gus Wiseman_, Oct 14 2020: (Start)
The a(6) = 1 through a(20) = 15 triples (empty column indicated by dot, A..H = 10..17):
321  .  431  531  532  731  543  751  743  753  754  971  765  B53  875
        521       541       651       752  951  853  B51  873  B71  974
                  721       732       761  B31  871  D31  954  D51  A73
                            741       851       952       972       A91
                            831       941       B32       981       B54
                            921       A31       B41       A71       B72
                                      B21       D21       B43       B81
                                                          B52       C71
                                                          B61       D43
                                                          C51       D52
                                                          D32       D61
                                                          D41       E51
                                                          E31       F41
                                                          F21       G31
                                                                    H21
(End)

Fausto A. C. Cariboni, Table of n, a(n) for n = 6..10000 (terms 6..1000 from Seiichi Manyama)

Cf. A015617, A300815.
A023022 is the 2-part version.
A101271 is the relative prime instead of pairwise coprime version.
A220377*6 is the ordered version.
A305713 counts these partitions of any length, with Heinz numbers A302797.
A307719 is the non-strict version.
A337461 is the non-strict ordered version.
A337563 is the case with no 1's.
A337605 is the pairwise non-coprime instead of pairwise coprime version.
A001399(n-6) counts strict 3-part partitions, with Heinz numbers A007304.
A008284 counts partitions by sum and length, with strict case A008289.
A318717 counts pairwise non-coprime strict partitions.
A326675 ranks pairwise coprime sets.
A327516 counts pairwise coprime partitions.
A337601 counts 3-part partitions whose distinct parts are pairwise coprime.
Cf. A000217, A007360, A023023, A051424, A078374, A087087, A302696, A333227, A337485, A337561.

Table[Length@Select[ IntegerPartitions[ n, {3}], #[[1]] != #[[2]] != #[[3]] && GCD[#[[1]], #[[2]]] == 1 && GCD[#[[1]], #[[3]]] == 1 && GCD[#[[2]], #[[3]]] == 1 &], {n, 6, 100}]
Table[Count[IntegerPartitions[n,{3}],?(CoprimeQ@@#&&Length[ Union[#]] == 3&)],{n,6,100}] (* _Harvey P. Dale, May 22 2020 *)

a(n)=my(P=partitions(n));sum(i=1,#P,#P[i]==3&&P[i][1]Charles R Greathouse IV, Dec 14 2012

a(n > 2) = A307719(n) - 1. - Gus Wiseman, Oct 15 2020

A302698 Number of integer partitions of n into relatively prime parts that are all greater than 1.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 3, 2, 5, 4, 13, 7, 23, 18, 32, 33, 65, 50, 104, 92, 148, 153, 252, 226, 376, 376, 544, 570, 846, 821, 1237, 1276, 1736, 1869, 2552, 2643, 3659, 3887, 5067, 5509, 7244, 7672, 10086, 10909, 13756, 15168, 19195, 20735, 26237, 28708, 35418, 39207

Offset: 1

Gus Wiseman, Apr 11 2018

nonn

Two or more numbers are relatively prime if they have no common divisor other than 1. A single number is not considered relatively prime unless it is equal to 1 (which is impossible in this case).

The Heinz numbers of these partitions are given by A302697.

			The a(5) = 1 through a(12) = 7 partitions (empty column indicated by dot):
  (32)  .  (43)   (53)   (54)    (73)    (65)     (75)
           (52)   (332)  (72)    (433)   (74)     (543)
           (322)         (432)   (532)   (83)     (552)
                         (522)   (3322)  (92)     (732)
                         (3222)          (443)    (4332)
                                         (533)    (5322)
                                         (542)    (33222)
                                         (632)
                                         (722)
                                         (3332)
                                         (4322)
                                         (5222)
                                         (32222)

Alois P. Heinz, Table of n, a(n) for n = 1..1000

A000837 is the version allowing 1's.
A002865 does not require relative primality.
A302697 gives the Heinz numbers of these partitions.
A337450 is the ordered version.
A337451 is the ordered strict version.
A337452 is the strict version.
A337485 is the pairwise coprime instead of relatively prime version.
A000740 counts relatively prime compositions.
A078374 counts relatively prime strict partitions.
A212804 counts compositions with no 1's.
A291166 appears to rank relatively prime compositions.
A332004 counts strict relatively prime compositions.
A337561 counts pairwise coprime strict compositions.
A338332 is the case of length 3, with strict case A338333.
Cf. A007359, A018783, A051424, A101268, A289508, A289509, A302568, A337563, A337984, A338468.

b:= proc(n, i, g) option remember; `if`(n=0, `if`(g=1, 1, 0),
      `if`(i<2, 0, b(n, i-1, g)+b(n-i, min(n-i, i), igcd(g, i))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=1..60);  # Alois P. Heinz, Apr 12 2018

Table[Length[Select[IntegerPartitions[n],FreeQ[#,1]&&GCD@@#===1&]],{n,30}]
(* Second program: *)
b[n_, i_, g_] := b[n, i, g] = If[n == 0, If[g == 1, 1, 0], If[i < 2, 0, b[n, i - 1, g] + b[n - i, Min[n - i, i], GCD[g, i]]]];
a[n_] := b[n, n, 0];
Array[a, 60] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

a(n) = A002865(n) - A018783(n).

Extended by Gus Wiseman, Oct 29 2020

A305148 Number of integer partitions of n whose distinct parts are pairwise indivisible.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 12, 12, 17, 20, 22, 28, 35, 39, 48, 55, 65, 79, 90, 105, 121, 143, 166, 190, 219, 254, 290, 332, 382, 436, 493, 567, 637, 729, 824, 931, 1052, 1186, 1334, 1504, 1691, 1894, 2123, 2380, 2664, 2968, 3319, 3704, 4119, 4586, 5110

Offset: 0

Gus Wiseman, May 26 2018

nonn

			The a(9) = 7 integer partitions are (9), (72), (54), (522), (333), (3222), (111111111).

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..360 (terms 0..300 from Alois P. Heinz)

Cf. A000837, A001055, A001970, A007359, A007716, A051424, A078374, A100953, A285572, A285573, A302696, A303362, A305079, A305149, A305150.

Table[Length[Select[IntegerPartitions[n],Select[Tuples[Union[#],2],UnsameQ@@#&&Divisible@@#&]=={}&]],{n,20}]

More terms from Alois P. Heinz, May 26 2018

A307719 Number of partitions of n into 3 mutually coprime parts.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 1, 3, 2, 4, 2, 7, 2, 8, 4, 8, 4, 15, 4, 16, 7, 15, 7, 26, 7, 23, 11, 26, 10, 43, 9, 35, 16, 38, 16, 54, 14, 49, 23, 54, 18, 79, 18, 66, 31, 64, 25, 100, 25, 89, 36, 85, 31, 127, 35, 104, 46, 104, 39, 167, 36, 125, 58, 129, 52, 185, 45

Offset: 0

Wesley Ivan Hurt, Apr 24 2019

The Heinz numbers of these partitions are the intersection of A014612 (triples) and A302696 (pairwise coprime). - Gus Wiseman, Oct 16 2020

			There are 2 partitions of 9 into 3 mutually coprime parts: 7+1+1 = 5+3+1, so a(9) = 2.
There are 4 partitions of 10 into 3 mutually coprime parts: 8+1+1 = 7+2+1 = 5+4+1 = 5+3+2, so a(10) = 4.
There are 2 partitions of 11 into 3 mutually coprime parts: 9+1+1 = 7+3+1, so a(11) = 2.
There are 7 partitions of 12 into 3 mutually coprime parts: 10+1+1 = 9+2+1 = 8+3+1 = 7+4+1 = 6+5+1 = 7+3+2 = 5+4+3, so a(12) = 7.

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10000 (terms 0..2000 from Robert Israel)
Index entries for sequences related to partitions

A023022 is the version for pairs.
A220377 is the strict case, with ordered version A220377*6.
A327516 counts these partitions of any length, with strict version A305713 and Heinz numbers A302696.
A337461 is the ordered version.
A337563 is the case with no 1's.
A337599 is the pairwise non-coprime instead of pairwise coprime version.
A337601 only requires the distinct parts to be pairwise coprime.
A001399(n-3) = A069905(n) = A211540(n+2) counts 3-part partitions.
A002865 counts partitions with no 1's, with strict case A025147.
A007359 and A337485 count pairwise coprime partitions with no 1's.
A200976 and A328673 count pairwise non-coprime partitions.
Cf. A007304, A014612, A078374, A082024, A284825, A304709, A337462, A337605.

N:= 200: # to get a(0)..a(N)
A:= Array(0..N):
for a from 1 to N/3 do
  for b from a to (N-a)/2 do
    if igcd(a,b) > 1 then next fi;
    ab:= a*b;
    for c from b to N-a-b do
       if igcd(ab,c)=1 then A[a+b+c]:= A[a+b+c]+1 fi
od od od:
convert(A,list); # Robert Israel, May 09 2019

Table[Sum[Sum[Floor[1/(GCD[i, j] GCD[j, n - i - j] GCD[i, n - i - j])], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 100}]
Table[Length[Select[IntegerPartitions[n,{3}],CoprimeQ@@#&]],{n,0,100}] (* Gus Wiseman, Oct 15 2020 *)

a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} [gcd(i,j) * gcd(j,n-i-j) * gcd(i,n-i-j) = 1], where [] is the Iverson bracket.

a(n > 2) = A220377(n) + 1. - Gus Wiseman, Oct 15 2020

A304711 Heinz numbers of integer partitions whose distinct parts are pairwise coprime.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 32, 33, 34, 35, 36, 38, 40, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 58, 60, 62, 64, 66, 68, 69, 70, 72, 74, 75, 76, 77, 80, 82, 85, 86, 88, 90, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 106, 108, 110

Offset: 1

Gus Wiseman, May 17 2018

nonn

Two parts are coprime if they have no common divisor greater than 1. For partitions of length 1 note that (1) is coprime but (x) is not coprime for x > 1.

First differs from A289509 at a(24) = 44, A289509(24) = 42.

			Sequence of all partitions whose distinct parts are pairwise coprime begins (1), (11), (21), (111), (31), (211), (41), (32), (1111), (221), (311), (51), (2111), (61), (411), (321), (11111), (52), (71), (43), (2211), (81), (3111).

Cf. A000837, A007359, A018783, A051424, A056239, A078374, A101268, A289508, A289509, A298748, A300486, A302569, A302696, A302698, A302796, A302797, A304709.

Select[Range[200],CoprimeQ@@PrimePi/@FactorInteger[#][[All,1]]&]

A337485 Number of pairwise coprime integer partitions of n with no 1's, where a singleton is not considered coprime unless it is (1).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 2, 1, 2, 2, 4, 3, 5, 4, 4, 7, 8, 9, 10, 10, 9, 13, 17, 18, 17, 19, 19, 24, 29, 34, 33, 31, 31, 42, 42, 56, 55, 50, 54, 66, 77, 86, 86, 79, 81, 96, 124, 127, 126, 127, 126, 145, 181, 190, 184, 183, 192, 212, 262, 289, 278, 257, 270, 311

Offset: 0

Gus Wiseman, Sep 21 2020

nonn

Such a partition is necessarily strict.

The Heinz numbers of these partitions are the intersection of A005408 (no 1's), A005117 (strict), and A302696 (coprime).

			The a(n) partitions for n = 5, 7, 12, 13, 16, 17, 18, 19 (A..H = 10..17):
  (3,2)  (4,3)  (7,5)    (7,6)  (9,7)    (9,8)      (B,7)    (A,9)
         (5,2)  (5,4,3)  (8,5)  (B,5)    (A,7)      (D,5)    (B,8)
                (7,3,2)  (9,4)  (D,3)    (B,6)      (7,6,5)  (C,7)
                         (A,3)  (7,5,4)  (C,5)      (8,7,3)  (D,6)
                         (B,2)  (8,5,3)  (D,4)      (9,5,4)  (E,5)
                                (9,5,2)  (E,3)      (9,7,2)  (F,4)
                                (B,3,2)  (F,2)      (B,4,3)  (G,3)
                                         (7,5,3,2)  (B,5,2)  (H,2)
                                                    (D,3,2)  (B,5,3)
                                                             (7,5,4,3)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..750

A005408 intersected with A302696 ranks these partitions.
A007359 considers all singletons to be coprime.
A327516 allows 1's, with non-strict version A305713.
A337452 is the relatively prime instead of pairwise coprime version, with non-strict version A302698.
A337563 is the restriction to partitions of length 3.
A002865 counts partitions with no 1's.
A078374 counts relatively prime strict partitions.
A200976 and A328673 count pairwise non-coprime partitions.
Cf. A101268, A220377, A302696, A304709, A332004, A337450, A337451, A337462, A337561, A337605.

Table[Length[Select[IntegerPartitions[n],!MemberQ[#,1]&&CoprimeQ@@#&]],{n,0,30}]

a(n) = A007359(n) - 1 for n > 1.

A324751 Number of strict integer partitions of n containing no prime indices of the parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 3, 2, 4, 5, 5, 6, 8, 8, 12, 10, 14, 13, 18, 19, 26, 25, 30, 34, 39, 40, 51, 55, 60, 71, 77, 90, 97, 111, 123, 136, 153, 170, 179, 216, 230, 264, 282, 322, 345, 385, 423, 470, 513, 573, 629, 686, 755, 834, 910, 1005, 1095, 1194, 1303, 1433

Offset: 0

Gus Wiseman, Mar 16 2019

nonn

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 a(1) = 1 through a(13) = 8 strict integer partitions (A...D = 10...13):
  1   2   3   4    5   6    7    8    9    A    B     C     D
              31       42   43   71   54   64   65    75    76
                       51   52        63   73   83    84    85
                                      72   82   542   93    94
                                           91   731   A2    B2
                                                      B1    643
                                                            751
                                                            931

The subset version is A324741, with maximal case A324743. The non-strict version is A324756. The Heinz number version is A324758. An infinite version is A304360.
Cf. A000720, A001462, A007097, A074971, A078374, A112798, A276625, A290822, A305713, A306844, A324764.
Cf. A324695, A324742, A324748.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Intersection[#,PrimePi/@First/@Join@@FactorInteger/@#]=={}&]],{n,0,30}]

cp's OEIS Frontend

A072774 Powers of squarefree numbers.

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A007360 Number of partitions of n into distinct and pairwise relatively prime parts.

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A305713 Number of strict integer partitions of n into pairwise coprime parts.

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A220377 Number of partitions of n into three distinct and mutually relatively prime parts.

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A302698 Number of integer partitions of n into relatively prime parts that are all greater than 1.

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A305148 Number of integer partitions of n whose distinct parts are pairwise indivisible.

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A307719 Number of partitions of n into 3 mutually coprime parts.

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