A018783 -id:A018783 - cp's OEIS frontend

A366847 Numbers whose halved even prime indices are nonempty and relatively prime.

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 91, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 138, 141, 144, 147, 150, 153, 156, 159, 162, 165, 168, 171, 174

Offset: 1

Gus Wiseman, Oct 31 2023

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.

Consists of powers of 2 times elements of the odd restriction A366849.

			The even prime indices of 91 are {4,6}, halved {2,3}, which are relatively prime, so 91 is in the sequence.
The prime indices of 665 are {3,4,8}, even {4,8}, halved {2,4}, which are not relatively prime, so 665 is not in the sequence.
The terms together with their prime indices begin:
    3: {2}
    6: {1,2}
    9: {2,2}
   12: {1,1,2}
   15: {2,3}
   18: {1,2,2}
   21: {2,4}
   24: {1,1,1,2}
   27: {2,2,2}
   30: {1,2,3}
   33: {2,5}
   36: {1,1,2,2}
   39: {2,6}
   42: {1,2,4}
   45: {2,2,3}
   48: {1,1,1,1,2}

Including odd indices gives A289509, ones of A289508, counted by A000837.
The complement including odd indices is A318978, counted by A018783.
The partitions with these ranks are counted by A366845.
A version for odd indices A366846, counted by A366850.
The odd restriction is A366849.
A000041 counts integer partitions, strict A000009 (also into odds).
A035363 counts partitions into all even parts, ranks A066207.
A112798 lists prime indices, length A001222, sum A056239.
A162641 counts even prime exponents, odd A162642.
A257992 counts even prime indices, odd A257991.
A366528 adds up odd prime indices, partition triangle A113685.
A366531 = 2*A366533 adds up even prime indices, triangle A113686/A174713.
Cf. A000720, A055396, A061395, A066208, A168532, A302696, A302697, A325698, A366842, A366843, A366844, A366848.

Select[Range[100],GCD@@Select[PrimePi/@First/@FactorInteger[#],EvenQ]/2==1&]

A202385 Number of partitions of n into distinct parts having pairwise common factors but no overall common factor.

Original entry on oeis.org

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, 2, 13, 2, 29, 0, 34, 2, 17, 2, 34, 1, 49, 2, 21, 3, 58, 2, 63, 3, 20, 7, 72, 2, 81, 3

Offset: 31

Alois P. Heinz, Dec 18 2011

nonn

			a(31) = 1: [6,10,15] = [2*3,2*5,3*5].
a(37) = 1: [10,12,15] = [2*5,2*2*3,3*5].
a(41) = 2: [6,15,20], [6,14,21].
a(43) = 2: [6,10,12,15], [10,15,18].
a(53) = 4: [6,12,15,20], [15,18,20], [6,12,14,21], [14,18,21].
a(55) = 3: [10,12,15,18], [6,10,15,24], [6,21,28].

Alois P. Heinz, Table of n, a(n) for n = 31..251

Cf. A200976, A018783.

with(numtheory):
w:= (m, h)-> mul(`if`(j>=h, 1, j), j=factorset(m)):
b:= proc(n, i, g, s) option remember; local j, ok;
      if n<0 then 0
    elif n=0 then `if`(g>1, 0, 1)
    elif i<2 then 0
    else ok:= evalb(i<=n);
         for j in s while ok do ok:= igcd(i, j)>1 od;
         b(n, i-1, g, map(x->w(x, i), s)) +`if`(ok,
         b(n-i, i-1, igcd(i, g), map(x->w(x, i), {s[], i}) ), 0)
      fi
    end:
a:= n-> b(n, n, 0, {}):
seq(a(n), n=31..100);

w[m_, h_] := Product[If[j >= h, 1, j], {j, FactorInteger[m][[All, 1]]}]; b[n_, i_, g_, s_] := b[n, i, g, s] = Module[{j, ok}, Which[n<0, 0, n==0, If[g>1, 0, 1], i<2, 0, True, ok = i <= n; For[j = 1, ok && j <= Length[s], j++, ok = GCD[i, s[[j]]]>1]; b[n, i-1, g, Map[w[#, i]&, s]] + If[ok, b[n-i, i-1, GCD[i, g], Map[w[#, i]&, Union @ Append[s, i]]], 0]]]; a[n_] := b[n, n, 0, {}]; Table[a[n], {n, 31, 100}] (* Jean-François Alcover, Feb 15 2017, translated from Maple *)

A303553 Number of periodic factorizations of n > 1 into positive factors greater than 1; a(1) = 1 by convention.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Apr 26 2018

nonn

A multiset is periodic if its multiplicities have a common divisor greater than 1.

			The a(64)  = 4 periodic factorizations are (2*2*2*2*2*2), (2*2*4*4), (4*4*4), (8*8).
The a(144) = 4 periodic factorizations are (2*2*2*2*3*3), (2*2*6*6), (3*3*4*4), (12*12).
The a(256) = 5 periodic factorizations are (2*2*2*2*2*2*2*2), (2*2*2*2*4*4), (2*2*8*8), (4*4*4*4), (16*16).
The a(576) = 7 periodic factorizations are (2*2*2*2*2*2*3*3), (2*2*2*2*6*6), (2*2*3*3*4*4), (2*2*12*12), (3*3*8*8), (4*4*6*6), (24*24).

Antti Karttunen, Table of n, a(n) for n = 1..20736
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Cf. A000740, A000837, A001055, A007916, A018783(n) = a(2^n), A052409, A052410, A275870, A303386, A303431, A303547, A303552, A303709.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],GCD@@Length/@Split[#]>1&]],{n,2,100}]

gcd_of_multiplicities(lista) = { my(u=length(lista)); if(u<2, u, my(g=0, pe = lista[1], j=1); for(i=2,u,if(lista[i]==pe, j++, g = gcd(j,g); j=1; pe = lista[i])); gcd(g,j)); }; \\ the supplied lista (newfacs) should be monotonic
A303553(n, m=n, facs=List([])) = if(1==n, (gcd_of_multiplicities(facs)!=1), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A303553(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Dec 06 2018

a(n) >= A303709(n). - Antti Karttunen, Dec 06 2018

a(1) = 1 prepended by Antti Karttunen, Dec 06 2018

A305735 Number of integer partitions of n whose greatest common divisor is a prime number.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 3, 2, 7, 1, 10, 1, 15, 8, 17, 1, 34, 1, 37, 16, 56, 1, 80, 6, 101, 27, 122, 1, 208, 1, 209, 57, 297, 20, 410, 1, 490, 102, 599, 1, 901, 1, 948, 194, 1255, 1, 1690, 14, 1985, 298, 2337, 1, 3327, 61, 3597, 491, 4565, 1, 6031, 1, 6842, 802

Offset: 1

Gus Wiseman, Jun 22 2018

nonn

			The a(10) = 7 integer partitions are (82), (64), (622), (55), (442), (4222), (22222).

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000040, A000041, A000837, A018783, A100953, A143519, A289508, A302698, A305563, A305731.

Table[Length[Select[IntegerPartitions[n],PrimeQ[GCD@@#]&]],{n,20}]

seq(n)={dirmul(vector(n, n, numbpart(n)), dirmul(vector(n, n, moebius(n)), vector(n, n, isprime(n))))} \\ Andrew Howroyd, Jun 22 2018

a(n) = Sum_{d|n} A143519(d) * A000041(n/d). - Andrew Howroyd, Jun 22 2018

A318979 Number of divisors of n with relatively prime prime indices, meaning they belong to A289509.

Original entry on oeis.org

0, 1, 0, 2, 0, 2, 0, 3, 0, 2, 0, 4, 0, 2, 1, 4, 0, 3, 0, 4, 0, 2, 0, 6, 0, 2, 0, 4, 0, 5, 0, 5, 1, 2, 1, 6, 0, 2, 0, 6, 0, 4, 0, 4, 2, 2, 0, 8, 0, 3, 1, 4, 0, 4, 1, 6, 0, 2, 0, 9, 0, 2, 0, 6, 0, 5, 0, 4, 1, 5, 0, 9, 0, 2, 2, 4, 1, 4, 0, 8, 0, 2, 0, 8, 1, 2, 0

Offset: 1

Gus Wiseman, Sep 06 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

			The divisors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36, corresponding to the prime index multisets (), (1), (2), (11), (12), (22), (112), (122), (1122) respectively. Of these, only (1), (11), (12), (112), (122), (1122) are relatively prime, corresponding to the divisors 2, 4, 6, 12, 18, 36, so a(36) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Cf. A000005, A000837, A001221, A018783, A056239, A289508, A289509, A296150, A298748, A318978, A327657.

Table[Length[Select[Divisors[n],GCD@@PrimePi/@FactorInteger[#][[All,1]]==1&]],{n,100}]

a(n) = sumdiv(n, d, gcd(apply(x->primepi(x), factor(d)[,1])) == 1); \\ Michel Marcus, Jan 09 2019

a(n) = A000005(n) - A327657(n). - Antti Karttunen, Dec 05 2021

A327685 Nonprime numbers whose prime indices have a common divisor > 1.

Original entry on oeis.org

9, 21, 25, 27, 39, 49, 57, 63, 65, 81, 87, 91, 111, 115, 117, 121, 125, 129, 133, 147, 159, 169, 171, 183, 185, 189, 203, 213, 235, 237, 243, 247, 259, 261, 267, 273, 289, 299, 301, 303, 305, 319, 321, 325, 333, 339, 343, 351, 361, 365, 371, 377, 387, 393, 399

Offset: 1

Gus Wiseman, Sep 22 2019

nonn

First differs from A322336 in lacking 2535 = prime(2)*prime(3)*prime(6)^2.

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. Heinz numbers of integer partitions with a common divisor > 1 are A318978, and the enumeration of these partitions by sum is A108572.

			The sequence of terms together with their prime indices begins:
    9: {2,2}
   21: {2,4}
   25: {3,3}
   27: {2,2,2}
   39: {2,6}
   49: {4,4}
   57: {2,8}
   63: {2,2,4}
   65: {3,6}
   81: {2,2,2,2}
   87: {2,10}
   91: {4,6}
  111: {2,12}
  115: {3,9}
  117: {2,2,6}
  121: {5,5}
  125: {3,3,3}
  129: {2,14}
  133: {4,8}
  147: {2,4,4}

Cf. A000837, A007916, A018783, A056239, A108572, A112798, A289509, A318978, A327407, A327658.

Select[Range[100],#>1&&!PrimeQ[#]&&GCD@@PrimePi/@First/@FactorInteger[#]>1&]

Complement of A000040 in A318978.

A338552 Non-powers of primes whose prime indices have a common divisor > 1.

Original entry on oeis.org

21, 39, 57, 63, 65, 87, 91, 111, 115, 117, 129, 133, 147, 159, 171, 183, 185, 189, 203, 213, 235, 237, 247, 259, 261, 267, 273, 299, 301, 303, 305, 319, 321, 325, 333, 339, 351, 365, 371, 377, 387, 393, 399, 417, 427, 441, 445, 453, 477, 481, 489, 497, 507

Offset: 1

Gus Wiseman, Nov 03 2020

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.

Also Heinz numbers of non-constant, non-relatively prime partitions. 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 sequence of terms together with their prime indices begins:
     21: {2,4}      183: {2,18}       305: {3,18}
     39: {2,6}      185: {3,12}       319: {5,10}
     57: {2,8}      189: {2,2,2,4}    321: {2,28}
     63: {2,2,4}    203: {4,10}       325: {3,3,6}
     65: {3,6}      213: {2,20}       333: {2,2,12}
     87: {2,10}     235: {3,15}       339: {2,30}
     91: {4,6}      237: {2,22}       351: {2,2,2,6}
    111: {2,12}     247: {6,8}        365: {3,21}
    115: {3,9}      259: {4,12}       371: {4,16}
    117: {2,2,6}    261: {2,2,10}     377: {6,10}
    129: {2,14}     267: {2,24}       387: {2,2,14}
    133: {4,8}      273: {2,4,6}      393: {2,32}
    147: {2,4,4}    299: {6,9}        399: {2,4,8}
    159: {2,16}     301: {4,14}       417: {2,34}
    171: {2,2,8}    303: {2,26}       427: {4,18}

A318978 allows prime powers, counted by A018783, with complement A289509.
A327685 allows nonprime prime powers.
A338330 is the coprime instead of relatively prime version.
A338554 counts the partitions with these Heinz numbers.
A338555 is the complement.
A000740 counts relatively prime compositions.
A000961 lists powers of primes, with complement A024619.
A051424 counts pairwise coprime or singleton partitions.
A108572 counts nontrivial periodic partitions, with Heinz numbers A001597.
A291166 ranks relatively prime compositions, with complement A291165.
A302696 gives the Heinz numbers of pairwise coprime partitions.
A327516 counts pairwise coprime partitions, with Heinz numbers A302696.
Cf. A000005, A000837, A007916, A056239, A112798, A289508, A302569, A302796, A318716, A327658, A328867, A328677, A338331, A338553.

Select[Range[100],!(#==1||PrimePowerQ[#]||GCD@@PrimePi/@First/@FactorInteger[#]==1)&]

Equals A024619 /\ A318978.

Complement of A000961 \/ A289509.

A338555 Numbers that are either a power of a prime or have relatively prime prime indices.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 64, 66, 67, 68, 69, 70, 71, 72

Offset: 1

Gus Wiseman, Nov 03 2020

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.

Also Heinz numbers of partitions either constant or relatively prime (A338553). 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.

A327534 uses primes instead of prime powers.
A338331 is the pairwise coprime version, with complement A338330.
A338552 is the complement.
A338553 counts the partitions with these Heinz numbers.
A000837 counts relatively prime partitions, with Heinz numbers A289509.
A000961 lists powers of primes.
A018783 counts partitions whose prime indices are not relatively prime, with Heinz numbers A318978.
A051424 counts pairwise coprime or singleton partitions.
A291166 ranks relatively prime compositions, with complement A291165.
A327516 counts pairwise coprime partitions, with Heinz numbers A302696.
Cf. A000740, A056239, A108572, A112798, A302569, A302796, A327685, A328677.

Select[Range[100],#==1||PrimePowerQ[#]||GCD@@PrimePi/@First/@FactorInteger[#]==1&]

Equals A000961 \/ A289509.

Complement of A024619 /\ A318978.

A366853 Number of integer partitions of n into odd, pairwise coprime parts.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 17, 18, 20, 22, 25, 29, 33, 36, 39, 43, 49, 55, 61, 66, 69, 75, 85, 94, 104, 113, 120, 129, 143, 159, 172, 183, 193, 207, 226, 251, 272, 288, 304, 325, 350, 383, 414, 437, 460, 494, 532, 577, 622, 655, 684

Offset: 0

Gus Wiseman, Nov 01 2023

nonn

			The a(1) = 1 through a(10) = 7 partitions:
1  11  3    31    5      51      7        53        9          73
       111  1111  311    3111    511      71        531        91
                  11111  111111  31111    5111      711        5311
                                 1111111  311111    51111      7111
                                          11111111  3111111    511111
                                                    111111111  31111111
                                                               1111111111

Partitions into odd parts are counted by A000009, ranks A066208.
Allowing even parts gives A051424.
For relatively prime (not pairwise coprime): A366843, with evens A000837.
A000041 counts integer partitions, strict A000009 (also into odds).
A101268 counts pairwise coprime compositions.
A168532 counts partitions by gcd.
Cf. A007359, A018783, A055922, A078374, A337485, A366844, A366852.

pwcop[y_]:=And@@(GCD@@#==1&)/@Subsets[y,{2}]
Table[Length[Select[IntegerPartitions[n],And@@OddQ/@#&&pwcop[#]&]],{n,0,30}]

A319179 Number of integer partitions of n that are relatively prime but not aperiodic. Number of integer partitions of n that are aperiodic but not relatively prime.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 3, 2, 6, 1, 9, 1, 14, 7, 17, 1, 32, 1, 36, 15, 55, 1, 77, 6, 100, 27, 121, 1, 200, 1, 209, 56, 296, 19, 403, 1, 489, 101, 596, 1, 885, 1, 947, 192, 1254, 1, 1673, 14, 1979, 297, 2336, 1, 3300, 60, 3594, 490, 4564, 1, 5988, 1, 6841, 800

Offset: 1

Gus Wiseman, Sep 12 2018

nonn

An integer partition is aperiodic if its multiplicities are relatively prime.

			The a(12) = 9 integer partitions that are relatively prime but not aperiodic:
  (5511),
  (332211), (333111), (441111),
  (22221111), (33111111),
  (222111111),
  (2211111111),
  (111111111111).
The a(12) = 9 integer partitions that are aperiodic but not relatively prime:
  (12),
  (8,4), (9,3), (10,2),
  (6,3,3), (6,4,2), (8,2,2),
  (6,2,2,2),
  (4,2,2,2,2).

Cf. A000837, A018783, A047966, A100953, A305563, A319149, A319160, A319162, A319164.

Table[Length[Select[IntegerPartitions[n],And[GCD@@#==1,GCD@@Length/@Split[#]>1]&]],{n,30}]

cp's OEIS Frontend

A366847 Numbers whose halved even prime indices are nonempty and relatively prime.

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A202385 Number of partitions of n into distinct parts having pairwise common factors but no overall common factor.

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A303553 Number of periodic factorizations of n > 1 into positive factors greater than 1; a(1) = 1 by convention.

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A305735 Number of integer partitions of n whose greatest common divisor is a prime number.

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A318979 Number of divisors of n with relatively prime prime indices, meaning they belong to A289509.

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A327685 Nonprime numbers whose prime indices have a common divisor > 1.

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A338552 Non-powers of primes whose prime indices have a common divisor > 1.

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A338555 Numbers that are either a power of a prime or have relatively prime prime indices.

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