A318978 -id:A318978 - cp's OEIS frontend

A327659 Number of factorizations of A318978(n - 1), the n-th number that is 1 or whose prime indices have a common divisor > 1, into numbers > 1 satisfying the same conditions.

1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 4, 2, 1, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 4, 2, 3, 1, 2, 1, 2, 1, 1, 4, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 2, 7, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 7, 2, 1, 1

Offset: 1

Gus Wiseman, Sep 21 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. Numbers whose prime indices have a common divisor > 1 are listed in A318978.

Gus Wiseman, Sequences counting and encoding certain classes of multisets

See link for additional cross-references.

Cf. A056239, A112798, A281116, A289509, A318721, A327406.

nn=100;
facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
y=Select[Range[1000],GCD@@PrimePi/@First/@FactorInteger[#]!=1&];
Table[Length[facsusing[Rest[y],n]],{n,y}]

a(n) = A001055(A318978(n - 1)).

A328336 Numbers with no consecutive prime indices relatively prime.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 37, 39, 41, 43, 47, 49, 53, 57, 59, 61, 63, 65, 67, 71, 73, 79, 81, 83, 87, 89, 91, 97, 101, 103, 107, 109, 111, 113, 115, 117, 121, 125, 127, 129, 131, 133, 137, 139, 147, 149, 151, 157, 159, 163, 167

Offset: 1

Gus Wiseman, Oct 14 2019

nonn

First differs from A318978 in having 897, with prime indices {2, 6, 9}.
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 Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of partitions no consecutive parts relatively prime (A328187).
Besides the initial 1 this differs from A305078: 47541=897*prime(16) is in A305078 but not in this set. - Andrey Zabolotskiy, Nov 13 2019

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   5: {3}
   7: {4}
   9: {2,2}
  11: {5}
  13: {6}
  17: {7}
  19: {8}
  21: {2,4}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  29: {10}
  31: {11}
  37: {12}
  39: {2,6}
  41: {13}
  43: {14}

Numbers with consecutive prime indices relatively prime are A328335.
Strict partitions with no consecutive parts relatively prime are A328220.
Numbers with relatively prime prime indices are A289509.
Cf. A000837, A056239, A078374, A112798, A281116, A289508, A318981, A328168, A328169, A328172, A328187, A328188.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!MatchQ[primeMS[#],{_,x_,y_,_}/;GCD[x,y]==1]&]

A328167 GCD of the prime indices of n, all minus 1.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 3, 0, 1, 2, 4, 1, 5, 3, 1, 0, 6, 1, 7, 2, 1, 4, 8, 1, 2, 5, 1, 3, 9, 1, 10, 0, 1, 6, 1, 1, 11, 7, 1, 2, 12, 1, 13, 4, 1, 8, 14, 1, 3, 2, 1, 5, 15, 1, 2, 3, 1, 9, 16, 1, 17, 10, 1, 0, 1, 1, 18, 6, 1, 1, 19, 1, 20, 11, 1, 7, 1, 1, 21, 2, 1, 12

Offset: 1

Gus Wiseman, Oct 08 2019

nonn

Zeros are ignored when computing GCD, and the empty set has GCD 0.

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.

			85 has prime indices {3,7}, so a(85) = GCD(2,6) = 2.

Positions of 0's are A000079.
Positions of 1's are A328168.
Positions of records (first appearances) are A006005.
The GCD of the prime indices of n is A289508(n).
The GCD of the prime indices of n, all plus 1, is A328169(n).
Looking at divisors instead of prime indices gives A258409.
Partitions whose parts minus 1 are relatively prime are A328170.
Cf. A000837, A001222, A007359, A056239, A112798, A289509, A318978, A318981, A328163, A328164.

Table[GCD@@(PrimePi/@First/@If[n==1,{},FactorInteger[n]]-1),{n,100}]

A318981 Numbers whose prime indices plus 1 are relatively prime.

Original entry on oeis.org

1, 6, 12, 14, 15, 18, 21, 24, 26, 28, 30, 35, 36, 38, 39, 42, 45, 48, 51, 52, 54, 56, 58, 60, 63, 65, 66, 69, 70, 72, 74, 75, 76, 77, 78, 84, 86, 87, 90, 91, 95, 96, 98, 102, 104, 105, 106, 108, 111, 112, 114, 116, 117, 119, 120, 122, 123, 126, 130, 132, 133

Offset: 1

Gus Wiseman, Sep 06 2018

nonn

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

			The sequence of integer partitions whose Heinz numbers are in the sequence begins: (), (21), (211), (41), (32), (221), (42), (2111), (61), (411), (321), (43), (2211), (81), (62), (421), (322), (21111).

Cf. A000837, A007359, A018783, A281116, A289508, A289509, A318978, A318980.

Select[Range[100],GCD@@(PrimePi/@FactorInteger[#][[All,1]]+1)==1&]

A318980 Number of integer partitions of n whose parts plus 1 are relatively prime.

Original entry on oeis.org

0, 0, 1, 1, 4, 5, 9, 13, 21, 29, 43, 56, 79, 109, 146, 192, 254, 329, 428, 553, 707, 900, 1139, 1434, 1800, 2251, 2799, 3472, 4286, 5275, 6469, 7918, 9655, 11755, 14252, 17248, 20817, 25084, 30134, 36142, 43235, 51644, 61548, 73241, 86961, 103108, 122010

Offset: 1

Gus Wiseman, Sep 06 2018

nonn

			The a(7) = 9 partitions are (61), (43), (421), (4111), (322), (3211), (2221), (22111), (211111).
The a(8) = 13 partitions:
  (62),
  (332), (422), (431), (521), (611),
  (3221), (4211),
  (22211), (32111), (41111),
  (221111),
  (2111111).

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

Cf. A000837, A007359, A018783, A051424, A281116, A289508, A289509, A318978.

Table[Length[Select[IntegerPartitions[n],GCD@@(#+1)==1&]],{n,30}]

seq(n)={Vec(sum(d=1, n+1, moebius(d)*(-1 + 1/prod(k=ceil(2/d), (n+1)\d, 1 - x^(k*d-1) + O(x*x^n)))), -n)} \\ Andrew Howroyd, Oct 17 2019

G.f.: Sum_{d>=1} mu(d)*(-1 + 1/(Prod_{k>=2/d} 1 - x^(k*d - 1))). - Andrew Howroyd, Oct 17 2019

A327656 Maximum divisor of n that is 1 or whose prime indices have a common divisor > 1.

Original entry on oeis.org

1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, 3, 13, 7, 5, 1, 17, 9, 19, 5, 21, 11, 23, 3, 25, 13, 27, 7, 29, 5, 31, 1, 11, 17, 7, 9, 37, 19, 39, 5, 41, 21, 43, 11, 9, 23, 47, 3, 49, 25, 17, 13, 53, 27, 11, 7, 57, 29, 59, 5, 61, 31, 63, 1, 65, 11, 67, 17, 23, 7, 71, 9, 73

Offset: 1

Gus Wiseman, Sep 21 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. Numbers whose prime indices have a common divisor > 1 are listed in A318978, which is the union of this sequence without 1.

			The divisors of 90 that are 1 or whose prime indices have a common divisor > 1 are {1, 3, 5, 9}, so a(90) = 9.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Gus Wiseman, Sequences counting and encoding certain classes of multisets
Index entries for sequences computed from indices in prime factorization

The union consists of {1} followed by A318978.
See link for additional cross-references.
Cf. A000005, A006530, A056239, A112798, A281116, A289509, A327406, A327657.

Table[Max[Select[Divisors[n],GCD@@PrimePi/@First/@FactorInteger[#]!=1&]],{n,100}]

A327656(n) = vecmax(select(d -> (1==d)||(gcd(apply(primepi,factor(d)[, 1]~))>1), divisors(n))); \\ Antti Karttunen, Dec 06 2021

a(n) = n/A327405(n).

n belongs to A318978 iff a(n) = 1.

A366846 Numbers whose odd prime indices are relatively prime.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 55, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 85, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122

Offset: 1

Gus Wiseman, Oct 29 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.

			The odd prime indices of 115 are {3,9}, and these are not relatively prime, so 115 is not in the sequence.
The odd prime indices of 825 are {3,3,5}, and these are relatively prime, so 825 is in the sequence.

Including even indices gives A289509, ones of A289508, counted by A000837.
The complement when including even indices is A318978, counted by A018783.
The nonzero complement ranks the partitions counted by A366842.
The version for halved even indices is A366847.
The odd case is A366848.
The partitions with these Heinz numbers are counted by A366850.
A000041 counts integer partitions, strict A000009 (also into odds).
A112798 lists prime indices, length A001222, sum A056239.
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, A087436, A302696, A302697, A325698, A366843, A366844, A366849.

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

A327406 Number of steps to reach a fixed point starting with n and repeatedly taking the quotient by the maximum divisor that is 1 or whose prime indices have a common divisor > 1 (A327405, A327656).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 21 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. Numbers whose prime indices have a common divisor > 1 are listed in A318978.

Note that A318978 includes also all odd primes and their powers, thus the only numbers for which a maximum such divisor is 1 are the powers of 2. Therefore A000079 gives the indices of zeros in this sequence. - Antti Karttunen, Dec 06 2021

			We have 5115 -> 165 -> 15 -> 3 -> 1, so a(5115) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Gus Wiseman, Sequences counting and encoding certain classes of multisets
Index entries for sequences computed from indices in prime factorization

First appearance of n is A080696(n).
See link for additional cross-references.
Cf. A000005, A000079 (positions of 0's), A056239, A112798, A281116, A289509, A302569, A318978.

Table[Length[FixedPointList[#/Max[Select[Divisors[#],GCD@@PrimePi/@First/@FactorInteger[#]!=1&]]&,n]]-2,{n,100}]

A327405(n) = (n / vecmax(select(d -> (1==d)||(gcd(apply(primepi,factor(d)[, 1]~))>1), divisors(n))));
A327406(n) = { my(u = A327405(n), k=0); while(u!=n, k++; n = u; u = A327405(n)); (k); }; \\ Antti Karttunen, Dec 06 2021

Data section extended up to 105 terms by Antti Karttunen, Dec 06 2021

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

Original entry on oeis.org

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&]

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

cp's OEIS Frontend

A327659 Number of factorizations of A318978(n - 1), the n-th number that is 1 or whose prime indices have a common divisor > 1, into numbers > 1 satisfying the same conditions.

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A328336 Numbers with no consecutive prime indices relatively prime.

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A328167 GCD of the prime indices of n, all minus 1.

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A318981 Numbers whose prime indices plus 1 are relatively prime.

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A318980 Number of integer partitions of n whose parts plus 1 are relatively prime.

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A327656 Maximum divisor of n that is 1 or whose prime indices have a common divisor > 1.

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A366846 Numbers whose odd prime indices are relatively prime.

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A327406 Number of steps to reach a fixed point starting with n and repeatedly taking the quotient by the maximum divisor that is 1 or whose prime indices have a common divisor > 1 (A327405, A327656).

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