A302243 -id:A302243 - cp's OEIS frontend

A302242 Total weight of the n-th multiset multisystem. Totally additive with a(prime(n)) = Omega(n).

0, 0, 1, 0, 1, 1, 2, 0, 2, 1, 1, 1, 2, 2, 2, 0, 1, 2, 3, 1, 3, 1, 2, 1, 2, 2, 3, 2, 2, 2, 1, 0, 2, 1, 3, 2, 3, 3, 3, 1, 1, 3, 2, 1, 3, 2, 2, 1, 4, 2, 2, 2, 4, 3, 2, 2, 4, 2, 1, 2, 3, 1, 4, 0, 3, 2, 1, 1, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 2, 1, 4, 1, 1, 3, 2, 2, 3, 1, 4

Offset: 1

Gus Wiseman, Apr 03 2018

nonn

A multiset multisystem is a finite multiset of finite multisets of positive integers. The n-th multiset multisystem is constructed by factoring n into prime numbers and then factoring each prime index into prime numbers and taking their prime indices. This produces a unique multiset multisystem for each n, and every possible multiset multisystem is so constructed as n ranges over all positive integers.

			Sequence of finite multisets of finite multisets of positive integers begins: (), (()), ((1)), (()()), ((2)), (()(1)), ((11)), (()()()), ((1)(1)), (()(2)), ((3)), (()()(1)), ((12)), (()(11)), ((1)(2)), (()()()()), ((4)), (()(1)(1)), ((111)), (()()(2)).

Alois P. Heinz, Table of n, a(n) for n = 1..65536

Cf. A001222, A003963, A007716, A034691, A056239, A061775, A063834, A096443, A249620, A255397, A255906, A275024, A279789, A281113, A299757, A302243.

with(numtheory):
a:= n-> add(add(j[2], j=ifactors(pi(i[1]))[2])*i[2], i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Sep 07 2018

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[PrimeOmega/@primeMS[n]],{n,100}]

a(n,f=factor(n))=sum(i=1,#f~, bigomega(primepi(f[i,1]))*f[i,2]) \\ Charles R Greathouse IV, Nov 10 2021

A302478 Products of prime numbers of squarefree index.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 20, 22, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 36, 39, 40, 41, 43, 44, 45, 47, 48, 50, 51, 52, 54, 55, 58, 59, 60, 62, 64, 65, 66, 67, 68, 72, 73, 75, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 90, 93, 94

Offset: 1

Gus Wiseman, Apr 08 2018

nonn

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

			Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of set multisystems.
01:  {}
02:  {{}}
03:  {{1}}
04:  {{},{}}
05:  {{2}}
06:  {{},{1}}
08:  {{},{},{}}
09:  {{1},{1}}
10:  {{},{2}}
11:  {{3}}
12:  {{},{},{1}}
13:  {{1,2}}
15:  {{1},{2}}
16:  {{},{},{},{}}
17:  {{4}}
18:  {{},{1},{1}}
20:  {{},{},{2}}
22:  {{},{3}}
24:  {{},{},{},{1}}
25:  {{2},{2}}
26:  {{},{1,2}}
27:  {{1},{1},{1}}
29:  {{1,3}}
30:  {{},{1},{2}}
31:  {{5}}
32:  {{},{},{},{},{}}

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

Cf. A000961, A001222, A003963, A005117, A007716, A050320, A056239, A063834, A076610, A270995, A275024, A281113, A296119, A301753, A302242, A302243, A302491.

Select[Range[100],Or[#===1,And@@SquareFreeQ/@PrimePi/@FactorInteger[#][[All,1]]]&]

ok(n)={!#select(p->!issquarefree(primepi(p)), factor(n)[,1])} \\ Andrew Howroyd, Aug 26 2018

A302494 Products of distinct primes of squarefree index.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 11, 13, 15, 17, 22, 26, 29, 30, 31, 33, 34, 39, 41, 43, 47, 51, 55, 58, 59, 62, 65, 66, 67, 73, 78, 79, 82, 83, 85, 86, 87, 93, 94, 101, 102, 109, 110, 113, 118, 123, 127, 129, 130, 134, 137, 139, 141, 143, 145, 146, 149, 155, 157, 158, 163

Offset: 1

Gus Wiseman, Apr 08 2018

nonn

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

			Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of set systems.
01: {}
02: {{}}
03: {{1}}
05: {{2}}
06: {{},{1}}
10: {{},{2}}
11: {{3}}
13: {{1,2}}
15: {{1},{2}}
17: {{4}}
22: {{},{3}}
26: {{},{1,2}}
29: {{1,3}}
30: {{},{1},{2}}
31: {{5}}
33: {{1},{3}}
34: {{},{4}}
39: {{1},{1,2}}

Cf. A000961, A001222, A003963, A005117, A007716, A056239, A270995, A275024, A276625, A277098, A279785, A281113, A296120, A301754, A302242, A302243.

Select[Range[100],Or[#===1,SquareFreeQ[#]&&And@@SquareFreeQ/@PrimePi/@FactorInteger[#][[All,1]]]&]

is(n) = if(bigomega(n)!=omega(n), return(0), my(f=factor(n)[, 1]~); for(k=1, #f, if(!issquarefree(primepi(f[k])) && primepi(f[k])!=1, return(0)))); 1 \\ Felix Fröhlich, Apr 10 2018

A302505 Numbers whose prime indices are squarefree and have disjoint prime indices.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, 15, 16, 17, 20, 22, 24, 26, 29, 30, 31, 32, 33, 34, 40, 41, 43, 44, 47, 48, 51, 52, 55, 58, 59, 60, 62, 64, 66, 67, 68, 73, 79, 80, 82, 83, 85, 86, 88, 93, 94, 96, 101, 102, 104, 109, 110, 113, 116, 118, 120, 123, 124, 127

Offset: 1

Gus Wiseman, Apr 09 2018

nonn

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

			Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of set multisystems.
01: {}
02: {{}}
03: {{1}}
04: {{},{}}
05: {{2}}
06: {{},{1}}
08: {{},{},{}}
10: {{},{2}}
11: {{3}}
12: {{},{},{1}}
13: {{1,2}}
15: {{1},{2}}
16: {{},{},{},{}}
17: {{4}}
20: {{},{},{2}}
22: {{},{3}}
24: {{},{},{},{1}}
26: {{},{1,2}}
29: {{1,3}}
30: {{},{1},{2}}
31: {{5}}
32: {{},{},{},{},{}}

Cf. A000961, A001222, A003963, A005117, A007716, A056239, A275024, A279790, A281113, A294788, A301750, A302242, A302243.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],UnsameQ@@Join@@primeMS/@primeMS[#]&]

A302590 Squarefree numbers whose prime indices are prime numbers.

Original entry on oeis.org

1, 3, 5, 11, 15, 17, 31, 33, 41, 51, 55, 59, 67, 83, 85, 93, 109, 123, 127, 155, 157, 165, 177, 179, 187, 191, 201, 205, 211, 241, 249, 255, 277, 283, 295, 327, 331, 335, 341, 353, 367, 381, 401, 415, 431, 451, 461, 465, 471, 509, 527, 537, 545, 547, 561, 563

Offset: 1

Gus Wiseman, Apr 10 2018

nonn

A prime index of n is a number m such that prime(m) divides n.
From David A. Corneth, Feb 05 2021: (Start)
Product_{p in A006450} (p + 1)/p where primepi(p) <= 10^k for k = 3..9 respectively is
3221793975627545730894469494385382768...
3962097386916566795581118542505513350...
4423525010102788492232765893521739629...
4739349879225654126399615785205666552...
4969363158706022367680967716958174889...
5144436325229538304870684054018856517...
5282263225826916578696019016723107071... (End)

			Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of set systems.
001: {}
003: {{1}}
005: {{2}}
011: {{3}}
015: {{1},{2}}
017: {{4}}
031: {{5}}
033: {{1},{3}}
041: {{6}}
051: {{1},{4}}
055: {{2},{3}}
059: {{7}}
067: {{8}}
083: {{9}}
085: {{2},{4}}
093: {{1},{5}}
109: {{10}}
123: {{1},{6}}
127: {{11}}
155: {{2},{5}}
157: {{12}}
165: {{1},{2},{3}}

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

Cf. A000961, A001222, A003963, A005117, A006450, A007716, A056239, A076610, A275024, A281113, A302242, A302243, A302568.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[600],SquareFreeQ[#]&&And@@PrimeQ/@primeMS[#]&]

ok(n)={issquarefree(n) && !#select(p->!isprime(primepi(p)), factor(n)[,1])} \\ Andrew Howroyd, Aug 26 2018

Intersection of A005117 and A076610.

Sum_{n>=1} 1/a(n) = Product_{p in A006450} (1 + 1/p) converges since the sum of the reciprocals of A006450 converges. - Amiram Eldar, Feb 02 2021

A302540 Numbers whose prime indices other than 1 are prime numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 15, 16, 17, 18, 20, 22, 24, 25, 27, 30, 31, 32, 33, 34, 36, 40, 41, 44, 45, 48, 50, 51, 54, 55, 59, 60, 62, 64, 66, 67, 68, 72, 75, 80, 81, 82, 83, 85, 88, 90, 93, 96, 99, 100, 102, 108, 109, 110, 118, 120, 121, 123, 124

Offset: 1

Gus Wiseman, Apr 09 2018

nonn

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

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

Cf. A000961, A001222, A003963, A005117, A006450, A007716, A056239, A076610, A275024, A281113, A291686, A302242, A302243, A302534, A302539.

Select[Range[400],#===1||And@@(#===1||PrimeQ[#]&)/@PrimePi/@FactorInteger[#][[All,1]]&]

ok(n)={!#select(p->p>2 && !isprime(primepi(p)), factor(n)[,1])} \\ Andrew Howroyd, Aug 26 2018

Sum_{n>=1} 1/a(n) = 2 * Sum_{n>=1} 1/A076610(n) = 2 * Product_{p in A006450} p/(p-1) converges since the sum of the reciprocals of A006450 converges. - Amiram Eldar, Feb 02 2021

A302492 Products of any power of 2 with prime numbers of prime-power index, i.e., prime numbers p of the form p = prime(q^k), for q prime, k >= 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 30, 31, 32, 33, 34, 35, 36, 38, 40, 41, 42, 44, 45, 46, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 60, 62, 63, 64, 66, 67, 68, 69, 70, 72, 75, 76, 77, 80, 81, 82, 83

Offset: 1

Gus Wiseman, Apr 08 2018

nonn

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

			Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of multiset multisystems.
01: {}
02: {{}}
03: {{1}}
04: {{},{}}
05: {{2}}
06: {{},{1}}
07: {{1,1}}
08: {{},{},{}}
09: {{1},{1}}
10: {{},{2}}
11: {{3}}
12: {{},{},{1}}
14: {{},{1,1}}
15: {{1},{2}}
16: {{},{},{},{}}
17: {{4}}
18: {{},{1},{1}}
19: {{1,1,1}}
20: {{},{},{2}}
21: {{1},{1,1}}
22: {{},{3}}
23: {{2,2}}
24: {{},{},{},{1}}

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

Cf. A000961, A001222, A003963, A005117, A007716, A050361, A056239, A076610, A270995, A275024, A279784, A281113, A295935, A301762, A302242, A302243, A302493.

Select[Range[100],Or[#===1,And@@PrimePowerQ/@PrimePi/@DeleteCases[FactorInteger[#][[All,1]],2]]&]

ok(n)={!#select(p->p<>2&&!isprimepower(primepi(p)), factor(n)[,1])} \\ Andrew Howroyd, Aug 26 2018

A318995 Totally additive with a(prime(n)) = n - 1.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 07 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A000040, A001222, A003963, A056239, A275024, A299757, A302242, A302243, A318994.

a:= n-> add((-1+numtheory[pi](i[1]))*i[2], i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Sep 07 2018

Table[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>(PrimePi[p]-1)*k]//Total,{n,200}]

a(n)={my(f=factor(n)); sum(i=1, #f~, my([p, e]=f[i, ]); (primepi(p)-1)*e)} \\ Andrew Howroyd, Sep 07 2018

A358334 Number of twice-partitions of n into odd-length partitions.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 25, 43, 77, 137, 241, 410, 720, 1209, 2073, 3498, 5883, 9768, 16413, 26978, 44741, 73460, 120462, 196066, 320389, 518118, 839325, 1353283, 2178764, 3490105, 5597982, 8922963, 14228404, 22609823, 35875313, 56756240, 89761600, 141410896, 222675765

Offset: 0

Gus Wiseman, Dec 01 2022

nonn

A twice-partition of n (A063834) is a sequence of integer partitions, one of each part of an integer partition of n.

			The a(0) = 1 through a(5) = 13 twice-partitions:
  ()  ((1))  ((2))     ((3))        ((4))           ((5))
             ((1)(1))  ((111))      ((211))         ((221))
                       ((2)(1))     ((2)(2))        ((311))
                       ((1)(1)(1))  ((3)(1))        ((3)(2))
                                    ((111)(1))      ((4)(1))
                                    ((2)(1)(1))     ((11111))
                                    ((1)(1)(1)(1))  ((111)(2))
                                                    ((211)(1))
                                                    ((2)(2)(1))
                                                    ((3)(1)(1))
                                                    ((111)(1)(1))
                                                    ((2)(1)(1)(1))
                                                    ((1)(1)(1)(1)(1))

Andrew Howroyd, Table of n, a(n) for n = 0..1000

For multiset partitions of integer partitions: A356932, ranked by A356935.
For odd length instead of lengths we have A358824.
For odd sums instead of lengths we have A358825.
For odd sums also we have A358827.
For odd length also we have A358834.
A000041 counts integer partitions.
A027193 counts odd-length partitions, ranked by A026424.
A055922 counts partitions with odd multiplicities, also odd parts A117958.
A063834 counts twice-partitions, strict A296122, row-sums of A321449.
Cf. A000219, A001970, A072233, A078408, A270995, A279374, A298118, A300300, A300301, A300647, A302243.

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],OddQ[Times@@Length/@#]&]],{n,0,10}]

P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
R(u,y) = {1/prod(k=1, #u, 1 - u[k]*y*x^k + O(x*x^#u))}
seq(n) = {my(u=Vec(P(n,1)-P(n,-1))/2); Vec(R(u, 1), -(n+1))} \\ Andrew Howroyd, Dec 30 2022

G.f.: 1/Product_{k>=1} (1 - A027193(k)*x^k). - Andrew Howroyd, Dec 30 2022

Terms a(21) and beyond from Andrew Howroyd, Dec 30 2022

A302593 Numbers whose prime indices are powers of a common prime number.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 31, 32, 34, 36, 38, 40, 41, 42, 44, 46, 48, 49, 50, 53, 54, 56, 57, 59, 62, 63, 64, 67, 68, 72, 76, 80, 81, 82, 83, 84, 88, 92, 96, 97, 98, 100, 103, 106, 108, 109, 112

Offset: 1

Gus Wiseman, Apr 10 2018

nonn

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

			Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of set systems.
01: {}
02: {{}}
03: {{1}}
04: {{},{}}
05: {{2}}
06: {{},{1}}
07: {{1,1}}
08: {{},{},{}}
09: {{1},{1}}
10: {{},{2}}
11: {{3}}
12: {{},{},{1}}
14: {{},{1,1}}
16: {{},{},{},{}}
17: {{4}}
18: {{},{1},{1}}
19: {{1,1,1}}
20: {{},{},{2}}
21: {{1},{1,1}}
22: {{},{3}}
23: {{2,2}}
24: {{},{},{},{1}}
25: {{2},{2}}
27: {{1},{1},{1}}
28: {{},{},{1,1}}
31: {{5}}
32: {{},{},{},{},{}}
34: {{},{4}}
36: {{},{},{1},{1}}
38: {{},{1,1,1}}
40: {{},{},{},{2}}

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

Cf. A000961, A001222, A003963, A005117, A007716, A056239, A275024, A047968, A281113, A295924, A301760, A302242, A302243.

filter:= proc(n) local F,q;
  uses numtheory;
  F:= map(pi, factorset(n));
  nops(`union`(op(map(factorset,F)))) <= 1
end proc:
select(filter, [$1..200]); # Robert Israel, Oct 22 2020

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],SameQ@@Join@@primeMS/@primeMS[#]&]

cp's OEIS Frontend

A302242 Total weight of the n-th multiset multisystem. Totally additive with a(prime(n)) = Omega(n).

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A302478 Products of prime numbers of squarefree index.

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A302494 Products of distinct primes of squarefree index.

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A302505 Numbers whose prime indices are squarefree and have disjoint prime indices.

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A302590 Squarefree numbers whose prime indices are prime numbers.

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A302540 Numbers whose prime indices other than 1 are prime numbers.

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A302492 Products of any power of 2 with prime numbers of prime-power index, i.e., prime numbers p of the form p = prime(q^k), for q prime, k >= 1.

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A318995 Totally additive with a(prime(n)) = n - 1.

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