A353333 -id:A353333

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A340784

Heinz numbers of even-length integer partitions of even numbers.

+10
16

1, 4, 9, 10, 16, 21, 22, 25, 34, 36, 39, 40, 46, 49, 55, 57, 62, 64, 81, 82, 84, 85, 87, 88, 90, 91, 94, 100, 111, 115, 118, 121, 129, 133, 134, 136, 144, 146, 155, 156, 159, 160, 166, 169, 183, 184, 187, 189, 194, 196, 198, 203, 205, 206, 210, 213, 218, 220

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are positive integers whose number of prime indices and sum of prime indices are both even, counting multiplicity in both cases.

A multiplicative semigroup: if m and n are in the sequence, then so is m*n. - Antti Karttunen, Jul 28 2024

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..20000

Index entries for sequences computed from indices in prime factorization

FORMULA

Intersection of A028260 and A300061.

EXAMPLE

The sequence of partitions together with their Heinz numbers begins:

1: () 57: (8,2) 118: (17,1)

4: (1,1) 62: (11,1) 121: (5,5)

9: (2,2) 64: (1,1,1,1,1,1) 129: (14,2)

10: (3,1) 81: (2,2,2,2) 133: (8,4)

16: (1,1,1,1) 82: (13,1) 134: (19,1)

21: (4,2) 84: (4,2,1,1) 136: (7,1,1,1)

22: (5,1) 85: (7,3) 144: (2,2,1,1,1,1)

25: (3,3) 87: (10,2) 146: (21,1)

34: (7,1) 88: (5,1,1,1) 155: (11,3)

36: (2,2,1,1) 90: (3,2,2,1) 156: (6,2,1,1)

39: (6,2) 91: (6,4) 159: (16,2)

40: (3,1,1,1) 94: (15,1) 160: (3,1,1,1,1,1)

46: (9,1) 100: (3,3,1,1) 166: (23,1)

49: (4,4) 111: (12,2) 169: (6,6)

55: (5,3) 115: (9,3) 183: (18,2)

MATHEMATICA

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

Select[Range[100], EvenQ[PrimeOmega[#]]&&EvenQ[Total[primeMS[#]]]&]

PROG

(PARI)

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }

A353331(n) = ((!(bigomega(n)%2)) && (!(A056239(n)%2)));

isA340784(n) = A353331(n); \\ Antti Karttunen, Apr 14 2022

CROSSREFS

Note: A-numbers of Heinz-number sequences are in parentheses below.

The case of prime powers is A056798.

These partitions are counted by A236913.

The odd version is A160786 (A340931).

A000009 counts partitions into odd parts (A066208).

A001222 counts prime factors.

A047993 counts balanced partitions (A106529).

A056239 adds up prime indices.

A058695 counts partitions of odd numbers (A300063).

A061395 selects the maximum prime index.

A072233 counts partitions by sum and length.

A112798 lists the prime indices of each positive integer.

- Even -

A027187 counts partitions of even length/maximum (A028260/A244990).

A034008 counts compositions of even length.

A035363 counts partitions into even parts (A066207).

A058696 counts partitions of even numbers (A300061).

A067661 counts strict partitions of even length (A030229).

A339846 counts factorizations of even length.

A340601 counts partitions of even rank (A340602).

A340785 counts factorizations into even factors.

A340786 counts even-length factorizations into even factors.

Cf. A026424, A257541, A300272, A326837, A326845, A340385 (A340386), A340604, A353331 (characteristic function), A353332, A353333, A353334.

Squares (A000290) is a subsequence.

Not a subsequence of A329609 (30 is the first term of A329609 not occurring here, and 210 is the first term here not present in A329609).

Positions of even terms in A373381.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 30 2021

STATUS

approved

A353331

a(n) = 1 if A001222(n) [bigomega(n)] and A056239(n) are both even, otherwise a(n) = 0.

+10
8

1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for characteristic functions

Index entries for sequences computed from indices in prime factorization

FORMULA

a(n) = [A008836(n) = +1] * [A056239(n) = 0 (mod 2)], where [ ] is the Iverson bracket.

a(n^2) = 1 for all n.

a(n) = a(A003961(n)) = a(A348717(n)), for all n >= 1.

PROG

(PARI)

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }

A353331(n) = ((!(bigomega(n)%2)) && (!(A056239(n)%2)));

CROSSREFS

Characteristic function of A340784.

Cf. A000290, A001222, A003961, A008836, A056239, A348717, A353332 (inverse Möbius transform), A353333, A353334.

Differs from A353269 for the first time at n=30.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Apr 14 2022

STATUS

approved

A353334

Number of factorizations of the square of n into factors k > 1 for which both A001222(k) and A056239(k) are even.

+10
7

1, 1, 1, 2, 1, 2, 1, 3, 2, 3, 1, 4, 1, 2, 2, 5, 1, 4, 1, 6, 3, 3, 1, 7, 2, 2, 3, 4, 1, 7, 1, 7, 2, 3, 2, 9, 1, 2, 3, 12, 1, 7, 1, 6, 4, 3, 1, 12, 2, 6, 2, 4, 1, 7, 3, 7, 3, 2, 1, 17, 1, 3, 6, 11, 2, 7, 1, 6, 2, 7, 1, 16, 1, 2, 4, 4, 2, 7, 1, 21, 5, 3, 1, 16, 3, 2, 3, 12, 1, 16, 3, 6, 2, 3, 2, 19, 1, 4, 4, 16, 1, 7

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

Number of factorizations of n^2 into terms of A340784 that are larger than one.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10080

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Index entries for sequences computed from indices in prime factorization

FORMULA

a(n) = A353333(A000290(n)).

a(n) = a(A003961(n)) = a(A348717(n)), for all n >= 1.

a(p) = 1 for all primes p.

PROG

(PARI)

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }

A353331(n) = ((!(bigomega(n)%2)) && (!(A056239(n)%2)));

A353333(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1) && (d<=m) && A353331(d), s += A353333(n/d, d))); (s));

A353334(n) = A353333(n^2);

CROSSREFS

Cf. A000290, A001222, A003961, A056239, A340784, A348717, A353331, A353333.

Differs from A353304 for the first time at n=30, where a(30) = 7, while A353304(30) = 8.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Apr 14 2022

STATUS

approved

A353353

Number of ways to write n as a product of the terms of A332820 larger than 1; a(1) = 1 by convention (an empty product).

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,36

COMMENTS

Number of factorizations of n into factors k > 1 for which A048675(k) is a multiple of three.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences computed from indices in prime factorization

FORMULA

a(n) = a(A003961(n)) = a(A348717(n)), for all n >= 1.

a(p) = 0 for all primes p.

EXAMPLE

Of the eight divisors of 36 larger than 1, [2, 3, 4, 6, 9, 12, 18, 36], only 6 and 36 are in A332820, and because these allow two different factorizations as 36 = 6*6, we have a(36) = 2.

PROG

(PARI)

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };

A353350(n) = (0==(A048675(n)%3));

A353353(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&A353350(d), s += A353353(n/d, d))); (s));

CROSSREFS

Cf. A003961, A048675, A332820, A348717, A353350, A353359 [= a(n^3)].

Cf. also A353303, A353333.

KEYWORD

nonn,easy

AUTHOR

Antti Karttunen, Apr 15 2022

STATUS

approved

A353332

Number of divisors d of n for which both A001222(d) and A056239(d) are even.

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

Number of terms of A340784 that divide n.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences computed from indices in prime factorization

FORMULA

a(n) = Sum_{d|n} A353331(d).

a(n) = a(A003961(n)) = a(A348717(n)), for all n >= 1.

EXAMPLE

Of the 9 divisors of 36, only divisors 1, 4, 9 and 36 satisfy the condition, therefore a(36) = 4.

PROG

(PARI)

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }

A353331(n) = ((!(bigomega(n)%2)) && (!(A056239(n)%2)));

A353332(n) = sumdiv(n, d, A353331(d));

CROSSREFS

Cf. A001222, A003961, A056239, A340784, A348717.

Inverse Möbius transform of A353331. Cf. also A353333, A353334.

Differs from A353362 for the first time at n=30, where a(30) = 2, while A353362(30) = 3.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Apr 14 2022

STATUS

approved

A353377

Number of ways to write n as a product of the terms of A345452 larger than 1; a(1) = 1 by convention (an empty product).

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,16

COMMENTS

Number of factorizations of n into factors k > 1 for which there is an even number of primes (when counted with multiplicity, A001222) in their prime factorization, and the 2-adic valuation of k (A007814) is also even.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

FORMULA

For all n >= 1, a(n) <= A353337(n).

EXAMPLE

Of the 19 divisors of 240 larger than 1, the following: [4, 15, 16, 60, 240] are found in A345452. Using them, we can factor 240 in four possible ways, as 240 = 60*4 = 16*15 = 15*4*4, therefore a(240) = 4.

Of the 23 divisors of 540 larger than 1, the following: [4, 9, 15, 36, 60, 135, 540] are found in A345452. Using them, we can factor 540 in five possible ways, as 540 = 135*4 = 60*9 = 36*15 = 15*9*4, therefore a(540) = 5.

PROG

(PARI)

A353374(n) = (!(bigomega(n)%2) && !(valuation(n, 2)%2));

A353377(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&A353374(d), s += A353377(n/d, d))); (s));

CROSSREFS

Cf. A001222, A007814, A345452, A353374, A353378 [= a(n^2)].

Cf. also A353333, A353337, A353353, A353373.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Apr 17 2022

STATUS

approved

A353373

Number of ways to write n as a product of the terms of A325698 larger than 1; a(1) = 1 by convention (an empty product).

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,36

COMMENTS

Number of factorizations of n into factors k > 1 for which there are equal number of odd-indexed primes (A031368) as there are even-indexed primes (A031215) in their prime factorization.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences computed from indices in prime factorization

FORMULA

a(p) = 0 for all primes p.

a(n) = a(A003961(n)) = a(A348717(n)), for all n >= 1.

EXAMPLE

Of the 23 divisors of 936 larger than 1, only [6, 26, 36, 156, 936] are in A325698. These can be combined to yield a product yielding 936 in four possible ways as 936 = 156*6 = 36*26 = 26*6*6, therefore a(936) = 4.

Of the 31 divisors of 2184 larger than 1, only [6, 14, 26, 84, 156, 364, 2184] are in A325698. These can be combined to yield a product yielding 2184 in five possible ways as 2184 = 364*6 = 156*14 = 84*26, therefore a(2184) = 5.

PROG

(PARI)

A353370(n) = { my(f = factor(n)); (0==sum(i=1, #f~, f[i, 2]*((-1)^(primepi(f[i, 1])%2)))); }; \\ Antti Karttunen, Apr 16 2022

A353373(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&A353370(d), s += A353373(n/d, d))); (s));

CROSSREFS

Cf. A031215, A031368, A325698, A353370.

Cf. also A353333, A353353, A353377.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Apr 16 2022

STATUS

approved

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