A316430 -id:A316430 - cp's OEIS frontend

A316431 Least common multiple divided by greatest common divisor of the integer partition with Heinz number n > 1.

1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 4, 6, 1, 1, 2, 1, 3, 2, 5, 1, 2, 1, 6, 1, 4, 1, 6, 1, 1, 10, 7, 12, 2, 1, 8, 3, 3, 1, 4, 1, 5, 6, 9, 1, 2, 1, 3, 14, 6, 1, 2, 15, 4, 4, 10, 1, 6, 1, 11, 2, 1, 2, 10, 1, 7, 18, 12, 1, 2, 1, 12, 6, 8, 20, 6, 1, 3, 1, 13, 1, 4, 21, 14, 5, 5, 1, 6, 6, 9, 22, 15, 24, 2, 1, 4, 10, 3, 1, 14, 1, 6, 12

Offset: 2

Gus Wiseman, Jul 02 2018

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			63 is the Heinz number of (4,2,2), which has LCM 4 and GCD 2, so a(63) = 4/2 = 2.
91 is the Heinz number of (6,4), which has LCM 12 and GCD 2, so a(91) = 12/2 = 6.

Cf. A005117, A056239, A074761, A289508, A289509, A290103, A296150, A316429, A316430, A316437.

Table[With[{pms=Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]]},LCM@@pms/GCD@@pms],{n,2,100}]

A316431(n) = if(1==n,1,my(pis = apply(p -> primepi(p), factor(n)[, 1]~)); lcm(pis)/gcd(pis)); \\ Antti Karttunen, Sep 06 2018

a(n) = A290103(n)/A289508(n).

a(n) = a(A005117(n)). - David A. Corneth, Sep 06 2018

More terms from Antti Karttunen, Sep 06 2018

A316429 Heinz numbers of integer partitions whose length is equal to their LCM.

Original entry on oeis.org

2, 6, 9, 20, 50, 56, 84, 125, 126, 176, 189, 196, 240, 294, 360, 416, 441, 540, 600, 624, 686, 810, 900, 936, 968, 1029, 1040, 1088, 1215, 1350, 1404, 1500, 1560, 2025, 2106, 2250, 2340, 2401, 2432, 2600, 2704, 3159, 3375, 3510, 3648, 3750, 3900, 4056, 5265

Offset: 1

Gus Wiseman, Jul 02 2018

A110295 is a subsequence.

			3750 is the Heinz number of (3,3,3,3,2,1), whose length and lcm are both 6.

David A. Corneth, Table of n, a(n) for n = 1..10822 (Terms <= 5 * 10^11)

Cf. A056239, A074761, A110295, A143773, A237984, A289508, A289509, A290103, A296150, A316413, A316428, A316430, A316431.

Select[Range[2,200],PrimeOmega[#]==LCM@@Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]]&]

heinz(n) = my(f=factor(n), pr=f[,1]~,exps=f[,2], res=vector(vecsum(exps)), t=0); for(i = 1, #pr, pr[i] = primepi(pr[i]); for(j=1, exps[i],t++; res[t] = pr[i])); res
is(n) = my(h = heinz(n)); lcm(h)==#h \\ David A. Corneth, Jul 05 2018

A316432 Number of integer partitions of n whose length is equal to the GCD of all parts.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 1, 2, 0, 2, 0, 3, 2, 3, 0, 5, 0, 3, 4, 5, 0, 8, 1, 6, 6, 6, 0, 11, 0, 8, 10, 8, 2, 18, 0, 9, 14, 15, 0, 19, 0, 16, 21, 11, 0, 34, 1, 16, 24, 24, 0, 30, 10, 27, 30, 14, 0, 71, 0, 15, 34, 38, 18, 47, 0, 47, 44, 36, 0, 88, 0, 18, 79, 63, 5

Offset: 1

Gus Wiseman, Jul 02 2018

nonn

			The a(24) = 8 partitions:
(14,10), (22,2),
(9,9,6), (12,9,3), (15,6,3), (18,3,3),
(8,8,4,4), (12,4,4,4).

Cf. A000837, A067538, A074761, A289508, A289509, A290103, A316429, A316430, A316431, A316433.

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

a(n) = {my(nb = 0); forpart(p=n, if (gcd(p)==#p, nb++);); nb;} \\ Michel Marcus, Jul 03 2018

A316433 Number of integer partitions of n whose length is equal to the LCM of all parts.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 2, 1, 4, 3, 4, 4, 8, 5, 7, 8, 10, 8, 13, 13, 20, 18, 25, 25, 36, 34, 48, 52, 64, 64, 85, 85, 108, 106, 129, 133, 160, 158, 189, 194, 229, 228, 276, 279, 332, 336, 394, 402, 476, 489, 572, 599, 699, 728, 845, 889, 1032, 1094, 1251, 1332, 1523

Offset: 1

Gus Wiseman, Jul 02 2018

nonn

			The a(13) = 8 partitions are (4441), (55111), (322222), (332221), (333211), (622111), (631111), (7111111).

Cf. A067538, A074761, A143773, A285572, A289508, A290103, A305566, A316429, A316430, A316431, A316432.

Table[Length[Select[IntegerPartitions[n],LCM@@#==Length[#]&]],{n,30}]

a(n) = {my(nb = 0); forpart(p=n, if (lcm(Vec(p))==#p, nb++);); nb;} \\ Michel Marcus, Jul 03 2018

A316556 Number of distinct LCMs of nonempty submultisets of the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 1, 3, 2, 3, 2, 1, 2, 2, 2, 1, 3, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 1, 2, 3, 2, 2, 2, 1, 4, 1, 2, 2, 1, 2, 4, 1, 2, 3, 4, 1, 2, 1, 2, 3, 2, 3, 3, 1, 2, 1, 2, 1, 3, 3, 2, 2, 2, 1, 4, 3, 2, 3, 2, 3, 2, 1, 2, 3, 2, 1, 4, 1, 2, 5

Offset: 1

Gus Wiseman, Jul 06 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Number of distinct values obtained when A290103 is applied to all divisors of n larger than one. - Antti Karttunen, Sep 25 2018

			462 is the Heinz number of (5,4,2,1) which has possible LCMs of nonempty submultisets {1,2,4,5,10,20} so a(462) = 6.

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

Cf. A056239, A074761, A108917, A122768, A275972, A290103, A296150, A301957, A316313, A316314, A316430, A316555, A316557.

Cf. also A304793, A305611, A319685, A319695 for other similarly constructed sequences.

Table[Length[Union[LCM@@@Rest[Subsets[If[n==1,{},Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]]]]]]],{n,100}]

A290103(n) = lcm(apply(p->primepi(p),factor(n)[,1]));
A316556(n) = { my(m=Map(),s,k=0); fordiv(n,d,if((d>1)&&!mapisdefined(m,s=A290103(d)), mapput(m,s,s); k++)); (k); }; \\ Antti Karttunen, Sep 25 2018

More terms from Antti Karttunen, Sep 25 2018

A319567 Product of y divided by the GCD of y to the power of the length of y, where y is the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 4, 6, 1, 1, 4, 1, 3, 2, 5, 1, 2, 1, 6, 1, 4, 1, 6, 1, 1, 10, 7, 12, 4, 1, 8, 3, 3, 1, 8, 1, 5, 12, 9, 1, 2, 1, 9, 14, 6, 1, 8, 15, 4, 4, 10, 1, 6, 1, 11, 2, 1, 2, 10, 1, 7, 18, 12, 1, 4, 1, 12, 18, 8, 20, 12, 1, 3, 1, 13

Offset: 0

Gus Wiseman, Sep 23 2018

nonn

The Heinz number of a partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Cf. A003963, A001222, A056239, A067538, A112798, A143773, A289508, A289509, A316430.

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

a(n) = A003963(n) / A289508(n) ^ A001222(n).

A316437 Take the integer partition with Heinz number n, divide all parts by the GCD of the parts, then take the Heinz number of the resulting partition.

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 2, 8, 4, 10, 2, 12, 2, 14, 15, 16, 2, 18, 2, 20, 6, 22, 2, 24, 4, 26, 8, 28, 2, 30, 2, 32, 33, 34, 35, 36, 2, 38, 10, 40, 2, 42, 2, 44, 45, 46, 2, 48, 4, 50, 51, 52, 2, 54, 55, 56, 14, 58, 2, 60, 2, 62, 12, 64, 6, 66, 2, 68, 69, 70, 2, 72, 2, 74, 75, 76, 77, 78, 2, 80, 16, 82, 2, 84, 85, 86, 22, 88, 2, 90, 15

Offset: 1

Gus Wiseman, Jul 03 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
This sequence is idempotent, meaning a(a(n)) = a(n) for all n.
All terms belong to A289509.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Wikipedia, Idempotence
Index entries for sequences computed from indices in prime factorization

Cf. A000720, A056239, A289508, A289509, A290103, A296150, A316430, A316431, A316432, A316438.

f[n_]:=If[n==1,1,With[{pms=Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]},Times@@Prime/@(pms/GCD@@pms)]];
Table[f[n],{n,100}]

A316437(n) = if(1==n,1,my(f = factor(n), pis = apply(p -> primepi(p), f[, 1]~), es = f[, 2]~, g = gcd(pis)); factorback(vector(#f~, k, prime(pis[k]/g)^es[k]))); \\ Antti Karttunen, Aug 06 2018

More terms from Antti Karttunen, Aug 06 2018

A316555 Number of distinct GCDs of nonempty submultisets of the integer partition with Heinz number n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 06 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Number of distinct values obtained when A289508 is applied to all divisors of n larger than one. - Antti Karttunen, Sep 28 2018

			455 is the Heinz number of (6,4,3) which has possible GCDs of nonempty submultisets {1,2,3,4,6} so a(455) = 5.

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

Cf. A056239, A108917, A122768, A275972, A289508, A289509, A296150, A316313, A316314, A316430, A316556, A316557.

Cf. also A304793, A305611, A319685, A319695 for other similarly constructed sequences.

Table[Length[Union[GCD@@@Rest[Subsets[If[n==1,{},Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]]]]]]],{n,100}]

A289508(n) = gcd(apply(p->primepi(p),factor(n)[,1]));
A316555(n) = { my(m=Map(),s,k=0); fordiv(n,d,if((d>1)&&!mapisdefined(m,s=A289508(d)), mapput(m,s,s); k++)); (k); }; \\ Antti Karttunen, Sep 28 2018

More terms from Antti Karttunen, Sep 28 2018

A316436 Sum divided by GCD of the integer partition with Heinz number n > 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 3, 2, 4, 1, 4, 1, 5, 5, 4, 1, 5, 1, 5, 3, 6, 1, 5, 2, 7, 3, 6, 1, 6, 1, 5, 7, 8, 7, 6, 1, 9, 4, 6, 1, 7, 1, 7, 7, 10, 1, 6, 2, 7, 9, 8, 1, 7, 8, 7, 5, 11, 1, 7, 1, 12, 4, 6, 3, 8, 1, 9, 11, 8, 1, 7, 1, 13, 8, 10, 9, 9, 1, 7, 4, 14, 1, 8, 10, 15, 6, 8, 1, 8, 5, 11, 13, 16, 11, 7, 1, 9, 9, 8, 1, 10, 1, 9, 9

Offset: 2

Gus Wiseman, Jul 03 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Cf. A056239, A289508, A289509, A290103, A290104, A296150, A316430, A316431, A316432, A316437.

a:= n-> (l-> add(i, i=l)/igcd(l[]))(map(i->
      numtheory[pi](i[1])$i[2], ifactors(n)[2])):
seq(a(n), n=2..100);  # Alois P. Heinz, Jul 03 2018

Table[With[{pms=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]},Total[pms]/GCD@@pms],{n,2,100}]

A316436(n) = { my(f = factor(n), pis = apply(p -> primepi(p), f[, 1]~), es = f[, 2]~, g = gcd(pis)); sum(i=1, #f~, pis[i]*es[i])/g; }; \\ Antti Karttunen, Sep 10 2018

More terms from Antti Karttunen, Sep 10 2018

A319329 Heinz numbers of integer partitions, whose length is equal to the GCD of the parts and whose sum is equal to the LCM of the parts, in increasing order.

Original entry on oeis.org

2, 1495, 179417, 231133, 727531, 1378583, 1787387, 3744103, 4556993, 7566167, 18977519, 29629391, 30870587, 34174939, 39973571, 53508983, 70946617, 110779141, 138820187, 139681069, 170583017, 225817751, 409219217, 441317981, 493580417, 539462099, 544392433, 712797613, 802903541

Offset: 1

Gus Wiseman, Sep 17 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The corresponding sequence of partitions, whose length is equal to their GCD and whose sum is equal to their LCM: (1), (9,6,3), (20,8,8,4), (24,16,4,4), (16,16,12,4).

Subsequence of A316430.

Cf. A056239, A067538, A074761, A143773, A289508, A289509, A290103, A290104, A316431, A316432, A319328, A319330, A319333.

Select[Range[2,10000],With[{m=If[#==1,{},Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]},And[LCM@@m==Total[m],GCD@@m==Length[m]]]&]

More terms from Max Alekseyev, Jul 25 2024

cp's OEIS Frontend

A316431 Least common multiple divided by greatest common divisor of the integer partition with Heinz number n > 1.

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A316429 Heinz numbers of integer partitions whose length is equal to their LCM.

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A316432 Number of integer partitions of n whose length is equal to the GCD of all parts.

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A316433 Number of integer partitions of n whose length is equal to the LCM of all parts.

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A316556 Number of distinct LCMs of nonempty submultisets of the integer partition with Heinz number n.

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A319567 Product of y divided by the GCD of y to the power of the length of y, where y is the integer partition with Heinz number n.

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A316437 Take the integer partition with Heinz number n, divide all parts by the GCD of the parts, then take the Heinz number of the resulting partition.

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A316555 Number of distinct GCDs of nonempty submultisets of the integer partition with Heinz number n.

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