A316431 -id:A316431 - cp's OEIS frontend

A319320 Number of integer partitions of n such that every distinct submultiset has a different LCM.

1, 1, 1, 1, 2, 1, 3, 2, 3, 4, 5, 4, 6, 7, 7, 9, 11, 12, 12, 15, 17, 20, 22, 24, 25, 31, 35, 39, 40, 48, 51, 55, 64, 73, 77, 85, 92, 104, 115, 126, 136, 147, 157, 176, 198, 211, 234, 246, 269, 294, 326, 350, 375, 403, 443, 475, 526, 560, 600, 650

Offset: 1

Gus Wiseman, Sep 17 2018

nonn

Note that such partitions are necessarily strict.

			The a(19) = 12 partitions:
  (19),
  (10,9), (11,8), (12,7), (13,6), (14,5), (15,4), (16,3), (17,2),
  (8,6,5), (11,5,3),
  (7,5,4,3).

Cf. A074761, A108917, A275972, A290103, A316313, A316429, A316431, A319315, A319318, A319327.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&UnsameQ@@LCM@@@Union[Rest[Subsets[#]]]&]],{n,30}]

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

A316438 Heinz numbers of integer partitions whose product is strictly greater than the LCM of the parts.

Original entry on oeis.org

9, 18, 21, 25, 27, 36, 39, 42, 45, 49, 50, 54, 57, 63, 65, 72, 75, 78, 81, 84, 87, 90, 91, 98, 99, 100, 105, 108, 111, 114, 115, 117, 121, 125, 126, 129, 130, 133, 135, 144, 147, 150, 153, 156, 159, 162, 168, 169, 171, 174, 175, 180, 182, 183, 185, 189, 195

Offset: 1

Gus Wiseman, Jul 03 2018

nonn

Also numbers n > 1 such that A290104(n) > 1.

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

			Sequence of partitions whose product is greater than their LCM begins: (22), (221), (42), (33), (222), (2211), (62), (421), (322), (44), (331), (2221), (82), (422), (63), (22111), (332), (621), (2222), (4211).

Cf. A056239, A074761, A143773, A285572, A289508, A290103, A296150, A316429, A316431, A316433, A316437.

Select[Range[2,300],With[{pms=Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]},Times@@pms/LCM@@pms>1]&]

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

A319334 Nonprime Heinz numbers of integer partitions whose sum is equal to their LCM.

Original entry on oeis.org

30, 198, 264, 273, 364, 490, 525, 630, 700, 840, 918, 1120, 1224, 1495, 1632, 1794, 2392, 2420, 2750, 3105, 3450, 3726, 4140, 4263, 4400, 4466, 4921, 4968, 5481, 5520, 5684, 6327, 6624, 7030, 7040, 7308, 8436, 8832, 9744, 11248, 12992, 14079, 14450, 14993

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 sequence of all non-singleton integer partitions whose sum is equal to their LCM begins: (321), (5221), (52111), (642), (6411), (4431), (4332), (43221), (43311), (432111), (72221), (4311111), (722111), (963), (7211111), (9621).

Cf. A067538, A074761, A143773, A290103, A305566, A316429, A316431, A316432, A316433, A317624, A319315, A319333.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,1000],And[!PrimeQ[#],LCM@@primeMS[#]==Total[primeMS[#]]]&]

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A319320 Number of integer partitions of n such that every distinct submultiset has a different LCM.

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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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A316438 Heinz numbers of integer partitions whose product is strictly greater than the LCM of the parts.

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

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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.

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A319334 Nonprime Heinz numbers of integer partitions whose sum is equal to their LCM.

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