A290103 -id:A290103 - cp's OEIS frontend

A328169 GCD of the prime indices of n, all plus 1.

0, 2, 3, 2, 4, 1, 5, 2, 3, 2, 6, 1, 7, 1, 1, 2, 8, 1, 9, 2, 1, 2, 10, 1, 4, 1, 3, 1, 11, 1, 12, 2, 3, 2, 1, 1, 13, 1, 1, 2, 14, 1, 15, 2, 1, 2, 16, 1, 5, 2, 1, 1, 17, 1, 2, 1, 3, 1, 18, 1, 19, 2, 1, 2, 1, 1, 20, 2, 1, 1, 21, 1, 22, 1, 1, 1, 1, 1, 23, 2, 3, 2

Offset: 1

Gus Wiseman, Oct 09 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(4,8) = 4.

Positions of 0's and 1's are A318981.
Positions of records (first appearances) appear to be A116974.
The GCD of the prime indices of n, all minus 1, is A328167(n).
The LCM of the prime indices of n, all plus 1, is A328219(n).
Partitions whose parts plus 1 are relatively prime are A318980.
Cf. A000837, A056239, A112798, A258409, A289508, A289509, A290103, A328168.

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

a(n) = A289508(A003961(n)).

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

A319333 Heinz numbers of integer partitions whose sum is equal to their LCM.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 198, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263

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 partitions whose Heinz numbers are in the sequence begins: (1), (2), (3), (4), (5), (6), (7), (8), (9), (10), (3,2,1), (11), (12).

Cf. A067538, A074761, A143773, A290103, A305566, A316429, A316431, A316432, A316433, A317624, A319315, A319334.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],LCM@@primeMS[#]==Total[primeMS[#]]&]

A320459 MM-numbers of labeled multigraphs spanning an initial interval of positive integers.

Original entry on oeis.org

1, 13, 169, 377, 611, 1363, 1937, 2021, 2117, 2197, 4901, 7943, 10933, 16211, 17719, 25181, 26273, 27521, 28561, 28717, 39527, 44603, 56173, 58609, 61393, 63713, 64061, 83291, 86903, 91031, 91039, 94987, 99499, 103259, 141401, 142129, 143663, 146653, 147533

Offset: 1

Gus Wiseman, Oct 13 2018

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 multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}.

			The sequence of terms together with their multiset multisystems begins:
      1: {}
     13: {{1,2}}
    169: {{1,2},{1,2}}
    377: {{1,2},{1,3}}
    611: {{1,2},{2,3}}
   1363: {{1,3},{2,3}}
   1937: {{1,2},{3,4}}
   2021: {{1,4},{2,3}}
   2117: {{1,3},{2,4}}
   2197: {{1,2},{1,2},{1,2}}
   4901: {{1,2},{1,2},{1,3}}
   7943: {{1,2},{1,2},{2,3}}
  10933: {{1,2},{1,3},{1,3}}
  16211: {{1,2},{1,3},{1,4}}
  17719: {{1,2},{1,3},{2,3}}
  25181: {{1,2},{1,2},{3,4}}
  26273: {{1,2},{1,4},{2,3}}
  27521: {{1,2},{1,3},{2,4}}
  28561: {{1,2},{1,2},{1,2},{1,2}}
  28717: {{1,2},{2,3},{2,3}}
  39527: {{1,3},{1,3},{2,3}}
  44603: {{1,2},{2,3},{2,4}}

Eric Weisstein's World of Mathematics, Simple Graph

Cf. A003963, A055932, A056239, A112798, A255906, A290103, A302242, A302478, A305052, A305078.

Cf. A320456, A320458, A320461, A320462, A320464.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
Select[Range[100000],And[normQ[primeMS/@primeMS[#]],And@@(And[SquareFreeQ[#],Length[primeMS[#]]==2]&/@primeMS[#])]&]

A320532 MM-numbers of labeled hypergraphs with multiset edges and no singletons spanning an initial interval of positive integers.

Original entry on oeis.org

1, 7, 13, 19, 37, 53, 61, 89, 91, 113, 131, 133, 151, 161, 223, 247, 251, 259, 281, 299, 311, 329, 359, 371, 377, 427, 437, 463, 481, 503, 593, 611, 623, 659, 667, 689, 703, 719, 721, 791, 793, 827, 851, 863, 893, 917, 923, 953, 1007, 1057, 1069, 1073, 1157

Offset: 1

Gus Wiseman, Oct 14 2018

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 multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}.

			The sequence of terms together with their multiset multisystems begins:
    1: {}
    7: {{1,1}}
   13: {{1,2}}
   19: {{1,1,1}}
   37: {{1,1,2}}
   53: {{1,1,1,1}}
   61: {{1,2,2}}
   89: {{1,1,1,2}}
   91: {{1,1},{1,2}}
  113: {{1,2,3}}
  131: {{1,1,1,1,1}}
  133: {{1,1},{1,1,1}}
  151: {{1,1,2,2}}
  161: {{1,1},{2,2}}
  223: {{1,1,1,1,2}}
  247: {{1,2},{1,1,1}}
  251: {{1,2,2,2}}
  259: {{1,1},{1,1,2}}
  281: {{1,1,2,3}}
  299: {{1,2},{2,2}}
  311: {{1,1,1,1,1,1}}
  329: {{1,1},{2,3}}
  359: {{1,1,1,2,2}}
  371: {{1,1},{1,1,1,1}}

Wikipedia, Hypergraph

Cf. A003963, A055932, A056239, A112798, A255906, A290103, A302242, A302478, A305052.

Cf. A320456, A320461, A320463, A320464, A320533.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
Select[Range[1000],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],And@@(And[PrimeOmega[#]>1]&/@primeMS[#])]&]

A320533 MM-numbers of labeled multi-hypergraphs with multiset edges and no singletons spanning an initial interval of positive integers.

Original entry on oeis.org

1, 7, 13, 19, 37, 49, 53, 61, 89, 91, 113, 131, 133, 151, 161, 169, 223, 247, 251, 259, 281, 299, 311, 329, 343, 359, 361, 371, 377, 427, 437, 463, 481, 503, 593, 611, 623, 637, 659, 667, 689, 703, 719, 721, 791, 793, 827, 851, 863, 893, 917, 923, 931, 953

Offset: 1

Gus Wiseman, Oct 14 2018

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 multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}.

			The sequence of terms together with their multiset multisystems begins:
    1: {}
    7: {{1,1}}
   13: {{1,2}}
   19: {{1,1,1}}
   37: {{1,1,2}}
   49: {{1,1},{1,1}}
   53: {{1,1,1,1}}
   61: {{1,2,2}}
   89: {{1,1,1,2}}
   91: {{1,1},{1,2}}
  113: {{1,2,3}}
  131: {{1,1,1,1,1}}
  133: {{1,1},{1,1,1}}
  151: {{1,1,2,2}}
  161: {{1,1},{2,2}}
  169: {{1,2},{1,2}}
  223: {{1,1,1,1,2}}
  247: {{1,2},{1,1,1}}
  251: {{1,2,2,2}}
  259: {{1,1},{1,1,2}}
  281: {{1,1,2,3}}
  299: {{1,2},{2,2}}
  311: {{1,1,1,1,1,1}}
  329: {{1,1},{2,3}}
  343: {{1,1},{1,1},{1,1}}

Wikipedia, Hypergraph

Cf. A003963, A055932, A056239, A112798, A255906, A290103, A302242, A302478, A305052.

Cf. A320456, A320462, A320463, A320464, A320532.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
Select[Range[1000],And[normQ[primeMS/@primeMS[#]],And@@(And[PrimeOmega[#]>1]&/@primeMS[#])]&]

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

A319826 GCD of the strict integer partition with FDH number n; GCD of the indices (in A050376) of Fermi-Dirac prime factors of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 2, 9, 10, 1, 11, 1, 1, 1, 12, 1, 13, 1, 2, 1, 14, 1, 15, 1, 1, 1, 1, 3, 16, 1, 2, 1, 17, 1, 18, 1, 2, 1, 19, 1, 20, 1, 2, 1, 21, 1, 1, 1, 1, 1, 22, 1, 23, 1, 1, 3, 4, 1, 24, 1, 2, 1, 25, 1, 26, 1, 1, 1, 1, 1, 27, 1, 28

Offset: 1

Gus Wiseman, Sep 28 2018

nonn

Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. The FDH number of a strict integer partition (y_1, ..., y_k) is f(y_1) * ... * f(y_k).

			45 is the FDH number of (6,4), which has GCD 2, so a(45) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..16384

A left inverse of A050376.

Cf. A052331, A056239, A064547, A213925, A289508, A289509, A290103, A299755, A299757, A319825.

nn=200;
FDfactor[n_]:=If[n==1,{},Sort[Join@@Cases[FactorInteger[n],{p_,k_}:>Power[p,Cases[Position[IntegerDigits[k,2]//Reverse,1],{m_}->2^(m-1)]]]]];
FDprimeList=Array[FDfactor,nn,1,Union];FDrules=MapIndexed[(#1->#2[[1]])&,FDprimeList];
GCD@@@Table[Reverse[FDfactor[n]/.FDrules],{n,nn}]

A319826(n) = { my(i=1,g=0,x=A052331(n)); while(x,if(x%2,g = gcd(g,i)); x>>=1; i++); (g); }; \\ (Uses the program given in A052331) - Antti Karttunen, Feb 18 2023

For all n >= 1, a(A050376(n)) = n. - Antti Karttunen, Feb 18 2023

Secondary definition added by Antti Karttunen, Feb 18 2023

A327778 Number of integer partitions of n whose LCM is a multiple of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 1, 11, 1, 11, 23, 1, 1, 23, 1, 85, 85, 45, 1, 152, 1, 84, 1, 451, 1, 1787, 1, 1, 735, 260, 1925, 1908, 1, 437, 1877, 4623, 1, 14630, 1, 6934, 10519, 1152, 1, 6791, 1, 1817, 10159, 22556, 1, 2819, 47927, 69333, 22010, 4310, 1

Offset: 0

Gus Wiseman, Sep 25 2019

nonn

			The partitions of n = 6, 10, 12, and 15 whose LCM is a multiple of n:
  (6)      (10)         (12)             (15)
  (3,2,1)  (5,3,2)      (5,4,3)          (6,5,4)
           (5,4,1)      (6,4,2)          (7,5,3)
           (5,2,2,1)    (8,3,1)          (9,5,1)
           (5,2,1,1,1)  (4,3,3,2)        (10,3,2)
                        (4,4,3,1)        (5,4,3,3)
                        (6,4,1,1)        (5,5,3,2)
                        (4,3,2,2,1)      (6,5,2,2)
                        (4,3,3,1,1)      (6,5,3,1)
                        (4,3,2,1,1,1)    (10,3,1,1)
                        (4,3,1,1,1,1,1)  (5,3,3,2,2)
                                         (5,3,3,3,1)
                                         (5,4,3,2,1)
                                         (5,5,3,1,1)
                                         (6,5,2,1,1)
                                         (5,3,2,2,2,1)
                                         (5,3,3,2,1,1)
                                         (5,4,3,1,1,1)
                                         (6,5,1,1,1,1)
                                         (5,3,2,2,1,1,1)
                                         (5,3,3,1,1,1,1)
                                         (5,3,2,1,1,1,1,1)
                                         (5,3,1,1,1,1,1,1,1)

Alois P. Heinz, Table of n, a(n) for n = 0..1000

The Heinz numbers of these partitions are given by A327783.
Partitions whose LCM is equal to their sum are A074761.
Partitions whose LCM is greater than their sum are A327779.
Partitions whose LCM is less than their sum are A327781.
Cf. A000961, A018818, A067538, A290103, A319333, A326842, A326843, A327780.

a:= proc(m) option remember; local b; b:=
      proc(n, i, l) option remember; `if`(n=0 or i=1,
        `if`(l=m, 1, 0), `if`(i<2, 0, b(n, i-1, l))+
         b(n-i, min(n-i, i), igcd(m, ilcm(l, i))))
      end; `if`(isprime(m), 1, b(m$2, 1))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Sep 26 2019

Table[Length[Select[IntegerPartitions[n],Divisible[LCM@@#,n]&]],{n,30}]
(* Second program: *)
a[m_] := a[m] = Module[{b}, b[n_, i_, l_] := b[n, i, l] = If[n == 0 || i == 1, If[l == m, 1, 0], If[i<2, 0, b[n, i - 1, l]] + b[n - i, Min[n - i, i], GCD[m, LCM[l, i]]]]; If[PrimeQ[m], 1, b[m, m, 1]]];
a /@ Range[0, 60] (* Jean-François Alcover, May 18 2021, after Alois P. Heinz *)

a(n) = 1 <=> n in { A000961 }. - Alois P. Heinz, Sep 26 2019

A328219 LCM of the prime indices of n, all plus 1.

Original entry on oeis.org

1, 2, 3, 2, 4, 6, 5, 2, 3, 4, 6, 6, 7, 10, 12, 2, 8, 6, 9, 4, 15, 6, 10, 6, 4, 14, 3, 10, 11, 12, 12, 2, 6, 8, 20, 6, 13, 18, 21, 4, 14, 30, 15, 6, 12, 10, 16, 6, 5, 4, 24, 14, 17, 6, 12, 10, 9, 22, 18, 12, 19, 12, 15, 2, 28, 6, 20, 8, 30, 20, 21, 6, 22, 26

Offset: 1

Gus Wiseman, Oct 16 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.

Sorted positions of first appearances are A328451.
LCM of prime indices is A290103.
LCM of prime indices minus 1 is A328456.
GCD of prime indices plus 1 is A328169.
Partitions whose parts plus 1 are relatively prime are A318980.
Numbers whose prime indices plus 1 are relatively prime are A318981,
Cf. A003961, A056239, A112798, A258409, A289508, A289509, A328167, A328168.

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

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A290103(n) = lcm(apply(p->primepi(p),factor(n)[,1]));
A328219(n) = A290103(A003961(n)); \\ Antti Karttunen, Oct 18 2019

a(n) = A290103(A003961(n)).

If n = A000040(i_1) * ... * A000040(i_k), then a(n) = lcm(1+i_1,...,1+i_k).

cp's OEIS Frontend

A328169 GCD of the prime indices of n, all plus 1.

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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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A319333 Heinz numbers of integer partitions whose sum is equal to their LCM.

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A320459 MM-numbers of labeled multigraphs spanning an initial interval of positive integers.

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A320532 MM-numbers of labeled hypergraphs with multiset edges and no singletons spanning an initial interval of positive integers.

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A320533 MM-numbers of labeled multi-hypergraphs with multiset edges and no singletons spanning an initial interval of positive integers.

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Mathematica

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

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Extensions

A319826 GCD of the strict integer partition with FDH number n; GCD of the indices (in A050376) of Fermi-Dirac prime factors of n.

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