A371735 -id:A371735 - cp's OEIS frontend

A371733 Maximal length of a factorization of n into factors > 1 all having the same sum of prime indices.

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 1, 5, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 6, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4

Offset: 1

Gus Wiseman, Apr 13 2024

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. Sum of prime indices is given by A056239.

Factorizations into factors all having the same sum of prime indices are counted by A321455.

			The factorizations of 588 of this type are (7*7*12), (21*28), (588), so a(588) = 3.
The factorizations of 900 of this type are (5*5*6*6), (9*10*10), (25*36), (30*30), (900), so a(900) = 4.

Positions of 1's are A321453, counted by A321451.
Positions of terms > 1 are A321454, counted by A321452.
Factorizations of this type are counted by A321455, different sums A321469.
For different sums instead of same sums we have A371734.
For set partitions of binary indices we have A371735.
A001055 counts factorizations.
A002219 (aerated) counts biquanimous partitions, ranks A357976.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A321142 and A371794 count non-biquanimous strict partitions.
A371789 counts non-quanimous sets, differences A371790.
A371796 counts quanimous sets, differences A371797.
Cf. A035470, A279787, A305551, A322794, A326515, A326518, A326534, A336137, A371783, A371791.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
Table[Max[Length/@Select[facs[n],SameQ@@hwt/@#&]],{n,100}]

A056239(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1])));
all_have_same_sum_of_pis(facs) = if(!#facs, 1, (#Set(apply(A056239,facs)) == 1));
A371733(n, m=n, facs=List([])) = if(1==n, if(all_have_same_sum_of_pis(facs),#facs,0), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s = max(s, A371733(n/d, d, newfacs)))); (s)); \\ Antti Karttunen, Jan 20 2025

Data section extended to a(108) by Antti Karttunen, Jan 20 2025

A371734 Maximal length of a factorization of n into factors > 1 all having different sums of prime indices.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Apr 13 2024

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. Sum of prime indices is given by A056239.

Factorizations into factors > 1 all having different sums of prime indices are counted by A321469.

			The factorizations of 90 of this type are (2*3*15), (2*5*9), (2*45), (3*30), (5*18), (6*15), (90), so a(90) = 3.

For set partitions of binary indices we have A000120, same sums A371735.
Positions of 1's are A000430.
Positions of terms > 1 are A080257.
Factorizations of this type are counted by A321469, same sums A321455.
For same instead of different sums we have A371733.
A001055 counts factorizations.
A002219 (aerated) counts biquanimous partitions, ranks A357976.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
Cf. A035470, A279787, A305551, A321142, A322794, A326515, A326518, A326534, A336137, A371783, A371789, A371791.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
Table[Max[Length/@Select[facs[n],UnsameQ@@hwt/@#&]],{n,100}]

A056239(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1])));
all_have_different_sum_of_pis(facs) = if(!#facs, 1, (#Set(apply(A056239,facs)) == #facs));
A371734(n, m=n, facs=List([])) = if(1==n, if(all_have_different_sum_of_pis(facs),#facs,0), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s = max(s,A371734(n/d, d, newfacs)))); (s)); \\ Antti Karttunen, Jan 20 2025

Data section extended to a(105) by Antti Karttunen, Jan 20 2025

A371738 Numbers with non-quanimous binary indices. Numbers whose binary indices have only one set partition with all equal block-sums.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 16, 17, 18, 19, 20, 21, 23, 24, 26, 28, 29, 32, 33, 34, 35, 36, 37, 38, 40, 41, 43, 44, 46, 48, 50, 52, 53, 55, 56, 57, 58, 61, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 77, 78, 79, 80, 81, 83, 84, 86, 88, 89, 91, 92

Offset: 1

Gus Wiseman, Apr 14 2024

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The binary indices of 165 are {1,3,6,8}, with qualifying set partitions {{1,8},{3,6}}, and {{1,3,6,8}}, so 165 is not in the sequence.
The terms together with their binary expansions and binary indices begin:
   1:     1 ~ {1}
   2:    10 ~ {2}
   3:    11 ~ {1,2}
   4:   100 ~ {3}
   5:   101 ~ {1,3}
   6:   110 ~ {2,3}
   8:  1000 ~ {4}
   9:  1001 ~ {1,4}
  10:  1010 ~ {2,4}
  11:  1011 ~ {1,2,4}
  12:  1100 ~ {3,4}
  14:  1110 ~ {2,3,4}
  16: 10000 ~ {5}
  17: 10001 ~ {1,5}
  18: 10010 ~ {2,5}
  19: 10011 ~ {1,2,5}
  20: 10100 ~ {3,5}
  21: 10101 ~ {1,3,5}
  23: 10111 ~ {1,2,3,5}

Set partitions with all equal block-sums are counted by A035470.
Positions of 1's in A336137 and A371735.
The complement is A371784.
A000110 counts set partitions.
A002219 (aerated) counts biquanimous partitions, ranks A357976.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A070939 gives length of binary expansion.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
A371789 counts non-quanimous sets, differences A371790.
A371796 counts quanimous sets, differences A371797.
Cf. A001055, A006827, A038041, A305551, A321455, A326534, A371733.

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Select[Range[100],Length[Select[sps[bix[#]],SameQ@@Total/@#&]]==1&]

A371784 Numbers with quanimous binary indices. Numbers whose binary indices can be partitioned in more than one way into blocks with the same sum.

Original entry on oeis.org

7, 13, 15, 22, 25, 27, 30, 31, 39, 42, 45, 47, 49, 51, 54, 59, 60, 62, 63, 75, 76, 82, 85, 87, 90, 93, 94, 95, 97, 99, 102, 107, 108, 109, 110, 115, 117, 119, 120, 122, 125, 126, 127, 141, 143, 147, 148, 153, 155, 158, 162, 165, 167, 170, 173, 175, 179, 180

Offset: 1

Gus Wiseman, Apr 16 2024

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The binary indices of 165 are {1,3,6,8}, with qualifying set partitions {{1,8},{3,6}}, and {{1,3,6,8}}, so 165 is in the sequence.
The terms together with their binary expansions and binary indices begin:
   7:     111 ~ {1,2,3}
  13:    1101 ~ {1,3,4}
  15:    1111 ~ {1,2,3,4}
  22:   10110 ~ {2,3,5}
  25:   11001 ~ {1,4,5}
  27:   11011 ~ {1,2,4,5}
  30:   11110 ~ {2,3,4,5}
  31:   11111 ~ {1,2,3,4,5}
  39:  100111 ~ {1,2,3,6}
  42:  101010 ~ {2,4,6}
  45:  101101 ~ {1,3,4,6}
  47:  101111 ~ {1,2,3,4,6}
  49:  110001 ~ {1,5,6}
  51:  110011 ~ {1,2,5,6}
  54:  110110 ~ {2,3,5,6}
  59:  111011 ~ {1,2,4,5,6}
  60:  111100 ~ {3,4,5,6}
  62:  111110 ~ {2,3,4,5,6}
  63:  111111 ~ {1,2,3,4,5,6}

Set partitions with all equal block-sums are counted by A035470.
Positions of terms > 1 in A336137 and A371735.
The complement is A371738.
A000110 counts set partitions.
A002219 (aerated) counts biquanimous partitions, ranks A357976.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A070939 gives length of binary expansion.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
A371789 counts non-quanimous sets, differences A371790.
A371796 counts quanimous sets, differences A371797.
Cf. A001055, A006827, A038041, A305551, A321455, A326534, A371733.

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Select[Range[100],Length[Select[sps[bix[#]],SameQ@@Total/@#&]]>1&]

cp's OEIS Frontend

A371733 Maximal length of a factorization of n into factors > 1 all having the same sum of prime indices.

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A371734 Maximal length of a factorization of n into factors > 1 all having different sums of prime indices.

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A371738 Numbers with non-quanimous binary indices. Numbers whose binary indices have only one set partition with all equal block-sums.

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A371784 Numbers with quanimous binary indices. Numbers whose binary indices can be partitioned in more than one way into blocks with the same sum.

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