A319241 -id:A319241 - cp's OEIS frontend

A357854 Squarefree numbers with a divisor having the same sum of prime indices as their quotient.

1, 30, 70, 154, 165, 210, 273, 286, 390, 442, 462, 561, 595, 646, 714, 741, 858, 874, 910, 1045, 1155, 1173, 1254, 1326, 1330, 1334, 1495, 1653, 1771, 1794, 1798, 1870, 1938, 2139, 2145, 2294, 2415, 2465, 2470, 2530, 2622, 2639, 2730, 2926, 2945, 2958, 3034

Offset: 1

Gus Wiseman, Oct 27 2022

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 terms together with their prime indices begin:
     1: {}
    30: {1,2,3}
    70: {1,3,4}
   154: {1,4,5}
   165: {2,3,5}
   210: {1,2,3,4}
   273: {2,4,6}
   286: {1,5,6}
   390: {1,2,3,6}
For example, 210 has factorization 14*15, and both factors have the same sum of prime indices 5, so 210 is in the sequence.

The partitions with these Heinz numbers are counted by A237258.
A subset of A319241, squarefree case of A300061.
Squarefree positions of nonzero terms in A357879.
This is the squarefree case of A357976, counted by A002219.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798.
Cf. A033879, A033880, A064914, A181819, A235130, A237194, A276107, A300273, A321144, A357975, A357976.

sumprix[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimePi[p]]];
Select[Range[1000],SquareFreeQ[#]&&MemberQ[sumprix/@Divisors[#],sumprix[#]/2]&]

A338908 Squarefree semiprimes whose prime indices sum to an even number.

Original entry on oeis.org

10, 21, 22, 34, 39, 46, 55, 57, 62, 82, 85, 87, 91, 94, 111, 115, 118, 129, 133, 134, 146, 155, 159, 166, 183, 187, 194, 203, 205, 206, 213, 218, 235, 237, 247, 253, 254, 259, 267, 274, 295, 298, 301, 303, 314, 321, 334, 335, 339, 341, 358, 365, 371, 377, 382

Offset: 1

Gus Wiseman, Nov 28 2020

nonn

A squarefree semiprime is a product of any two distinct prime numbers. A prime index of n is a number m such that the m-th prime number divides n. The multiset of prime indices of n is row n of A112798.

			The sequence of terms together with their prime indices begins:
     10: {1,3}     115: {3,9}     213: {2,20}
     21: {2,4}     118: {1,17}    218: {1,29}
     22: {1,5}     129: {2,14}    235: {3,15}
     34: {1,7}     133: {4,8}     237: {2,22}
     39: {2,6}     134: {1,19}    247: {6,8}
     46: {1,9}     146: {1,21}    253: {5,9}
     55: {3,5}     155: {3,11}    254: {1,31}
     57: {2,8}     159: {2,16}    259: {4,12}
     62: {1,11}    166: {1,23}    267: {2,24}
     82: {1,13}    183: {2,18}    274: {1,33}
     85: {3,7}     187: {5,7}     295: {3,17}
     87: {2,10}    194: {1,25}    298: {1,35}
     91: {4,6}     203: {4,10}    301: {4,14}
     94: {1,15}    205: {3,13}    303: {2,26}
    111: {2,12}    206: {1,27}    314: {1,37}

A031215 looks at primes instead of semiprimes.
A300061 and A319241 (squarefree) look all numbers (not just semiprimes).
A338905 has this as union of even-indexed rows.
A338906 is the nonsquarefree version.
A338907 is the odd version.
A001358 lists semiprimes, with odd/even terms A046315/A100484.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
A024697 is the sum of semiprimes of weight n.
A025129 is the sum of squarefree semiprimes of weight n.
A056239 gives the sum of prime indices of n.
A289182/A115392 list the positions of odd/even terms in A001358.
A320656 counts factorizations into squarefree semiprimes.
A332765 gives the greatest squarefree semiprime of weight n.
A338898, A338912, and A338913 give the prime indices of semiprimes, with product A087794, sum A176504, and difference A176506.
A338899, A270650, and A270652 give the prime indices of squarefree semiprimes, with difference A338900.
A338904 groups semiprimes by weight.
A338911 lists products of pairs of primes both of even index.
A339114/A339115 give the least/greatest semiprime of weight n.
A339116 groups squarefree semiprimes by greater prime factor.
Cf. A000040, A001221, A001222, A087112, A098350, A112798, A168472, A338901, A338904, A339004, A339005.

Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]==2&& EvenQ[Total[PrimePi/@First/@FactorInteger[#]]]&]

A319242 Heinz numbers of strict integer partitions of odd numbers. Squarefree numbers whose prime indices sum to an odd number.

Original entry on oeis.org

2, 5, 6, 11, 14, 15, 17, 23, 26, 31, 33, 35, 38, 41, 42, 47, 51, 58, 59, 65, 67, 69, 73, 74, 77, 78, 83, 86, 93, 95, 97, 103, 105, 106, 109, 110, 114, 119, 122, 123, 127, 137, 141, 142, 143, 145, 149, 157, 158, 161, 167, 170, 174, 177, 178, 179, 182, 185, 191

Offset: 1

Gus Wiseman, Sep 15 2018

nonn

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

			105 is the Heinz number of (4,3,2), which is strict and has odd weight, so 105 belongs to the sequence.
The sequence of all odd-weight strict partitions begins: (1), (3), (2,1), (5), (4,1), (3,2), (7), (9), (6,1), (11), (5,2), (4,3), (8,1), (13), (4,2,1).

Complement of the union of A319241 and A013929.

Cf. A000041, A000720, A001222, A005117, A008683, A056239, A296150, A299202, A300061, A300063.

Select[Range[100],And[SquareFreeQ[#],OddQ[Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]]]&]

A319246 Sum of prime indices of the n-th squarefree number.

Original entry on oeis.org

0, 1, 2, 3, 3, 4, 4, 5, 6, 5, 5, 7, 8, 6, 6, 9, 7, 10, 6, 11, 7, 8, 7, 12, 9, 8, 13, 7, 14, 10, 15, 9, 16, 8, 10, 11, 17, 18, 12, 9, 8, 19, 11, 8, 20, 21, 13, 9, 9, 22, 14, 23, 10, 15, 12, 24, 10, 13, 16, 11, 25, 26, 10, 27, 9, 17, 28, 29, 9, 14, 30, 11, 12

Offset: 1

Gus Wiseman, Sep 15 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

			The 19th squarefree number is 30 with prime indices (3,2,1), so a(19) = 6.

Rows sums of A319247.

Cf. A000720, A001222, A005117, A056239, A072047, A265668, A299755, A299757, A319241, A319242.

Table[Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimePi[p]]],{n,Select[Range[100],SquareFreeQ]}]

A319829 FDH numbers of strict integer partitions of odd numbers.

Original entry on oeis.org

2, 4, 6, 7, 10, 11, 12, 16, 18, 19, 20, 21, 25, 26, 30, 31, 33, 34, 35, 36, 41, 46, 47, 48, 52, 53, 54, 55, 56, 57, 58, 60, 61, 63, 68, 71, 74, 75, 78, 79, 80, 83, 86, 88, 90, 91, 92, 93, 95, 97, 98, 99, 102, 103, 105, 108, 109, 116, 118, 119, 121, 123, 125

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).

			The sequence of all strict integer partitions of odd numbers begins: (1), (3), (2,1), (5), (4,1), (7), (3,2), (9), (6,1), (11), (4,3), (5,2), (13), (8,1), (4,2,1), (15), (7,2), (10,1), (5,4), (6,3), (17), (12,1), (19), (9,2), (8,3), (21), (6,2,1), (7,4), (5,3,1), (11,2), (14,1), (4,3,2).

Complement of A319828.

Cf. A050376, A064547, A213925, A299755, A299757, A300061, A300063, A319241, A319242, A319827.

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];
Select[Range[nn],OddQ[Total[FDfactor[#]/.FDrules]]&]

A319828 FDH numbers of strict integer partitions of even numbers.

Original entry on oeis.org

1, 3, 5, 8, 9, 13, 14, 15, 17, 22, 23, 24, 27, 28, 29, 32, 37, 38, 39, 40, 42, 43, 44, 45, 49, 50, 51, 59, 62, 64, 65, 66, 67, 69, 70, 72, 73, 76, 77, 81, 82, 84, 85, 87, 89, 94, 96, 100, 101, 104, 106, 107, 110, 111, 112, 113, 114, 115, 117, 120, 122, 124

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).

			The sequence of all strict integer partitions of even numbers begins: (), (2), (4), (3,1), (6), (8), (5,1), (4,2), (10), (7,1), (12), (3,2,1), (6,2), (5,3), (14), (9,1), (16).

Complement of A319827.

Cf. A050376, A064547, A213925, A299755, A299757, A300061, A300063, A319241, A319242, A319829.

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];
Select[Range[nn],EvenQ[Total[FDfactor[#]/.FDrules]]&]

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A357854 Squarefree numbers with a divisor having the same sum of prime indices as their quotient.

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A338908 Squarefree semiprimes whose prime indices sum to an even number.

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A319242 Heinz numbers of strict integer partitions of odd numbers. Squarefree numbers whose prime indices sum to an odd number.

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A319246 Sum of prime indices of the n-th squarefree number.

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A319829 FDH numbers of strict integer partitions of odd numbers.

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A319828 FDH numbers of strict integer partitions of even numbers.

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