A316428 -id:A316428 - cp's OEIS frontend

A353395 Numbers k such that the prime shadow of k equals the product of prime shadows of the prime indices of k.

1, 3, 5, 11, 15, 17, 26, 31, 33, 41, 51, 55, 58, 59, 67, 78, 83, 85, 86, 93, 94, 109, 123, 126, 127, 130, 146, 148, 155, 157, 158, 165, 174, 177, 179, 187, 191, 196, 201, 202, 205, 211, 241, 244, 249, 255, 258, 274, 277, 278, 282, 283, 284, 286, 290, 295, 298

Offset: 1

Gus Wiseman, May 17 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.

We define the prime shadow A181819(n) to be the product of primes indexed by the exponents in the prime factorization of n. For example, 90 = prime(1)*prime(2)^2*prime(3) has prime shadow prime(1)*prime(2)*prime(1) = 12.

			The terms together with their prime indices begin:
      1: {}         78: {1,2,6}      158: {1,22}
      3: {2}        83: {23}         165: {2,3,5}
      5: {3}        85: {3,7}        174: {1,2,10}
     11: {5}        86: {1,14}       177: {2,17}
     15: {2,3}      93: {2,11}       179: {41}
     17: {7}        94: {1,15}       187: {5,7}
     26: {1,6}     109: {29}         191: {43}
     31: {11}      123: {2,13}       196: {1,1,4,4}
     33: {2,5}     126: {1,2,2,4}    201: {2,19}
     41: {13}      127: {31}         202: {1,26}
     51: {2,7}     130: {1,3,6}      205: {3,13}
     55: {3,5}     146: {1,21}       211: {47}
     58: {1,10}    148: {1,1,12}     241: {53}
     59: {17}      155: {3,11}       244: {1,1,18}
     67: {19}      157: {37}         249: {2,23}
For example, 126 is in the sequence because its prime indices {1,2,2,4} have shadows {1,2,2,3}, with product 12, which is also the prime shadow of 126.

The prime terms are A006450.
The LHS (prime shadow) is A181819, with an inverse A181821.
The RHS (product of shadows) is A353394, first appearances A353397.
This is a ranking of the partitions counted by A353396.
Another related comparison is A353399, counted by A353398.
A001222 counts prime factors with multiplicity, distinct A001221.
A003963 gives product of prime indices.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914, product A005361.
A130091 lists numbers with distinct prime exponents, counted by A098859.
A324850 lists numbers divisible by the product of their prime indices.
Numbers divisible by their prime shadow:
- counted by A325702
- listed by A325755
- co-recursive version A325756
- nonprime recursive version A353389
- recursive version A353393, counted by A353426
Cf. A000005, A000961, A003586, A005117, A143773, A182850, A316428, A316438, A320325, A325131, A339095.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
Select[Range[100],Times@@red/@primeMS[#]==red[#]&]

A181819(a(n)) = A353394(a(n)) = Product_i A181819(A112798(a(n),i)).

A340856 Squarefree numbers whose greatest prime index (A061395) is divisible by their number of prime factors (A001222).

Original entry on oeis.org

2, 3, 5, 6, 7, 11, 13, 14, 17, 19, 21, 23, 26, 29, 30, 31, 35, 37, 38, 39, 41, 43, 47, 53, 57, 58, 59, 61, 65, 67, 71, 73, 74, 78, 79, 83, 86, 87, 89, 91, 95, 97, 101, 103, 106, 107, 109, 111, 113, 122, 127, 129, 130, 131, 133, 137, 138, 139, 142, 143, 145

Offset: 1

Gus Wiseman, Feb 05 2021

nonn

Also Heinz numbers of strict integer partitions whose greatest part is divisible by their number of parts. These partitions are counted by A340828.

			The sequence of terms together with their prime indices begins:
      2: {1}         31: {11}       71: {20}
      3: {2}         35: {3,4}      73: {21}
      5: {3}         37: {12}       74: {1,12}
      6: {1,2}       38: {1,8}      78: {1,2,6}
      7: {4}         39: {2,6}      79: {22}
     11: {5}         41: {13}       83: {23}
     13: {6}         43: {14}       86: {1,14}
     14: {1,4}       47: {15}       87: {2,10}
     17: {7}         53: {16}       89: {24}
     19: {8}         57: {2,8}      91: {4,6}
     21: {2,4}       58: {1,10}     95: {3,8}
     23: {9}         59: {17}       97: {25}
     26: {1,6}       61: {18}      101: {26}
     29: {10}        65: {3,6}     103: {27}
     30: {1,2,3}     67: {19}      106: {1,16}

Note: Heinz number sequences are given in parentheses below.
The case of equality, and the reciprocal version, are both A002110.
The non-strict reciprocal version is A168659 (A340609).
The non-strict version is A168659 (A340610).
These are the Heinz numbers of partitions counted by A340828.
A001222 counts prime factors.
A006141 counts partitions whose length equals their minimum (A324522).
A056239 adds up the prime indices.
A061395 selects the maximum prime index.
A067538 counts partitions whose length divides their sum (A316413/A326836).
A112798 lists the prime indices of each positive integer.
A200750 counts partitions with length coprime to maximum (A340608).
A257541 gives the rank of the partition with Heinz number n.
A340830 counts strict partitions whose parts are multiples of the length.
Cf. A039900, A047993 (A106529), A064174, A143773 (A316428), A326837, A326849 (A326848), A340599, A340691.

Select[Range[2,100],SquareFreeQ[#]&&Divisible[PrimePi[FactorInteger[#][[-1,1]]],PrimeOmega[#]]&]

A344413 Numbers n whose sum of prime indices A056239(n) is even and is at least twice the number of prime factors A001222(n).

Original entry on oeis.org

1, 3, 7, 9, 10, 13, 19, 21, 22, 25, 27, 28, 29, 30, 34, 37, 39, 43, 46, 49, 52, 53, 55, 57, 61, 62, 63, 66, 70, 71, 75, 76, 79, 81, 82, 84, 85, 87, 88, 89, 90, 91, 94, 100, 101, 102, 107, 111, 113, 115, 116, 117, 118, 121, 129, 130, 131, 133, 134, 136, 138

Offset: 1

Gus Wiseman, May 19 2021

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.

Also Heinz numbers of integer partitions of even numbers m with at most m/2 parts, counted by A209816 riffled with zeros, or A110618 with odd positions zeroed out.

			The sequence of terms together with their prime indices begins:
      1: {}          37: {12}        75: {2,3,3}
      3: {2}         39: {2,6}       76: {1,1,8}
      7: {4}         43: {14}        79: {22}
      9: {2,2}       46: {1,9}       81: {2,2,2,2}
     10: {1,3}       49: {4,4}       82: {1,13}
     13: {6}         52: {1,1,6}     84: {1,1,2,4}
     19: {8}         53: {16}        85: {3,7}
     21: {2,4}       55: {3,5}       87: {2,10}
     22: {1,5}       57: {2,8}       88: {1,1,1,5}
     25: {3,3}       61: {18}        89: {24}
     27: {2,2,2}     62: {1,11}      90: {1,2,2,3}
     28: {1,1,4}     63: {2,2,4}     91: {4,6}
     29: {10}        66: {1,2,5}     94: {1,15}
     30: {1,2,3}     70: {1,3,4}    100: {1,1,3,3}
     34: {1,7}       71: {20}       101: {26}
For example, 75 has 3 prime indices {2,3,3} with sum 8 >= 2*3, so 75 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

These are the Heinz numbers of partitions counted by A209816 and A110618.
A subset of A300061 (sum of prime indices is even).
The conjugate version appears to be A320924 (allowing odd weights: A322109).
The case of equality is A340387.
Allowing odd weights gives A344291.
The 5-smooth case is A344295, or A344293 allowing odd weights.
The opposite version allowing odd weights is A344296.
The conjugate opposite version allowing odd weights is A344414.
The case of equality in the conjugate case is A344415.
The conjugate opposite version is A344416, counted by A000070.
A001222 counts prime factors with multiplicity.
A027187 counts partitions of even length, ranked by A028260.
A056239 adds up prime indices, row sums of A112798.
A058696 counts partitions of even numbers, ranked by A300061.
A301987 lists numbers whose sum of prime indices equals their product.
A330950 counts partitions of n with Heinz number divisible by n.
A334201 adds up all prime indices except the greatest.
Cf. A001414, A025065, A067538, A301988, A316413, A316428, A325037, A325038, A325044, A338914, A344294, A344297.

filter:= proc(n) local F,a,t;
  F:= ifactors(n)[2];
  a:= add((numtheory:-pi(t[1])-2)*t[2],t=F);
  a::even and a >= 0
end proc:
select(filter, [$1..300]); # Robert Israel, Oct 10 2024

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

Members m of A300061 such that A056239(m) >= 2*A001222(m).

A353396 Number of integer partitions of n whose Heinz number has prime shadow equal to the product of prime shadows of its parts.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 0, 3, 1, 3, 4, 3, 7, 5, 9, 8, 12, 15, 15, 20, 21, 25, 31, 33, 38, 42, 46, 56, 61, 67, 78, 76, 96, 100, 114, 131, 130, 157, 157, 185, 200, 214, 236, 253, 275, 302, 333, 351, 386, 408, 440, 486, 515, 564, 596, 633, 691, 734, 800, 854, 899, 964

Offset: 0

Gus Wiseman, May 15 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

We define the prime shadow A181819(n) to be the product of primes indexed by the exponents in the prime factorization of n. For example, 90 = prime(1)*prime(2)^2*prime(3) has prime shadow prime(1)*prime(2)*prime(1) = 12.

			The a(8) = 1 through a(14) = 9 partitions (A..D = 10..13):
  (53)  (72)    (73)    (B)     (75)     (D)      (B3)
        (621)   (532)   (A1)    (651)    (B2)     (752)
        (4221)  (631)   (4331)  (732)    (A21)    (761)
                (4411)          (6321)   (43321)  (A31)
                                (6411)   (44311)  (C11)
                                (43221)           (6521)
                                (44211)           (9221)
                                                  (54221)
                                                  (64211)

The LHS (prime shadow) is A181819, with an inverse A181821.
The RHS (product of prime shadows) is A353394, first appearances A353397.
These partitions are ranked by A353395.
A related comparison is A353398, ranked by A353399.
A001222 counts prime factors with multiplicity, distinct A001221.
A003963 gives product of prime indices.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914, product A005361.
A239455 counts Look-and-Say partitions, ranked by A351294.
A324850 lists numbers divisible by the product of their prime indices.
Cf. A000005, A002033, A143773, A182850, A316428, A325131, A325702, A325755, A353389, A353426.

red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
Table[Length[Select[IntegerPartitions[n],Times@@red/@#==red[Times@@Prime/@#]&]],{n,0,15}]

A340829 Number of strict integer partitions of n whose Heinz number (product of primes of parts) is divisible by n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 2, 0, 0, 2, 3, 0, 4, 3, 4, 0, 8, 0, 10, 0, 11, 12, 19, 0, 0, 22, 0, 0, 46, 23, 56, 0, 64, 66, 86, 0, 125, 104, 135, 0, 196, 111, 230, 0, 0, 274, 353, 0, 0, 0, 563, 0, 687, 0, 974, 0, 1039, 1052, 1290, 0, 1473, 1511, 0, 0, 2707, 1614, 2664, 0

Offset: 1

Gus Wiseman, Feb 01 2021

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions. The Heinz numbers of these partitions are squarefree numbers divisible by the sum of their prime indices.

			The a(6) = 1 through a(19) = 10 partitions (empty columns indicated by dots, A = 10, B = 11):
  321  43   .  .  631   65    .  76    941   A32    .  A7     .  B8
       421        4321  542      643   6431  6432      764       865
                        5321     652   7421  9321      872       874
                                 6421        54321     971       982
                                                       7532      A81
                                                       7541      8542
                                                       7631      8632
                                                       74321     8641
                                                                 8731
                                                                 85321

Note: A-numbers of Heinz-number sequences are in parentheses below.
Positions of zeros are 2 and A013929.
The non-strict version is A330950 (A324851) q.v.
A000009 counts strict partitions.
A003963 multiplies together prime indices.
A018818 counts partitions into divisors (A326841).
A047993 counts balanced partitions (A106529).
A056239 adds up prime indices.
A057568 counts partitions whose product is divisible by their sum (A326149).
A067538 counts partitions whose length/max divides sum (A316413/A326836).
A072233 counts partitions by sum and length, with strict case A008289.
A102627 counts strict partitions whose length divides sum.
A112798 lists the prime indices of each positive integer.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
A324925 counts partitions whose Heinz number is divisible by their product.
A326842 counts partitions whose parts and length all divide sum (A326847).
A326850 counts strict partitions whose maximum part divides sum.
A326851 counts strict partitions with length and maximum dividing sum.
A330952 counts partitions whose Heinz number is divisible by all parts.
A340828 counts strict partitions with length divisible by maximum.
A340830 counts strict partitions with parts divisible by length.
Cf. A052335, A064173, A064174, A096401, A143773 (A316428), A168659 (A340609/A340610), A326843 (A326837).

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Divisible[Times@@Prime/@#,n]&]],{n,30}]

A344292 Numbers m whose sum of prime indices A056239(m) is even and is at most twice the number of prime factors counted with multiplicity A001222(m).

Original entry on oeis.org

1, 3, 4, 9, 10, 12, 16, 27, 28, 30, 36, 40, 48, 64, 81, 84, 88, 90, 100, 108, 112, 120, 144, 160, 192, 208, 243, 252, 256, 264, 270, 280, 300, 324, 336, 352, 360, 400, 432, 448, 480, 544, 576, 624, 640, 729, 756, 768, 784, 792, 810, 832, 840, 880, 900, 972

Offset: 1

Gus Wiseman, May 22 2021

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.

Also Heinz numbers of integer partitions of even numbers m with at least m/2 parts, counted by A000070 riffled with 0's, or A025065 with odd positions zeroed out.

			The sequence of terms together with their prime indices begins:
       1: {}                 84: {1,1,2,4}
       3: {2}                88: {1,1,1,5}
       4: {1,1}              90: {1,2,2,3}
       9: {2,2}             100: {1,1,3,3}
      10: {1,3}             108: {1,1,2,2,2}
      12: {1,1,2}           112: {1,1,1,1,4}
      16: {1,1,1,1}         120: {1,1,1,2,3}
      27: {2,2,2}           144: {1,1,1,1,2,2}
      28: {1,1,4}           160: {1,1,1,1,1,3}
      30: {1,2,3}           192: {1,1,1,1,1,1,2}
      36: {1,1,2,2}         208: {1,1,1,1,6}
      40: {1,1,1,3}         243: {2,2,2,2,2}
      48: {1,1,1,1,2}       252: {1,1,2,2,4}
      64: {1,1,1,1,1,1}     256: {1,1,1,1,1,1,1,1}
      81: {2,2,2,2}         264: {1,1,1,2,5}

These are the Heinz numbers of partitions counted by A000070 and A025065.
A subset of A300061 (sum of prime indices is even).
The conjugate opposite version is A320924, counted by A209816.
The conjugate opposite version allowing odds is A322109, counted by A110618.
The case of equality is A340387, counted by A000041.
The opposite version allowing odd weights is A344291, counted by A110618.
Allowing odd weights gives A344296, counted by A025065.
The opposite version is A344413, counted by A209816.
The conjugate version allowing odd weights is A344414, counted by A025065.
The case of equality in the conjugate case is A344415, counted by A035363.
The conjugate version is A344416, counted by A000070.
A001222 counts prime factors with multiplicity.
A027187 counts partitions of even length, ranked by A028260.
A056239 adds up prime indices, row sums of A112798.
A058696 counts partitions of even numbers, ranked by A300061.
A301987 lists numbers whose sum of prime indices equals their product.
A330950 counts partitions of n with Heinz number divisible by n.
A334201 adds up all prime indices except the greatest.
Cf. A001414, A067538, A316413, A316428, A325037, A325038, A325044, A338914, A344294, A344297.

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

Members m of A300061 such that A056239(m) <= 2*A001222(m).

cp's OEIS Frontend

A353395 Numbers k such that the prime shadow of k equals the product of prime shadows of the prime indices of k.

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A340856 Squarefree numbers whose greatest prime index (A061395) is divisible by their number of prime factors (A001222).

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A344413 Numbers n whose sum of prime indices A056239(n) is even and is at least twice the number of prime factors A001222(n).

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A353396 Number of integer partitions of n whose Heinz number has prime shadow equal to the product of prime shadows of its parts.

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A340829 Number of strict integer partitions of n whose Heinz number (product of primes of parts) is divisible by n.

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A344292 Numbers m whose sum of prime indices A056239(m) is even and is at most twice the number of prime factors counted with multiplicity A001222(m).

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