A324850 -id:A324850 - cp's OEIS frontend

A353398 Number of integer partitions of n where the product of multiplicities equals the product of prime shadows of the parts.

1, 1, 0, 0, 1, 1, 1, 2, 1, 2, 1, 2, 6, 5, 4, 4, 6, 6, 8, 8, 13, 16, 13, 16, 18, 16, 20, 21, 27, 30, 27, 33, 41, 44, 51, 48, 58, 61, 66, 66, 74, 83, 86, 99, 102, 111, 115, 126, 137, 147, 156

Offset: 0

Gus Wiseman, May 17 2022

nonn

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) = 4 partitions (A = 10, B = 11):
  3311  711     61111  521111   5511      B11       A1111
        321111         3221111  9111      721111    731111
                                531111    811111    33221111
                                3321111   5221111   422111111
                                22221111  43111111
                                42111111

The LHS (product of multiplicities) is A005361, counted by A266477.
The RHS (product of prime shadows) is A353394, first appearances A353397.
A related comparison is A353396, ranked by A353395.
These partitions are ranked by A353399.
A001222 counts prime factors with multiplicity, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A325131 lists numbers relatively prime to their prime shadow.
A325755 lists numbers divisible by their prime shadow, counted by A325702.
A339095 counts partitions by product (or factorizations by sum).
Cf. A002033, A003963, A143773, A182850, A239455, A324850, A325756, A353393, A353426.

red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
Table[Length[Select[IntegerPartitions[n],Times@@red/@#==Times@@Length/@Split[#]&]],{n,0,30}]

A324853 First number divisible by n of its own distinct prime indices.

Original entry on oeis.org

1, 2, 6, 30, 330, 4290, 60060, 1021020, 29609580, 917896980, 33962188260, 1290563153880, 52913089309080, 2275262840290440, 106937353493650680, 6309303856125390120, 422723358360401138040, 30013358443588480800840, 2190975166381959098461320

Offset: 0

Gus Wiseman, Mar 18 2019

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.

a(n) is the first position of n in A324852.

			a(6) = 60060 = 2^2 * 3 * 5 * 7 * 11 * 13 has prime indices {1,1,2,3,4,5,6}, and is less than any other number divisible by six of its own distinct prime indices.

Rémy Sigrist, C program for A324853

Cf. A000720, A001222, A003963, A056239, A120383, A323440.

Cf. A324771, A324846, A324847, A324848, A324849, A324850, A324852, A324856.

C
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See Links section.
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nn=10000;
With[{mgs=Table[Count[If[n==1,{},FactorInteger[n]],{p_,_}/;Divisible[n,PrimePi[p]]],{n,nn}]},Table[Position[mgs,i][[1,1]],{i,0,5}]]

isok(k,n) = {my(f=factor(k)[,1]); sum(j=1, #f, !(k % primepi(f[j]))) == n;}
a(n) = {my(k=1); while (!isok(k, n), k++); k;} \\ Michel Marcus, Mar 20 2019

a(8)-a(9) from Rémy Sigrist, Mar 19 2019

a(10)-a(18) from Michel Lagneau, Aug 19 2019

A331382 Numbers whose sum of prime factors is divisible by their product of prime indices.

Original entry on oeis.org

1, 2, 4, 8, 16, 18, 20, 32, 35, 44, 60, 62, 64, 65, 68, 72, 92, 95, 98, 128, 154, 160, 168, 256, 264, 288, 291, 303, 324, 364, 400, 476, 480, 512, 618, 623, 624, 642, 706, 763, 791, 812, 816, 826, 938, 994, 1024, 1036, 1064, 1068, 1106, 1144, 1148, 1152, 1162

Offset: 1

Gus Wiseman, Jan 16 2020

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 sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
     4: {1,1}
     8: {1,1,1}
    16: {1,1,1,1}
    18: {1,2,2}
    20: {1,1,3}
    32: {1,1,1,1,1}
    35: {3,4}
    44: {1,1,5}
    60: {1,1,2,3}
    62: {1,11}
    64: {1,1,1,1,1,1}
    65: {3,6}
    68: {1,1,7}
    72: {1,1,1,2,2}
    92: {1,1,9}
    95: {3,8}
    98: {1,4,4}
   128: {1,1,1,1,1,1,1}
For example, 60 has prime factors {2,2,3,5} and prime indices {1,1,2,3}, and 12 is divisible by 6, so 60 is in the sequence.

These are the Heinz numbers of the partitions counted by A331381.
Numbers divisible by the sum of their prime factors are A036844.
Partitions whose product is divisible by their sum are A057568.
Numbers divisible by the sum of their prime indices are A324851.
Product of prime indices is divisible by sum of prime indices: A326149.
Partitions whose Heinz number is divisible by their sum are A330950.
Sum of prime factors is divisible by sum of prime indices: A331380
Partitions whose product is equal to the sum of primes are A331383.
Product of prime indices equals sum of prime factors: A331384.
Cf. A000040, A001414, A324850, A330953, A330954, A331378, A331379.

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

A323440 Numbers divisible by exactly one of their distinct prime indices.

Original entry on oeis.org

2, 4, 8, 10, 14, 15, 16, 20, 22, 26, 32, 34, 38, 40, 44, 45, 46, 50, 52, 55, 58, 62, 64, 68, 70, 74, 75, 76, 80, 82, 86, 88, 92, 94, 98, 100, 104, 105, 106, 116, 118, 119, 122, 124, 128, 130, 134, 135, 136, 142, 146, 148, 154, 158, 160, 164, 166, 170, 172, 176

Offset: 1

Gus Wiseman, Mar 21 2019

nonn

Numbers n such that A324852(n) = 1.

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 sequence of terms together with their prime indices begins:
   2: {1}
   4: {1,1}
   8: {1,1,1}
  10: {1,3}
  14: {1,4}
  15: {2,3}
  16: {1,1,1,1}
  20: {1,1,3}
  22: {1,5}
  26: {1,6}
  32: {1,1,1,1,1}
  34: {1,7}
  38: {1,8}
  40: {1,1,1,3}
  44: {1,1,5}
  45: {2,2,3}
  46: {1,9}
  50: {1,3,3}
  52: {1,1,6}
  55: {3,5}

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000720, A003963, A112798, A120383, A324704, A324846, A324847, A324848, A324849, A324850, A324856, A324926, A324929.

Select[Range[100],Count[If[#==1,{},FactorInteger[#]],{p_,_}/;Divisible[#,PrimePi[p]]]==1&]

isok(n) = my(f=factor(n)[,1]); sum(k=1, #f, (n % primepi(f[k])) == 0) == 1; \\ Michel Marcus, Mar 22 2019

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

Original entry on oeis.org

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)).

A355736 Least k such that there are exactly n ways to choose a divisor of each prime index of k (taken in weakly increasing order) such that the result is also weakly increasing.

Original entry on oeis.org

1, 3, 7, 13, 21, 37, 39, 89, 133, 117, 111, 273, 351, 259, 267, 333, 453, 793, 669, 623, 999, 777, 843, 1491, 1157, 1561, 2863, 1443, 1963, 2331, 1977, 1869, 2899, 2529, 3207, 4107, 3171, 5073, 4329, 3653, 4667, 3471, 7399, 4613, 7587, 5931, 7269, 5889, 7483

Offset: 1

Gus Wiseman, Jul 21 2022

nonn

This is the position of first appearance of n in A355735.

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: {}
     3: {2}
     7: {4}
    13: {6}
    21: {2,4}
    37: {12}
    39: {2,6}
    89: {24}
   133: {4,8}
   117: {2,2,6}
   111: {2,12}
   273: {2,4,6}
   351: {2,2,2,6}
For example, the choices for a(12) = 273 are:
  {1,1,1}  {1,2,2}  {2,2,2}
  {1,1,2}  {1,2,3}  {2,2,3}
  {1,1,3}  {1,2,6}  {2,2,6}
  {1,1,6}  {1,4,6}  {2,4,6}

Allowing any choice of divisors gives A355732, firsts of A355731.
Choosing a multiset instead of sequence gives A355734, firsts of A355733.
Positions of first appearances in A355735.
The case of prime factors instead of divisors is counted by A355745.
The decreasing version is counted by A355749.
A000005 counts divisors.
A001414 adds up distinct prime divisors, counted by A001221.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
Cf. A000720, A076610, A355737, A355739, A355740, A355741, A355744.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
az=Table[Length[Select[Tuples[Divisors/@primeMS[n]],LessEqual@@#&]],{n,1000}];
Table[Position[az,k][[1,1]],{k,mnrm[az]}]

A355749 Number of ways to choose a weakly decreasing sequence of divisors, one of each prime index of n (with multiplicity, taken in weakly increasing order).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 18 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 a(2) = 1 through a(19) = 4 choices:
  1  1  11  1  11  1  111  11  11  1  111  1  11  11  1111  1  111  1
     2      3      2       21      5       2      21        7       2
                   4       22              3                        4
                                           6                        8

Wikipedia, Cartesian product.

Allowing any choice of divisors gives A355731, firsts A355732.
Choosing a multiset instead of sequence gives A355733, firsts A355734.
The reverse version is A355735, firsts A355736, only primes A355745.
A000005 counts divisors.
A001414 adds up distinct prime divisors, counted by A001221.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A061395 selects the maximum prime index.
Cf. A000720, A076610, A120383, A316524, A324850, A355737, A355739, A355740, A355741, A355742, A355744.

Table[Length[Select[Tuples[Divisors/@primeMS[n]], GreaterEqual@@#&]],{n,100}]

A324771 Numbers divisible by at least one of their prime indices > 1.

Original entry on oeis.org

6, 12, 15, 18, 24, 28, 30, 36, 42, 45, 48, 54, 55, 56, 60, 66, 72, 75, 78, 84, 90, 96, 102, 105, 108, 110, 112, 114, 119, 120, 126, 132, 135, 138, 140, 144, 150, 152, 156, 162, 165, 168, 174, 180, 186, 192, 195, 196, 198, 204, 207, 210, 216, 220, 222, 224, 225

Offset: 1

Gus Wiseman, Mar 18 2019

nonn

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

			The sequence of terms together with their prime indices begins:
   6: {1,2}
  12: {1,1,2}
  15: {2,3}
  18: {1,2,2}
  24: {1,1,1,2}
  28: {1,1,4}
  30: {1,2,3}
  36: {1,1,2,2}
  42: {1,2,4}
  45: {2,2,3}
  48: {1,1,1,1,2}
  54: {1,2,2,2}
  55: {3,5}
  56: {1,1,1,4}
  60: {1,1,2,3}
  66: {1,2,5}
  72: {1,1,1,2,2}
  75: {2,3,3}
  78: {1,2,6}
  84: {1,1,2,4}

Amiram Eldar, Table of n, a(n) for n = 1..10000

Complement of A324849.
Cf. A003963, A056239, A112798, A120383, A289509, A290822, A304360.
Cf. A324695, A324846, A324847, A324848, A324850, A324851, A324852, A324853.

Select[Range[100],Or@@Cases[If[#==1,{},FactorInteger[#]],{p_?(#>2&),_}:>Divisible[#,PrimePi[p]]]&]

A324968 Matula-Goebel numbers of rooted identity trees whose non-leaf terminal subtrees are all different.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 11, 13, 22, 26, 29, 31, 41, 58, 62, 79, 82, 101, 109, 127, 158, 179, 202, 218, 254, 271, 293, 358, 401, 421, 542, 547, 586, 599, 709, 802, 842, 929, 1063, 1094, 1198, 1231, 1361, 1418, 1609, 1741, 1858, 1913, 2126, 2411, 2462, 2722, 2749

Offset: 1

Gus Wiseman, Mar 21 2019

nonn

A rooted identity tree is an unlabeled rooted tree with no repeated branches directly under the same root. This sequence ranks rooted identity trees satisfying the additional condition that all non-leaf terminal subtrees are different.

			The sequence of trees together with the Matula-Goebel numbers begins:
    1: o
    2: (o)
    3: ((o))
    5: (((o)))
    6: (o(o))
   10: (o((o)))
   11: ((((o))))
   13: ((o(o)))
   22: (o(((o))))
   26: (o(o(o)))
   29: ((o((o))))
   31: (((((o)))))
   41: (((o(o))))
   58: (o(o((o))))
   62: (o((((o)))))
   79: ((o(((o)))))
   82: (o((o(o))))
  101: ((o(o(o))))
  109: (((o((o)))))
  127: ((((((o))))))

Gus Wiseman, The first 36 trees together with their Matula-Goebel numbers.

Cf. A000081, A004111, A007097, A196050, A276625, A317713, A324850, A324923, A324935, A324936, A324969, A324970, A324978.

mgtree[n_Integer]:=If[n==1,{},mgtree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],And[And@@Cases[mgtree[#],q:{}:>UnsameQ@@q,{0,Infinity}],UnsameQ@@Cases[mgtree[#],{},{0,Infinity}]]&]

Intersection of A324935 and A276625.

A324970 Matula-Goebel numbers of rooted identity trees where not all terminal subtrees are different.

Original entry on oeis.org

15, 30, 33, 39, 47, 55, 65, 66, 78, 87, 93, 94, 110, 113, 123, 130, 137, 141, 143, 145, 155, 165, 167, 174, 186, 195, 205, 211, 226, 235, 237, 246, 257, 274, 282, 286, 290, 303, 310, 313, 317, 319, 327, 330, 334, 339, 341, 377, 381, 390, 395, 397, 403, 410

Offset: 1

Gus Wiseman, Mar 21 2019

nonn

A rooted identity tree is an unlabeled rooted tree with no repeated branches directly under the same root.

			The sequence of trees together with the Matula-Goebel numbers begins:
   15: ((o)((o)))
   30: (o(o)((o)))
   33: ((o)(((o))))
   39: ((o)(o(o)))
   47: (((o)((o))))
   55: (((o))(((o))))
   65: (((o))(o(o)))
   66: (o(o)(((o))))
   78: (o(o)(o(o)))
   87: ((o)(o((o))))
   93: ((o)((((o)))))
   94: (o((o)((o))))
  110: (o((o))(((o))))
  113: ((o(o)((o))))
  123: ((o)((o(o))))
  130: (o((o))(o(o)))
  137: (((o)(((o)))))
  141: ((o)((o)((o))))
  143: ((((o)))(o(o)))
  145: (((o))(o((o))))
  155: (((o))((((o)))))
  165: ((o)((o))(((o))))
  167: (((o)(o(o))))
  174: (o(o)(o((o))))
  186: (o(o)((((o)))))
  195: ((o)((o))(o(o)))

Gus Wiseman, The first 36 trees together with their Matula-Goebel numbers.

Cf. A000081, A004111, A007097, A196050, A276625, A317713, A324850, A324923, A324935, A324936, A324968, A324971, A324978.

mgtree[n_Integer]:=If[n==1,{},mgtree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],And[And@@Cases[mgtree[#],q:{}:>UnsameQ@@q,{0,Infinity}],!UnsameQ@@Cases[mgtree[#],{},{0,Infinity}]]&]

Complement of A324935 in A276625.

cp's OEIS Frontend

A353398 Number of integer partitions of n where the product of multiplicities equals the product of prime shadows of the parts.

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A324853 First number divisible by n of its own distinct prime indices.

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A331382 Numbers whose sum of prime factors is divisible by their product of prime indices.

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A323440 Numbers divisible by exactly one of their distinct prime indices.

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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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Formula

A355736 Least k such that there are exactly n ways to choose a divisor of each prime index of k (taken in weakly increasing order) such that the result is also weakly increasing.

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A355749 Number of ways to choose a weakly decreasing sequence of divisors, one of each prime index of n (with multiplicity, taken in weakly increasing order).

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A324771 Numbers divisible by at least one of their prime indices > 1.

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