|
|
A353395
|
|
Numbers k such that the prime shadow of k equals the product of prime shadows of the prime indices of k.
|
|
7
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
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.
|
|
LINKS
|
|
|
FORMULA
|
|
|
EXAMPLE
|
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.
|
|
MATHEMATICA
|
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[#]&]
|
|
CROSSREFS
|
This is a ranking of the partitions counted by A353396.
A003963 gives product of prime indices.
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:
- nonprime recursive version A353389
Cf. A000005, A000961, A003586, A005117, A143773, A182850, A316428, A316438, A320325, A325131, A339095.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|