A353395 -id:A353395 - cp's OEIS frontend

A353393 Positive integers m > 1 that are prime or whose prime shadow A181819(m) is a divisor of m that is already in the sequence.

2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 29, 31, 36, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 225, 227, 229, 233, 239, 241, 251

Offset: 1

Gus Wiseman, May 15 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 terms together with their prime indices begin:
    2: {1}
    3: {2}
    5: {3}
    7: {4}
    9: {2,2}
   11: {5}
   13: {6}
   17: {7}
   19: {8}
   23: {9}
   29: {10}
   31: {11}
   36: {1,1,2,2}

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

The first term that is not a prime power A000961 is 36.
The first term that is not a prime or a perfect power A001597 is 1260. - Corrected by Robert Israel, Mar 10 2025
The non-recursive version is A325755, counted by A325702.
Removing all primes gives A353389.
These partitions are counted by A353426.
The version for compositions is A353431.
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.
A130091 lists numbers with all distinct prime exponents, counted by A098859.
A181819 gives prime shadow, with an inverse A181821.
A325131 lists numbers relatively prime to their prime shadow.
Cf. A000005, A000040, A047966, A182850, A316413, A316428, A325756, A353394, A353395, A353399.

pshadow:= proc(n) local F,i;
  F:= ifactors(n)[2];
  mul(ithprime(i),i=F[..,2])
end proc:
filter:= proc(n) local s;
  if isprime(n) then return true fi;
  s:= pshadow(n);
  n mod s = 0 and member(s,R)
end proc:
R:= {}:
for i from 2 to 2000 do if filter(i) then R:= R union {i} fi od:
sort(convert(R,list)); # Robert Israel, Mar 10 2025

red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
suQ[n_]:=PrimeQ[n]||Divisible[n,red[n]]&&suQ[red[n]];
Select[Range[2,200],suQ[#]&]

Equals A353389 U A000040.

A353394 Product of prime shadows of prime indices of n (with multiplicity).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 4, 2, 2, 2, 4, 3, 4, 1, 2, 4, 5, 2, 6, 2, 3, 2, 4, 4, 8, 3, 4, 4, 2, 1, 4, 2, 6, 4, 6, 5, 8, 2, 2, 6, 4, 2, 8, 3, 4, 2, 9, 4, 4, 4, 7, 8, 4, 3, 10, 4, 2, 4, 6, 2, 12, 1, 8, 4, 2, 2, 6, 6, 6, 4, 4, 6, 8, 5, 6, 8, 4, 2, 16, 2, 2, 6, 4, 4

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.

			We have 42 = prime(1)*prime(2)*prime(4), so a(42) = 1*2*3 = 6.

Positions of first appearances are A353397.
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.
A181819 gives prime shadow, with an inverse A181821.
A324850 lists numbers divisible by the product of their prime indices.
A325131 lists numbers relatively prime to their prime shadow.
A325755 lists numbers divisible by their prime shadow, quotient also A325756, with recursion A353393.
Cf. A003586, A005117, A008480, A182850, A266477, A289509, A316428, A320325, A325702, A353389, A353395, A353399.

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]];
Table[Times@@red/@primeMS[n],{n,100}]

a(n) = Product_i A181819(A112798(n,i)).
Positions where a(n) = A003963(n) are A003586.
Positions where a(n) = A005361(n) are A353399, counted by A353398.
Positions where a(n) = A181819(n) are A353395, counted by A353396.

A353399 Numbers whose product of prime exponents equals the product of prime shadows of its prime indices.

Original entry on oeis.org

1, 2, 12, 20, 36, 44, 56, 68, 100, 124, 164, 184, 208, 236, 240, 268, 332, 436, 464, 484, 508, 528, 608, 628, 688, 716, 720, 752, 764, 776, 816, 844, 880, 964, 1108, 1132, 1156, 1168, 1200, 1264, 1296, 1324, 1344, 1360, 1412, 1468, 1488, 1584, 1604, 1616, 1724

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: {}
     2: {1}
    12: {1,1,2}
    20: {1,1,3}
    36: {1,1,2,2}
    44: {1,1,5}
    56: {1,1,1,4}
    68: {1,1,7}
   100: {1,1,3,3}
   124: {1,1,11}
   164: {1,1,13}
   184: {1,1,1,9}
   208: {1,1,1,1,6}
   236: {1,1,17}
   240: {1,1,1,1,2,3}

Product of prime indices is A003963, counted by A339095.
The LHS (product of exponents) is A005361, counted by A266477.
The RHS (product of shadows) is A353394, first appearances A353397.
A related comparison is A353395, counted by A353396.
The partitions are counted by A353398.
Taking indices instead of exponents on the LHS gives A353503.
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.
A130091 lists numbers with distinct prime exponents, counted by A098859.
A181819 gives prime shadow, with an inverse A181821.
A325131 lists numbers relatively prime to their prime shadow.
Numbers divisible by their prime shadow:
- counted by A325702
- listed by A325755
- co-recursive version A325756
- nonprime recursive version A353389
- recursive version A353393
- recursive version counted by A353426
Cf. A000720, A003586, A005117, A143773, A182850, A316428, A320325, A324850.

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[#]==Times@@Last/@FactorInteger[#]&]

A005361(a(n)) = A353394(a(n)).

A353397 Replace prime(k) with prime(2^k) in the prime factorization of n.

Original entry on oeis.org

1, 3, 7, 9, 19, 21, 53, 27, 49, 57, 131, 63, 311, 159, 133, 81, 719, 147, 1619, 171, 371, 393, 3671, 189, 361, 933, 343, 477, 8161, 399, 17863, 243, 917, 2157, 1007, 441, 38873, 4857, 2177, 513, 84017, 1113, 180503, 1179, 931, 11013, 386093, 567, 2809, 1083

Offset: 1

Gus Wiseman, May 17 2022

			The terms together with their prime indices begin:
      1: {}
      3: {2}
      7: {4}
      9: {2,2}
     19: {8}
     21: {2,4}
     53: {16}
     27: {2,2,2}
     49: {4,4}
     57: {2,8}
    131: {32}
     63: {2,2,4}

Amiram Eldar, Table of n, a(n) for n = 1..397 (calculated using the b-file at A033844)

These are the positions of first appearances in A353394.
A001222 counts prime factors with multiplicity, distinct A001221.
A003963 gives product of prime indices, counted by A339095.
A033844 lists primes indexed by powers of 2.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914, product A005361.
A181819 gives prime shadow, firsts A181821, relatively prime A325131.
Equivalent sequence with prime(2*k) instead of prime(2^k): A297002.
Cf. A003586, A005117, A130091, A182850, A289509, A324850, A325755, A353393, A353395, A353399.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@Prime/@(2^primeMS[n]),{n,100}]

a(n) = my(f=factor(n)); for(k=1, #f~, f[k,1] = prime(2^primepi(f[k,1]))); factorback(f); \\ Michel Marcus, May 20 2022

from math import prod
from sympy import prime, primepi, factorint
def A353397(n): return prod(prime(2**primepi(p))**e for p, e in factorint(n).items()) # Chai Wah Wu, May 20 2022

If n = prime(e_1)...prime(e_k), then a(n) = prime(2^(e_1))...prime(2^(e_k)).

Sum_{n>=1} 1/a(n) = 1/Product_{k>=1} (1 - 1/prime(2^k)) = 1.90812936178871496289... . - Amiram Eldar, Dec 09 2022

A353389 Create the sequence of all positive integers > 1 that are prime or whose prime shadow (A181819) is a divisor that is already in the sequence. Then remove all the primes.

Original entry on oeis.org

9, 36, 125, 225, 441, 1089, 1260, 1521, 1980, 2340, 2401, 2601, 2772, 3060, 3249, 3276, 3420, 4140, 4284, 4761, 4788, 5148, 5220, 5580, 5796, 6660, 6732, 7308, 7380, 7524, 7569, 7740, 7812, 7956, 8460, 8649, 8892, 9108, 9324, 9540, 10332, 10620, 10764, 10836

Offset: 1

Gus Wiseman, May 15 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.

Said differently, these are nonprime numbers > 1 whose prime shadow is a divisor that is either a prime number or a number already in the sequence.

			The initial terms and their prime indices:
     9: {2,2}
    36: {1,1,2,2}
   125: {3,3,3}
   225: {2,2,3,3}
   441: {2,2,4,4}
  1089: {2,2,5,5}
  1260: {1,1,2,2,3,4}
  1521: {2,2,6,6}
  1980: {1,1,2,2,3,5}

The first term that is not a perfect power A001597 is 1260.
Without the recursion we have A325755 (a superset), counted by A325702.
Before removing the primes we had A353393.
These partitions are counted by A353426 minus one.
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.
A181819 gives prime shadow, with an inverse A181821.
A182850 and A323014 give frequency depth, counted by A225485 and A325280.
A325131 lists numbers relatively prime to their prime shadow.
Cf. A000005, A000040, A005117, A316413, A316428, A324850, A353394, A353395, A353397, A353399.

red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
suQ[n_]:=PrimeQ[n]||Divisible[n,red[n]]&&suQ[red[n]];
Select[Range[2,2000],suQ[#]&&!PrimeQ[#]&]

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

Original entry on oeis.org

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}]

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}]

cp's OEIS Frontend

A353393 Positive integers m > 1 that are prime or whose prime shadow A181819(m) is a divisor of m that is already in the sequence.

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A353394 Product of prime shadows of prime indices of n (with multiplicity).

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A353399 Numbers whose product of prime exponents equals the product of prime shadows of its prime indices.

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A353397 Replace prime(k) with prime(2^k) in the prime factorization of n.

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A353389 Create the sequence of all positive integers > 1 that are prime or whose prime shadow (A181819) is a divisor that is already in the sequence. Then remove all the primes.

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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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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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Mathematica