A106092 -id:A106092 - cp's OEIS frontend

A330949 Odd nonprime numbers whose prime indices are not all prime numbers.

21, 35, 39, 49, 57, 63, 65, 69, 77, 87, 91, 95, 105, 111, 115, 117, 119, 129, 133, 141, 143, 145, 147, 159, 161, 169, 171, 175, 183, 185, 189, 195, 203, 207, 209, 213, 215, 217, 219, 221, 231, 235, 237, 245, 247, 253, 259, 261, 265, 267, 273, 285, 287, 291

Offset: 1

Gus Wiseman, Jan 14 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.

Also MM-numbers of multiset partitions with at least two parts, not all of which are singletons (see example).

			The sequence of terms together with their prime indices of prime indices begins:
   21: {{1},{1,1}}
   35: {{2},{1,1}}
   39: {{1},{1,2}}
   49: {{1,1},{1,1}}
   57: {{1},{1,1,1}}
   63: {{1},{1},{1,1}}
   65: {{2},{1,2}}
   69: {{1},{2,2}}
   77: {{1,1},{3}}
   87: {{1},{1,3}}
   91: {{1,1},{1,2}}
   95: {{2},{1,1,1}}
  105: {{1},{2},{1,1}}
  111: {{1},{1,1,2}}
  115: {{2},{2,2}}
  117: {{1},{1},{1,2}}
  119: {{1,1},{4}}
  129: {{1},{1,4}}
  133: {{1,1},{1,1,1}}
  141: {{1},{2,3}}

Complement of A106092 in A330945.
Including even numbers gives A330948.
Including primes gives A330946.
The number of prime prime indices is given by A257994.
The number of nonprime prime indices is given by A330944.
Primes of prime index are A006450.
Primes of nonprime index are A007821.
Products of primes of prime index are A076610.
Products of primes of nonprime index are A320628.
The set S of numbers whose prime indices do not all belong to S is A324694.
Cf. A000040, A000720, A001222, A018252, A056239, A112798, A302242, A320629, A320633, A330943, A330947.

Select[Range[1,100,2],!PrimeQ[#]&&!And@@PrimeQ/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]&]

A370482 Characteristic function of primes plus characteristic function of even numbers.

Original entry on oeis.org

1, 0, 2, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1

Offset: 0

Jens Ahlström, Mar 31 2024

There is only one 2 in the sequence, so if the value 2 is blanked out, a riddle is created that demands some out-of-the-box thinking.

			1 is neither prime nor even so a(1) = 0 + 0 = 0.
2 is both a prime and even so a(2) = 1 + 1 = 2.
3 is a prime but odd so a(3) = 1 + 0 = 1.
4 is not a prime but even so a(4) = 0 + 1 = 1.

Antti Karttunen, Table of n, a(n) for n = 0..100000

Cf. A010051, A059841.

If a(2) were 1 instead of 2, then this would the characteristic function of {0} U A106092, whose complement A014076 gives the positions of 0's. - Antti Karttunen, Jan 17 2025

a[n_] := Boole[PrimeQ[n]] + Boole[EvenQ[n]]; Array[a, 100, 0] (* Amiram Eldar, Mar 31 2024 *)

A370482(n) = (!(n%2) + isprime(n)); \\ Antti Karttunen, Jan 17 2025

from sympy import isprime
def A370482(n): return isprime(n)+(n&1^1) # Chai Wah Wu, Apr 25 2024

a(n) = A010051(n) + A059841(n).

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A330949 Odd nonprime numbers whose prime indices are not all prime numbers.

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A370482 Characteristic function of primes plus characteristic function of even numbers.

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