A320628 -id:A320628 - cp's OEIS frontend

A368833 Numbers whose prime indices are not 1, prime, or semiprime.

19, 37, 38, 53, 57, 61, 71, 74, 76, 89, 95, 103, 106, 107, 111, 113, 114, 122, 131, 133, 142, 148, 151, 152, 159, 171, 173, 178, 181, 183, 185, 190, 193, 197, 206, 209, 212, 213, 214, 222, 223, 226, 228, 229, 239, 244, 247, 251, 259, 262, 263, 265, 266, 267

Offset: 1

Gus Wiseman, Jan 08 2024

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 terms together with their prime indices begin:
   19: {8}
   37: {12}
   38: {1,8}
   53: {16}
   57: {2,8}
   61: {18}
   71: {20}
   74: {1,12}
   76: {1,1,8}
   89: {24}
   95: {3,8}
  103: {27}
  106: {1,16}
  107: {28}
  111: {2,12}
  113: {30}
  114: {1,2,8}
  122: {1,18}
  131: {32}
  133: {4,8}
  142: {1,20}
  148: {1,1,12}

These are non-products of primes indexed by elements of A037143.
The complement for just primes is A076610, strict A302590.
The complement for just semiprimes is A339112, strict A340020.
The complement for just squarefree semiprimes is A339113, strict A309356.
The complement is A368728.
The complement for just primes and semiprimes is A368729, strict A340019.
A000607 counts partitions into primes, with ones allowed A034891.
A001358 lists semiprimes, squarefree A006881.
A006450, A106349, A322551, A368732 list selected primes.
A056239 adds up prime indices, row sums of A112798.
A101048 counts partitions into semiprimes.
Cf. A000040, A000720, A001222, A003963, A005117, A302242, A320628.

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

A371449 Numbers whose prime indices are not powers of 2.

Original entry on oeis.org

1, 5, 11, 13, 17, 23, 25, 29, 31, 37, 41, 43, 47, 55, 59, 61, 65, 67, 71, 73, 79, 83, 85, 89, 97, 101, 103, 107, 109, 113, 115, 121, 125, 127, 137, 139, 143, 145, 149, 151, 155, 157, 163, 167, 169, 173, 179, 181, 185, 187, 191, 193, 197, 199, 205, 211, 215

Offset: 1

Gus Wiseman, Mar 31 2024

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 terms together with their prime indices begin:
     1: {}        85: {3,7}      169: {6,6}     253: {5,9}
     5: {3}       89: {24}       173: {40}      257: {55}
    11: {5}       97: {25}       179: {41}      263: {56}
    13: {6}      101: {26}       181: {42}      269: {57}
    17: {7}      103: {27}       185: {3,12}    271: {58}
    23: {9}      107: {28}       187: {5,7}     275: {3,3,5}
    25: {3,3}    109: {29}       191: {43}      277: {59}
    29: {10}     113: {30}       193: {44}      281: {60}
    31: {11}     115: {3,9}      197: {45}      283: {61}
    37: {12}     121: {5,5}      199: {46}      289: {7,7}
    41: {13}     125: {3,3,3}    205: {3,13}    293: {62}
    43: {14}     127: {31}       211: {47}      295: {3,17}
    47: {15}     137: {33}       215: {3,14}    299: {6,9}
    55: {3,5}    139: {34}       221: {6,7}     305: {3,18}
    59: {17}     143: {5,6}      223: {48}      307: {63}
    61: {18}     145: {3,10}     227: {49}      313: {65}
    65: {3,6}    149: {35}       229: {50}      317: {66}
    67: {19}     151: {36}       233: {51}      319: {5,10}
    71: {20}     155: {3,11}     235: {3,15}    325: {3,3,6}
    73: {21}     157: {37}       239: {52}      331: {67}
    79: {22}     163: {38}       241: {53}      335: {3,19}
    83: {23}     167: {39}       251: {54}      337: {68}

Partitions of this type are counted by A101417.
For binary indices instead of prime indices we have A326781.
Requiring powers of two gives A318400, for binary indices A253317.
An opposite version is A371443.
For primes instead of powers of 2 we have A320628.
A000040 lists prime numbers, complement A018252.
A000961 lists prime-powers.
A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
A057716 lists non-powers of 2.
A070939 gives length of binary expansion.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
Cf. A005117, A326782, A371451, A371444.

Select[Range[100],And@@Not/@IntegerQ/@Log[2, PrimePi/@First/@FactorInteger[#]]&]

A322385 2 and prime numbers whose prime index is a product of at least two not necessarily distinct prime numbers already in the sequence.

Original entry on oeis.org

2, 7, 19, 43, 53, 107, 131, 163, 227, 263, 311, 383, 443, 521, 577, 613, 719, 751, 881, 1021, 1193, 1301, 1307, 1423, 1619, 1667, 1699, 1993, 2003, 2161, 2309, 2311, 2437, 2539, 2693, 2939, 2969, 3167, 3209, 3671, 3767, 3779, 3833, 4423, 4481, 4597, 4871, 5147

Offset: 1

Gus Wiseman, Dec 05 2018

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 have 1993 = prime(301) = prime(7 * 43). Since 7 and 43 already belong to the sequence, so does 1993.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000002, A001462, A007097, A079000, A079254, A277098, A280996, A291636, A304360, A320628, A322386.

ppQ[n_]:=And[PrimeQ[n],!PrimeQ[PrimePi[n]],And@@ppQ/@First/@If[n==2,{},FactorInteger[PrimePi[n]]]];
Select[Range[1000],ppQ]

A322386 Numbers whose prime indices are not prime and already belong to the sequence.

Original entry on oeis.org

1, 2, 4, 7, 8, 14, 16, 19, 28, 32, 38, 43, 49, 53, 56, 64, 76, 86, 98, 106, 107, 112, 128, 131, 133, 152, 163, 172, 196, 212, 214, 224, 227, 256, 262, 263, 266, 301, 304, 311, 326, 343, 344, 361, 371, 383, 392, 424, 428, 443, 448, 454, 512, 521, 524, 526, 532

Offset: 1

Gus Wiseman, Dec 05 2018

nonn

Union of A291636 (Matula-Goebel numbers of series-reduced rooted trees) and A322385.
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 multiplicative semigroup: if x and y are in the sequence, then so is x*y. - Robert Israel, Dec 06 2018

			1 has no prime indices, so the definition is satisfied vacuously. - _Robert Israel_, Dec 07 2018
We have 301 = prime(4) * prime(14). Since 4 and 14 already belong to the sequence, so does 301.

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

Cf. A000002, A001462, A007097, A079000, A079254, A214577, A276625, A291636, A304360, A320628, A322385.

Res:= 1: S:= {1}:
for n from 2 to 1000 do
  F:= map(numtheory:-pi, numtheory:-factorset(n));
  if F subset S then
    Res:= Res, n;
    if not isprime(n) then S:= S union {n} fi
fi
od:
Res; # Robert Israel, Dec 06 2018

tnpQ[n_]:=With[{m=PrimePi/@First/@If[n==1,{},FactorInteger[n]]},And[!MemberQ[m,_?PrimeQ],And@@tnpQ/@m]]
Select[Range[1000],tnpQ]

A360332 Numbers k such that A360331(k) > 2*k.

Original entry on oeis.org

56, 104, 112, 196, 208, 224, 304, 364, 368, 392, 416, 448, 464, 532, 608, 644, 728, 736, 784, 812, 832, 896, 928, 1036, 1064, 1184, 1204, 1216, 1288, 1316, 1352, 1372, 1376, 1456, 1472, 1484, 1504, 1568, 1624, 1664, 1696, 1708, 1792, 1856, 1952, 1976, 1988, 2044

Offset: 1

Amiram Eldar, Feb 03 2023

nonn

Analogous to abundant numbers (A005101) with divisors that are restricted to numbers that have only nonprime-indexed prime factors.
The least odd term is 7^4 * (13*19)^3 * (29*...*71)^2 * (73*...*281) = 2.411... * 10^105 (where the dots are for consecutive terms in A007821).
Includes all the abundant (A005101) terms of A320628.
There are terms that are not in A320628, and the least of them is 3 * m, where m is a term of A320628 with sigma(m) > 6. Such a number exists, and it should be a positive multiple of Product_{i=1..k} A007821(k) = 2 * 7 * ... * 11443 = 9.164... * 10^4148, where k = 1160 is the least number such that Product_{i=1..k} A007821(k)/(A007821(k)-1) > 6.
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 0, 1, 23, 215, 1997, 19231, 189457, 1873511, 18593697, ... . Apparently, the asymptotic density of this sequence equals 0.018... .

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

Subsequence of A005101.

Cf. A007821, A320628, A360331.

q:= n-> is(mul(`if`(isprime(numtheory[pi](i[1])), 1,
   (i[1]^(i[2]+1)-1)/(i[1]-1)), i=ifactors(n)[2])>2*n):
select(q, [$1..2050])[];  # Alois P. Heinz, Feb 03 2023

f[p_, e_] := If[PrimeQ[PrimePi[p]], 1, (p^(e+1)-1)/(p-1)]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[2000], s[#] > 2*# &]

is(n) = {my(f = factor(n), p = f[,1], e = f[,2]); prod(i = 1, #p, if(isprime(primepi(p[i])), 1, (p[i]^(e[i]+1)-1)/(p[i]-1))) > 2*n;}

A360356 Primitive terms of A360332: terms of A360332 with no proper divisor in A360332.

Original entry on oeis.org

56, 104, 196, 304, 364, 368, 464, 532, 644, 812, 1036, 1184, 1204, 1316, 1376, 1484, 1504, 1696, 1708, 1952, 1988, 2044, 2212, 2492, 2716, 2828, 2884, 2996, 3164, 3496, 3668, 3836, 3892, 4172, 4228, 4408, 4544, 4564, 4672, 4676, 4844, 5056, 5068, 5336, 5404, 5516

Offset: 1

Amiram Eldar, Feb 04 2023

nonn

If m is a term then k*m is a term of A360332 for all k in A320628.

Analogous to primitive abundant numbers (A091191) with divisors that are restricted to numbers that have only nonprime-indexed prime factors.

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

Subsequence of A360332.
Cf. A320628.
Similar sequences: A006038, A091191, A249263, A302574, A360355.

f[p_, e_] := If[PrimeQ[PrimePi[p]], 1, (p^(e + 1) - 1)/(p - 1)]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; primQ[n_] := s[n] > 2*n && AllTrue[Divisors[n], # == n || s[#] <= 2*# &]; Select[Range[6000], primQ]

isab(n) = {my(f = factor(n), p = f[,1], e = f[,2]); prod(i = 1, #p, if(isprime(primepi(p[i])), 1, (p[i]^(e[i]+1)-1)/(p[i]-1))) > 2*n;}
is(n) = {if(!isab(n), return(0)); fordiv(n, d, if(d < n && isab(d), return(0))); return(1)};

A379313 Positive integers whose prime indices are not all composite.

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82

Offset: 1

Gus Wiseman, Dec 28 2024

nonn

Or, positive integers whose prime indices include at least one 1 or prime number.

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:
     2: {1}
     3: {2}
     4: {1,1}
     5: {3}
     6: {1,2}
     8: {1,1,1}
     9: {2,2}
    10: {1,3}
    11: {5}
    12: {1,1,2}
    14: {1,4}
    15: {2,3}
    16: {1,1,1,1}
    17: {7}
    18: {1,2,2}
    20: {1,1,3}
    21: {2,4}
    22: {1,5}
    24: {1,1,1,2}

Partitions of this type are counted by A000041 - A023895.
The "old" primes are listed by A008578.
For no composite parts we have A302540, counted by A034891 (strict A036497).
The complement is A320629, counted by A023895 (strict A204389).
For a unique prime we have A331915, counted by A379304 (strict A379305).
Positions of nonzeros in A379311.
For a unique 1 or prime we have A379312, counted by A379314 (strict A379315).
A000040 lists the prime numbers, differences A001223.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A080339 is the characteristic function for the old prime numbers.
A376682 gives k-th differences of old prime numbers, see A030016, A075526.
A377033 gives k-th differences of composite numbers, see A073445, A377034.
Other counts of prime indices:
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A000586, A000607, A076610, A331386.
- A330944 nonprime, see A002095, A096258, A320628, A330945.
- A379306 squarefree, see A302478, A379308, A379309, A379316.
- A379310 nonsquarefree, see A114374, A256012, A379307.
`Cf. A038550, A087436, A173390, A379300, A379301.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!And@@CompositeQ/@prix[#]&]

cp's OEIS Frontend

A368833 Numbers whose prime indices are not 1, prime, or semiprime.

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A371449 Numbers whose prime indices are not powers of 2.

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A322385 2 and prime numbers whose prime index is a product of at least two not necessarily distinct prime numbers already in the sequence.

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A322386 Numbers whose prime indices are not prime and already belong to the sequence.

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A360332 Numbers k such that A360331(k) > 2*k.

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A360356 Primitive terms of A360332: terms of A360332 with no proper divisor in A360332.

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PARI

A379313 Positive integers whose prime indices are not all composite.

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