A134376 -id:A134376 - cp's OEIS frontend

A134333 Numbers n whose number of prime factors (counted with multiplicity) is a prime factor of n.

4, 6, 10, 12, 14, 18, 22, 26, 27, 30, 34, 38, 42, 45, 46, 58, 62, 63, 66, 74, 75, 78, 80, 82, 86, 94, 99, 102, 105, 106, 114, 117, 118, 120, 122, 134, 138, 142, 146, 147, 153, 158, 165, 166, 171, 174, 178, 180, 186, 194, 195, 200, 202, 206, 207, 214, 218, 222, 226

Offset: 1

Hieronymus Fischer, Oct 23 2007

nonn

			a(1) = 4, since 4 has 2 prime factors and 2 is a prime factor of 4.
a(4) = 12, since 12 = 2*2*3 has 3 prime factors, and 3 is a prime factor of 12.
a(21) = 75, since 75 = 3*3*5 has 3 prime factors. and 3 is a prime factor of 75.
9 = 3*3 is not a term, since the number of prime factors (=2) is not a divisor of 9.
28 = 2*2*7 is not a term, since the number of prime factors (=3) is not a divisor of 28.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Cf. A000040, A001222, A100118, A046363, A133620, A133621, A133880, A133890, A133900, A133910, A133911, A046346, A134331, A134332, A134334, A134344, A134376, A063989.

fQ[n_] := Module[{d = Total[Transpose[FactorInteger[n]][[2]]]}, PrimeQ[d] && Mod[n, d] == 0]; Select[Range[2, 226], fQ] (* T. D. Noe, Apr 05 2013 *)

a(n)=my(t=bigomega(n)); n%t==0 && isprime(t) \\ Charles R Greathouse IV, Sep 14 2015

a(n) << n log n/(log log n)^k for any fixed k. - Charles R Greathouse IV, Sep 14 2015

Sequence definition corrected and examples added by Hieronymus Fischer, Apr 05 2013

A134600 Composite numbers such that the square mean of their prime factors is an integer (where the prime factors are taken with multiplicity and the square mean of c and d is sqrt((c^2+d^2)/2)).

Original entry on oeis.org

4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 119, 121, 125, 128, 161, 169, 243, 256, 289, 343, 351, 361, 378, 455, 512, 527, 529, 595, 625, 721, 729, 841, 845, 918, 959, 961, 1024, 1045, 1081, 1241, 1265, 1323, 1331, 1369, 1375, 1547, 1615, 1681, 1792, 1849, 1855

Offset: 1

Hieronymus Fischer, Nov 11 2007

nonn

All perfect prime powers (A025475) with power > 0 are included.

Originally, the definition started with "Nonprime numbers ..." and the first term was equal to 1. This is misleading, since 1 has no prime factors. - Hieronymus Fischer, Apr 20 2013

			a(5) = 25, since 25=5*5 and sqrt((5^2+5^2)/2)=5;
a(23) = 378, since 378=2*3*3*3*7 and sqrt((2^2+3*3^2+7^2)/5)=sqrt(16)=4.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Cf. A001597, A025475, A134333, A134344, A134376.

Cf. A134601, A134605, A134608, A134611, A134617, A134619, A134621.

f[{a_,b_}]:=Table[a,b];Select[Range[2,1855],!PrimeQ[#]&&IntegerQ[ RootMeanSquare[f/@FactorInteger[#]//Flatten] ]&] (* James C. McMahon, Apr 08 2025 *)

Definition clarified and edited by Hieronymus Fischer, Apr 20 2013

A134605 Composite numbers such that the square root of the sum of squares of their prime factors (with multiplicity) is an integer.

Original entry on oeis.org

16, 48, 81, 320, 351, 486, 512, 625, 1080, 1260, 1350, 1375, 1792, 1836, 2070, 2145, 2175, 2401, 2730, 2772, 3072, 3150, 3510, 4104, 4305, 4625, 4650, 4655, 4998, 5880, 6000, 6174, 6545, 7098, 7128, 7182, 7650, 7791, 7889, 7956, 9030, 9108, 9295, 9324

Offset: 1

Hieronymus Fischer, Nov 11 2007

nonn

			a(2)=48 since 48=2*2*2*2*3 and sqrt(4*2^2+3^2)=sqrt(25)=5.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Cf. A001597, A025475, A134333, A134344, A134376, A134600, A134602, A134608, A134611, A134616, A134618, A134620.

srssQ[n_]:=IntegerQ[Sqrt[Total[Flatten[Table[#[[1]],#[[2]]]&/@ FactorInteger[ n]]^2]]]; Select[Range[10000],CompositeQ[#]&&srssQ[#]&] (* Harvey P. Dale, Jan 22 2019 *)

is(n)=my(f=factor(n)); issquare(sum(i=1,#f~,f[i,1]^2*f[i,2])) && !isprime(n) && n>1 \\ Charles R Greathouse IV, Apr 29 2015

A134608 Composite numbers such that the cube root of the sum of cubes of their prime factors is an integer.

Original entry on oeis.org

256, 588, 693, 3840, 6561, 14157, 17787, 141960, 178360, 313600, 337365, 350000, 387072, 390625, 407442, 432000, 466560, 531674, 535815, 541310, 664909, 697851, 1044582, 1262056, 1264640, 1299272, 1374327, 1547570, 1608575, 1660360

Offset: 1

Hieronymus Fischer, Nov 11 2007

nonn

			a(3)=693, since 693=3*3*7*11 and (2*3^3+7^3+11^3)^(1/3)=1728^(1/3)=12.

Hieronymus Fischer, Table of n, a(n) for n = 1..300

Cf. A001597, A025475, A134333, A134344, A134376.

Cf. A134600, A134602, A134605, A134611, A134616, A134618, A134620.

criQ[n_]:=IntegerQ[Surd[Total[Flatten[Table[#[[1]],#[[2]]]&/@ FactorInteger[ n]]^3],3]]; Select[Range[1670000],CompositeQ[#] && criQ[#]&] (* Harvey P. Dale, Sep 19 2021 *)

lista(m) = {for (i=2, m, if (! isprime(i), f = factor(i); s = sum (j=1, length(f~), f[j,1]^3*f[j,2]); if (ispower(s, 3), print1(i, ", "));););} \\ Michel Marcus, Apr 14 2013

Minor edits by Hieronymus Fischer, Apr 20 2013

A134602 Composite numbers such that the square mean of their prime factors is a nonprime integer (where the prime factors are taken with multiplicity and the square mean of c and d is sqrt((c^2+d^2)/2)).

Original entry on oeis.org

378, 455, 527, 918, 1265, 1615, 2047, 2145, 2175, 2345, 2665, 3713, 3835, 4207, 4305, 4633, 5000, 5117, 5382, 6061, 6678, 6887, 6965, 7055, 7267, 7327, 7497, 7685, 7791, 8470, 8785, 8918, 9641, 10205, 10545, 10647, 11137, 11543, 11713, 13482, 14079

Offset: 1

Hieronymus Fischer, Nov 11 2007

nonn

Numbers included in A134600, but not in A134601. a(1)=378 is the minimal number with this property.

Also numbers included in A134603, but not in A134604.

			a(2)=455, since 455=5*7*13 and sqrt((5^2+7^2+13^2)/3)=sqrt(81)=9.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Cf. A001597, A025475, A134333, A134344, A134376.

Cf. A134600, A134604, A134608, A134611, A134617, A134619, A134621.

f[{a_,b_}]:=Table[a,b];Select[Range[2,14079],!PrimeQ[#]&&!PrimeQ[ RootMeanSquare[f/@FactorInteger[#]//Flatten] ]&&IntegerQ[ RootMeanSquare[f/@FactorInteger[#]//Flatten] ]&] (* James C. McMahon, Apr 08 2025 *)

Definition clarified by Hieronymus Fischer, Apr 20 2013, Jun 01 2013

A134621 Numbers such that the arithmetic mean of the 4th power of their prime factors (taken with multiplicity) is a prime.

Original entry on oeis.org

15, 28, 39, 48, 51, 65, 68, 76, 77, 85, 87, 93, 111, 119, 133, 141, 143, 148, 155, 161, 175, 187, 189, 209, 212, 215, 221, 225, 235, 244, 275, 287, 295, 301, 315, 316, 320, 323, 329, 355, 393, 403, 404, 411, 428, 437, 447, 451, 455, 481, 505, 508, 515, 517

Offset: 1

Hieronymus Fischer, Nov 11 2007

nonn

			a(3)=39, since 39=3*13 and (3^4+13^4)/2=14321 which is prime.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Cf. A001597, A025475, A134333, A134344, A134376.

Cf. A134600, A134602, A134605, A134608, A134613, A134617, A134619.

Minor edits by Hieronymus Fischer, May 06 2013

A134619 Numbers such that the arithmetic mean of the cubes of their prime factors (taken with multiplicity) is a prime.

Original entry on oeis.org

20, 44, 188, 297, 336, 400, 425, 540, 575, 605, 704, 752, 764, 908, 912, 1025, 1053, 1124, 1172, 1183, 1365, 1380, 1412, 1420, 1452, 1475, 1484, 1519, 1604, 1625, 1809, 1844, 1856, 1936, 1953, 2107, 2192, 2205, 2255, 2320, 2325, 2348, 2368, 2372, 2468

Offset: 1

Hieronymus Fischer, Nov 11 2007

nonn

			a(10)=605, since 605=5*11*11 and (5^3+11^3+11^3)/3=929 which is prime.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Cf. A001597, A025475, A134333, A134344, A134376.

Cf. A134600, A134602, A134605, A134608, A134613, A134617, A134621.

amcpfQ[n_]:=PrimeQ[Mean[Flatten[PadRight[{},#[[2]],#[[1]]]&/@FactorInteger[n]]^3]]; Select[ Range[ 2500],amcpfQ] (* Harvey P. Dale, Jun 06 2023 *)

lista(m) = {for (i=2, m, f = factor(i); s = sum (j=1, length(f~), f[j,1]^3*f[j,2]); s /= bigomega(i); if (type(s) == "t_INT" && isprime(s), print1(i, ", ")););} \\ Michel Marcus, Apr 14 2013

Minor edits by Hieronymus Fischer, May 06 2013

A134334 Numbers which are not divisible by the number of their prime factors (counted with multiplicity).

Original entry on oeis.org

8, 9, 15, 20, 21, 25, 28, 32, 33, 35, 39, 44, 48, 49, 50, 51, 52, 54, 55, 57, 64, 65, 68, 69, 70, 72, 76, 77, 81, 85, 87, 90, 91, 92, 93, 95, 98, 108, 110, 111, 112, 115, 116, 119, 121, 123, 124, 125, 126, 128, 129, 130, 133, 135, 141, 143, 145, 148, 150, 154, 155, 159

Offset: 1

Hieronymus Fischer, Oct 23 2007

nonn

The asymptotic density of this sequence is 1 (Erdős and Pomerance, 1990). - Amiram Eldar, Jul 10 2020

			a(1) = 8, since 8 = 2*2*2 has 3 prime factors and 8 is not divisible by 3.
a(3) = 15, since 15 = 3*5 has 2 prime factors and 15 is not divisible by 2.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000
Paul Erdős and Carl Pomerance, On a theorem of Besicovitch: values of arithmetic functions that divide their arguments, Indian J. Math., Vol. 32 (1990), pp. 279-287.

Cf. A000040, A001222, A074946 (complement), A100118, A046363, A133620, A133621.

Cf. A133880, A133890, A133900, A133910, A133911, A046346, A134331, A134332, A134333, A134335, A134344, A134376.

Select[Range[2,200],Mod[#,PrimeOmega[#]]!=0&] (* Harvey P. Dale, May 13 2023 *)

isok(n) = (n % bigomega(n)) \\ Michel Marcus, Jul 15 2013

A134611 Nonprime numbers such that the root mean cube of their prime factors is an integer (where the root mean cube of c and d is ((c^3+d^3)/2)^(1/3)).

Original entry on oeis.org

1, 4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 121, 125, 128, 169, 243, 256, 289, 343, 361, 512, 529, 625, 729, 841, 961, 1024, 1331, 1369, 1512, 1681, 1849, 2048, 2187, 2197, 2209, 2401, 2809, 3125, 3481, 3721, 4096, 4489, 4913, 5041, 5329, 6241, 6561, 6859, 6889

Offset: 1

Hieronymus Fischer, Nov 11 2007

nonn

The prime factors are taken with multiplicity.
All perfect prime powers (A025475) are included. First term not included in A025475 is a(30) = 1512 = A134613(2) = A134613(1).
Most terms have a last digit of 1 or 9 (i.e., 8326 out of 9000 terms). Mainly, this comes from the fact that all squares of primes are included. Since each prime > 10 has a last digit of 1, 3, 7 or 9, its square has a last digit of 1 or 9. In addition, m-th powers of primes have a last digit of 1, if m == 0 (mod 4), and have a last digit of 1 or 9 if m == 2 (mod 4), and have a 50% chance, roughly, for a last digit of 1 or 9, if m == 1 (mod 4) or m == 3 (mod 4). Since the number of terms <= N which are squares of primes is PrimePi(sqrt(N)) = A000720(sqrt(N)), it follows that the number of terms <= N which have a last digit of 1 or 9 is greater than PrimePi(sqrt(N)). This can be estimated as 2*N^(1/2)/log(N), approximately.

			a(6) = 25, since 25 = 5*5 and ((5^3+5^3)/2)^(1/3) = 5.
a(30) = 1512, since 1512 = 2*2*2*3*3*3*7 and ((3*2^3+3*3^3+7^3)/7)^(1/3) = 64^(1/3) = 4.

Hieronymus Fischer, Table of n, a(n) for n = 1..9000

Cf. A001597, A025475, A134333, A134344, A134376.

Cf. A134600, A134602, A134605, A134608, A134617, A134619, A134621.

lista(m) = {for (i=2, m, if (! isprime(i), f = factor(i); s = sum (j=1, length(f~), f[j,1]^3*f[j,2]); s /= bigomega(i); if (type(s) == "t_INT" && ispower(s, 3), print1(i, ", "));););}  \\ Michel Marcus, Apr 14 2013

Edited by Hieronymus Fischer, May 30 2013

A134617 Numbers such that the arithmetic mean of the squares of their prime factors (taken with multiplicity) is a prime.

Original entry on oeis.org

15, 20, 21, 28, 35, 39, 44, 48, 51, 52, 55, 65, 69, 85, 91, 92, 95, 108, 112, 115, 116, 129, 135, 141, 145, 159, 164, 172, 188, 189, 205, 208, 209, 215, 221, 225, 235, 236, 245, 249, 259, 268, 272, 295, 297, 299, 305, 309, 315, 316, 320, 325, 329, 339, 341, 365

Offset: 1

Hieronymus Fischer, Nov 11 2007

nonn

			a(2)=20, since 20=2*2*5 and (2^2+2^2+5^2)/3=33/3=11.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Cf. A001597, A025475, A134333, A134344, A134376.

Cf. A134600, A134602, A134605, A134608, A134613, A134619, A134621.

amspQ[n_]:=PrimeQ[Mean[Flatten[Table[#[[1]],#[[2]]]&/@FactorInteger[ n]]^2]]; Select[Range[400],amspQ] (* Harvey P. Dale, Jan 21 2017 *)

Minor edits by the author, May 06 2013

cp's OEIS Frontend

A134333 Numbers n whose number of prime factors (counted with multiplicity) is a prime factor of n.

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A134600 Composite numbers such that the square mean of their prime factors is an integer (where the prime factors are taken with multiplicity and the square mean of c and d is sqrt((c^2+d^2)/2)).

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A134605 Composite numbers such that the square root of the sum of squares of their prime factors (with multiplicity) is an integer.

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A134608 Composite numbers such that the cube root of the sum of cubes of their prime factors is an integer.

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A134602 Composite numbers such that the square mean of their prime factors is a nonprime integer (where the prime factors are taken with multiplicity and the square mean of c and d is sqrt((c^2+d^2)/2)).

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A134621 Numbers such that the arithmetic mean of the 4th power of their prime factors (taken with multiplicity) is a prime.

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A134619 Numbers such that the arithmetic mean of the cubes of their prime factors (taken with multiplicity) is a prime.

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A134334 Numbers which are not divisible by the number of their prime factors (counted with multiplicity).

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