A134617 -id:A134617 - cp's OEIS frontend

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)).

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

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

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

A134612 Nonprime numbers such that the root mean cube of their prime factors is a prime (where the root mean cube of c and d is ((c^3+d^3)/2)^(1/3)).

Original entry on oeis.org

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, 1681, 1849, 2048, 2187, 2197, 2209, 2401, 2809, 3125, 3481, 3721, 4096, 4489, 4913, 5041, 5329, 6241, 6561, 6859, 6889, 7921

Offset: 1

Hieronymus Fischer, Nov 11 2007

nonn

The prime factors are taken with multiplicity.
All perfect prime powers (A025475) with power > 1 are included. First term not included in A025475 is a(211) = 707265 = A134614(5) = A134615(1).
Originally, the first term was 1. This was wrong, since the cube mean of the prime factors of 1 is zero, by definition of the empty sum.

			a(5) = 25, since 25 = 5*5 and ((5^3+5^3)/2)^(1/3) = 5.

Hieronymus Fischer, Table of n, a(n) for n = 1..8600

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

Cf. A134600, A134602, A134605, A134614, 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, &p) && isprime(p), print1(i, ", "));););}  \\ Michel Marcus, Apr 14 2013

Edited by Hieronymus Fischer, May 30 2013

A134613 Numbers such that the root mean cube of their prime factors is a nonprime integer (where the root mean cube of c and d is ((c^3+d^3)/2)^(1/3)).

Original entry on oeis.org

1, 1512, 337365, 375360, 523809, 1177176, 1255254, 1380918, 1549431, 2277345, 2286144, 2816883, 3320713, 3340428, 3838185, 4378333, 6726969, 7043655, 8311212, 10281284, 10323390, 10666227, 10708544, 12333468, 14185724, 15883803, 21432000, 25760763, 27111825

Offset: 1

Hieronymus Fischer, Nov 11 2007

nonn

The prime factors are taken with multiplicity.
Numbers included in A134611, but not in A134612.
For n > 1, also numbers included in A134614, but not in A134615; a(2) = 1512 is the minimal number with this property.
No prime number and no power (> 1) of a prime number can be a term.

			a(1) = 1, since 1 has no prime factors, and so the cube mean is zero (by definition of empty sums).
a(2) = 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..411

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

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

isok(n) = if (n==1, return(1)); sc = 0; nb = 0; f = factor(n); for (i=1, #f~, sc += f[i, 2]*f[i, 1]^3; nb += f[i, 2]; ); return (type(quot = sc/nb) == "t_INT" && ispower(quot, 3, &cr) && (! isprime(cr))); \\ Michel Marcus, Jul 15 2013; corrected Jun 13 2022

Extended, edited and added initial term a(1) = 1 by Hieronymus Fischer, May 30 2013

A134601 Composite numbers such that the square mean of their prime factors is a prime (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, 512, 529, 595, 625, 721, 729, 841, 845, 959, 961, 1024, 1045, 1081, 1241, 1323, 1331, 1369, 1375, 1547, 1681, 1792, 1849, 1855, 2048, 2187, 2197, 2209, 2401

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(13) = 119, since 119=7*17 and sqrt((7^2+17^2)/2)=sqrt(169)=13.

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

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

Cf. A134600, A134605, A134608, A134611, A134617, A134619, A134621.

f[{a_,b_}]:=Table[a,b];Select[Range[2,2401],!PrimeQ[#]&&PrimeQ[ RootMeanSquare[f/@FactorInteger[#]//Flatten] ]&] (* James C. McMahon, Apr 08 2025 *)

Definition clarified and edited by Hieronymus Fischer, Apr 20 2013

A134615 Numbers (excluding primes and powers of primes) such that the root mean cube of their prime factors is a prime (where the root mean cube of c and d is ((c^3+d^3)/2)^(1/3)).

Original entry on oeis.org

707265, 1922816, 2284389, 12023505, 14689836, 21150800, 29444140, 30682000, 36533504, 39372480, 46309837, 52163097, 67303740, 73558065, 85751055, 107366283, 115291904, 161976045, 190384425, 204399585, 218317275, 231443940, 274960400, 286618640

Offset: 1

Hieronymus Fischer, Nov 11 2007

nonn

The prime factors are taken with multiplicity.
Numbers included in A134612, but not in A025475.
a(1) = 707265 is the minimal number with this property. a(3) = 2284389 is the greatest such number < 10^7.

			a(1) = 707265, since 707265 = 3*3*3*5*13*13*31 and ((3*3^3+5^3+2*13^3+31^3)/7)^(1/3) = 4913^(1/3) = 17.

Hieronymus Fischer, Table of n, a(n) for n = 1..108

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

Cf. A134600, A134602, A134605, A134608, A134613, A134617, A134619, A134621.

isok(n) = {if (omega(n) == 1, return (0)); f = factor(n); s = sum(i=1, #f~, f[i,2]*f[i,1]^3); s = s/bigomega(n); if (type(s) != "t_INT", return (0)); if (! ispower(s, 3, &p), return (0)); isprime(p);} \\ Michel Marcus, Nov 03 2013

More terms and minor edits by Hieronymus Fischer, May 06 2013, May 30 2013

A134603 Numbers (excluding primes and powers of primes) such that the square mean of their prime factors is an integer (where the square mean of c and d is sqrt((c^2+d^2)/2)).

Original entry on oeis.org

119, 161, 351, 378, 455, 527, 595, 721, 845, 918, 959, 1045, 1081, 1241, 1265, 1323, 1375, 1547, 1615, 1792, 1855, 2047, 2145, 2175, 2345, 2457, 2645, 2665, 2737, 3281, 3367, 3509, 3713, 3835, 3887, 3995, 4207, 4305, 4347, 4625, 4633, 4655, 4681, 5000

Offset: 1

Hieronymus Fischer, Nov 11 2007

nonn

Numbers included in A134600, but not in A025475. a(1)=119 is the minimal number with this property.

			a(2) = 161, since 161 = 7*23 and sqrt((7^2+23^2)/2) = sqrt(289) = 17 is an integer.
a(4) = 378, since 378 = 2*3*3*3*7 and sqrt((2^2+3*3^2+7^2)/5) = sqrt(80/5) = 4 is an integer.
a(28519) = 114445555, since 114445555 = 5*7*41*173*461 and sqrt((5^2+7^2+41^2+173^2+461^2)/5) = sqrt(48841) = 221.

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

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

Cf. A134600, A134605, A134608, A134611, A134617, A134619, A134621.

f[{a_,b_}]:=Table[a,b];Select[Range[2,5000],!PrimePowerQ[#]&&IntegerQ[ RootMeanSquare[f/@FactorInteger[#]//Flatten] ]&] (* James C. McMahon, Apr 09 2025 *)

Minor edits by Hieronymus Fischer, Apr 21 2013

cp's OEIS Frontend

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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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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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)).

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A134612 Nonprime numbers such that the root mean cube of their prime factors is a prime (where the root mean cube of c and d is ((c^3+d^3)/2)^(1/3)).

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A134613 Numbers such that the root mean cube of their prime factors is a nonprime integer (where the root mean cube of c and d is ((c^3+d^3)/2)^(1/3)).

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A134601 Composite numbers such that the square mean of their prime factors is a prime (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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