A134600 -id:A134600 - cp's OEIS frontend

A134618 Numbers such that the sum of cubes of their prime factors (taken with multiplicity) is a prime.

12, 28, 40, 45, 48, 52, 54, 56, 63, 75, 80, 96, 104, 108, 117, 136, 152, 153, 165, 175, 210, 224, 232, 245, 250, 261, 268, 300, 320, 325, 333, 344, 350, 363, 384, 387, 390, 399, 405, 416, 432, 462, 464, 468, 475, 477, 504, 507, 531, 536, 539, 561, 570, 584

Offset: 1

Hieronymus Fischer, Nov 11 2007

nonn

			a(2) = 28, since 28 = 2*2*7 and 2^3 + 2^3 + 7^3 = 359 which is prime.

Harvey P. Dale and Hieronymus Fischer, Table of n, a(n) for n = 1..10000 (first 1000 terms from _Harvey P. Dale_)

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

Cf. A134600, A134602, A134605, A134608, A134612, A134616, A134620.

Select[Range[600],PrimeQ[Total[Flatten[Table[#[[1]],{#[[2]]}]&/@ FactorInteger[#]]^3]]&] (* Harvey P. Dale, Feb 01 2013 *)

from sympy import factorint, isprime
def ok(n): return isprime(sum(p**3 for p in factorint(n, multiple=True)))
print([k for k in range(585) if ok(k)]) # Michael S. Branicky, Dec 28 2021

{k: A224787(k) in A000040}. - R. J. Mathar, Mar 25 2025

Example clarified by Harvey P. Dale, Feb 01 2013

Minor edits by Hieronymus Fischer, May 06 2013

A134620 Numbers such that the sum of 4th power of their prime factors is a prime.

Original entry on oeis.org

6, 10, 12, 14, 22, 34, 38, 40, 45, 46, 74, 82, 117, 118, 122, 126, 142, 152, 158, 171, 194, 231, 262, 278, 296, 345, 358, 363, 376, 384, 387, 429, 432, 446, 454, 458, 482, 486, 490, 500, 507, 522, 536, 550, 566, 584, 626, 627, 634, 639, 663, 675, 686, 704, 705

Offset: 1

Hieronymus Fischer, Nov 11 2007

nonn

Prime factors must be taken with multiplicity. - Harvey P. Dale, May 23 2012

The calculation of higher terms is time-consuming, since for any number of the form 2*p with a prime number p > 10^5 the primality test have to be accomplished for a number > 10^20. - Hieronymus Fischer, May 21 2013

			a(2) = 10, since 10 = 2*5 and 2^4+5^4 = 641 which is prime.
a(9) = 45, since 45 = 3*3*5 and 3^4+3^4+5^4 = 787 which is prime.
a(9883) = 333314, since 333314 = 3*166657 and 2^4+166657^4 = 771425941499397811217 which is prime.

Harvey P. Dale and Hieronymus Fischer, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

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

Cf. A134600, A134602, A134605, A134608, A134612, A134616, A134618, A134621.

Select[Range[1000],PrimeQ[Total[Flatten[Table[First[#],{Last[#]}]&/@ FactorInteger[#]]^4]]&] (* Harvey P. Dale, May 23 2012 *)

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

A134614 Numbers (excluding primes and powers of primes) 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

1512, 337365, 375360, 523809, 707265, 1177176, 1255254, 1380918, 1549431, 1922816, 2277345, 2284389, 2286144, 2816883, 3320713, 3340428, 3838185, 4378333, 6726969, 7043655, 8311212, 10281284, 10323390, 10666227, 10708544

Offset: 1

Hieronymus Fischer, Nov 11 2007

nonn

The prime factors are taken with multiplicity.
Numbers included in A134611, but not in A025475.
a(1) = 1512 is the minimal number with this property.

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

Hieronymus Fischer, Table of n, a(n) for n = 1..520

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

Cf. A134600, A134602, A134605, A134608, A134611, A134619, A134621.

Minor edits and more terms added by Hieronymus Fischer, May 06 2013, May 30 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

A134604 Numbers (excluding primes and powers of primes) such that the square mean of their prime factors is a prime (where the square mean of c and d is sqrt((c^2+d^2)/2)).

Original entry on oeis.org

119, 161, 351, 595, 721, 845, 959, 1045, 1081, 1241, 1323, 1375, 1547, 1792, 1855, 2457, 2645, 2737, 3281, 3367, 3509, 3887, 3995, 4347, 4625, 4655, 4681, 5376, 5795, 6545, 6615, 6643, 6993, 7505, 7705, 7803, 7889, 8019, 9295, 9625, 10557, 11845

Offset: 1

Hieronymus Fischer, Nov 11 2007

nonn

Numbers included in A134601, 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 a prime.
a(10183) = 114383711 = 13*83*227*467 and sqrt((13^2+83^2+227^2+467^2)/4) = sqrt(69169) = 263 is a prime.

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

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

Cf. A134600, A134602, A134608, A134611, A134617, A134619, A134621.

f[{a_,b_}]:=Table[a,b];Select[Range[2,11845],!PrimePowerQ[#]&&PrimeQ[ RootMeanSquare[f/@FactorInteger[#]//Flatten] ]&] (* James C. McMahon, Apr 09 2025 *)

Minor edits by Hieronymus Fischer, Apr 22 2013

A134606 Numbers such that the square root of the sum of squares of their prime factors is a nonprime integer.

Original entry on oeis.org

16, 81, 351, 512, 625, 1080, 1260, 1350, 1375, 1792, 1836, 2070, 2145, 2175, 2401, 2730, 2772, 3510, 4104, 4305, 4625, 4650, 4655, 4998, 5880, 6000, 6545, 7098, 7182, 7791, 7889, 7956, 9030, 9108, 9295, 9324, 10098, 10368, 10545, 11628, 11935, 12096

Offset: 1

Hieronymus Fischer, Nov 11 2007

nonn

No prime number is a term. - Hieronymus Fischer, Apr 19 2011

			a(3)=351, since 351=3*3*3*13 and sqrt(3*3^2+13^2)=sqrt(196)=14.

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

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

Cf. A134600, A134602, A134608, A134611, A134616, A134618, A134620.

f[{a_,b_}]:=Table[a,b];Select[Range[2,12096],!PrimeQ[Sqrt[Total[Flatten[f/@FactorInteger[#]]^2]]]&&IntegerQ[Sqrt[Total[Flatten[f/@FactorInteger[#]]^2]]]&] (* James C. McMahon, Apr 09 2025 *)

cp's OEIS Frontend

A134618 Numbers such that the sum of cubes of their prime factors (taken with multiplicity) is a prime.

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A134620 Numbers such that the sum of 4th power of their prime factors is a prime.

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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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A134614 Numbers (excluding primes and powers of primes) 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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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)).

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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)).

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