cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 12 results. Next

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

Views

Author

Hieronymus Fischer, Nov 11 2007

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

Cf. A001597, A025475, A134333, A134344, A134376.
Cf. A134601, A134605, A134608, A134611, A134617, A134619, A134621.

Programs

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

Extensions

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

Views

Author

Hieronymus Fischer, Nov 11 2007

Keywords

Examples

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

Links

Crossrefs

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

Programs

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

Views

Author

Hieronymus Fischer, Nov 11 2007

Keywords

Examples

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

Links

Crossrefs

Cf. A001597, A025475, A134333, A134344, A134376.
Cf. A134600, A134602, A134605, A134611, A134616, A134618, A134620.

Programs

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

Extensions

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

Views

Author

Hieronymus Fischer, Nov 11 2007

Keywords

Comments

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.

Examples

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

Links

Crossrefs

Cf. A001597, A025475, A134333, A134344, A134376.
Cf. A134600, A134604, A134608, A134611, A134617, A134619, A134621.

Programs

  • Mathematica
    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 *)

Extensions

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

Views

Author

Hieronymus Fischer, Nov 11 2007

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

Cf. A001597, A025475, A134333, A134344, A134376.
Cf. A134600, A134602, A134605, A134608, A134617, A134619, A134621.

Programs

  • PARI
    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

Extensions

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

Views

Author

Hieronymus Fischer, Nov 11 2007

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

Cf. A001597, A025475, A134333, A134344, A134376.
Cf. A134600, A134605, A134608, A134611, A134617, A134619, A134621.

Programs

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

Extensions

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

Views

Author

Hieronymus Fischer, Nov 11 2007

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

Cf. A001597, A025475, A134333, A134344, A134376.
Cf. A134600, A134602, A134605, A134608, A134611, A134619, A134621.

Extensions

Minor edits and more terms added 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

Views

Author

Hieronymus Fischer, Nov 11 2007

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

Cf. A001597, A025475, A134333, A134344, A134376.
Cf. A134600, A134605, A134608, A134611, A134617, A134619, A134621.

Programs

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

Extensions

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

Views

Author

Hieronymus Fischer, Nov 11 2007

Keywords

Comments

Numbers included in A134601, but not in A025475. a(1)=119 is the minimal number with this property.

Examples

			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.

Links

Crossrefs

Cf. A001597, A025475, A134333, A134344, A134376.
Cf. A134600, A134602, A134608, A134611, A134617, A134619, A134621.

Programs

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

Extensions

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

Views

Author

Hieronymus Fischer, Nov 11 2007

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
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