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.

Previous Showing 11-18 of 18 results.

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

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

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

Views

Author

Hieronymus Fischer, Nov 11 2007

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

Cf. A001597, A025475, A134333, A134344, A134376.
Cf. A134600, A134602, A134605, A134608, A134613, A134617, A134619, A134621.

Programs

  • PARI
    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

Extensions

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

A134609 Numbers such that the cube root of the sum of cubes of their prime factors is a nonprime integer.

Original entry on oeis.org

256, 588, 693, 3840, 6561, 17787, 178360, 313600, 337365, 350000, 387072, 390625, 407442, 432000, 531674, 535815, 541310, 664909, 697851, 1044582, 1262056, 1264640, 1299272, 1374327, 1547570, 1660360, 1740024, 2160756, 2578968
Offset: 1

Views

Author

Hieronymus Fischer, Nov 11 2007

Keywords

Comments

No prime number is a term. Hieronymus Fischer, Apr 20 2013

Examples

			a(2)=588, since 588=2*2*3*7*7 and (2*2^3+3^3+2*7^3)^(1/3)=729^(1/3)=81.

Links

Crossrefs

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

Extensions

Minor Edits by Hieronymus Fischer, Apr 20 2013

A134607 Composite numbers such that the square root of the sum of squares of their prime factors is a prime.

Original entry on oeis.org

48, 320, 486, 3072, 3150, 6174, 7128, 7650, 10890, 11466, 15000, 18018, 18810, 25578, 27846, 29400, 30240, 39546, 40590, 45056, 45927, 53010, 54600, 55062, 59202, 73440, 75582, 77418, 80910, 85800, 90552, 92106, 95238, 96642, 98838
Offset: 1

Views

Author

Hieronymus Fischer, Nov 11 2007

Keywords

Comments

Numbers included in A134605, but not in A134606. a(1)=48 is the minimal number with this property.

Examples

			a(2)=320, since 320=2*2*2*2*2*2*5 and sqrt(6*2^2+5^2)=sqrt(49)=7.

Links

Crossrefs

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

Programs

  • Mathematica
    sspfpQ[n_]:=PrimeQ[Sqrt[Total[Flatten[Table[#[[1]],{#[[2]]}]&/@ FactorInteger[ n]]^2]]]; upto=100000;With[{comps=Complement[ Range[ upto],Prime[ Range[PrimePi[upto]]]]},Select[comps,sspfpQ]] (* Harvey P. Dale, Jul 10 2013 *)

Extensions

Minor edits by Hieronymus Fischer, Apr 19 2013

A134610 Composite numbers such that the cube root of the sum of cubes of their prime factors is a prime.

Original entry on oeis.org

14157, 141960, 466560, 1608575, 3097055, 5338710, 6235076, 16017300, 22353408, 24948000, 25073792, 25564544, 27843750, 29761408, 30570408, 31894350, 40837825, 44175248, 46120064, 49867818, 55814400, 56141963, 71214803, 77450890, 92682405
Offset: 1

Views

Author

Hieronymus Fischer, Nov 11 2007

Keywords

Comments

Numbers included in A134608, but not in A134609. a(1)=14157 is the minimal number with this property.
The prime factors are taken by multiplicity.

Examples

			a(2)=141960 since 141960=2*2*2*3*5*7*13*13 and (3*2^3+3^3+5^3+7^3+2*13^3)^(1/3)=4913^(1/3)=17.

Links

Crossrefs

Cf. A001597, A025475, A134333, A134344, A134376.
Cf. A134600, A134602, A134605, A134613, A134616, A134618, A134620.

Extensions

Minor edits and additional terms by the author, Apr 15 2013
Previous Showing 11-18 of 18 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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