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.

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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)).

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

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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

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
Previous Showing 11-13 of 13 results.

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