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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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 *)

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

A275384 Composite squarefree numbers such that the arithmetic mean of its prime factors is an integer.

Original entry on oeis.org

15, 21, 33, 35, 39, 42, 51, 55, 57, 65, 69, 77, 78, 85, 87, 91, 93, 95, 105, 110, 111, 114, 115, 119, 123, 129, 133, 141, 143, 145, 155, 159, 161, 170, 177, 183, 185, 186, 187, 195, 201, 203, 205, 209, 213, 215, 217, 219, 221, 222, 230, 231, 235, 237, 247, 249, 253, 258, 259, 265, 267
Offset: 1

Views

Author

Antonio Roldán, Jul 25 2016

Keywords

Comments

Sopf(a(n)) is multiple of omega(a(n)) (sopf(n) is the sum of the distinct prime factors of n, and omega(n) is the number of distinct primes dividing n).
This sequence is subsequence of A078177 and supersequence of A187073.

Examples

			170 is in the sequence because 170 = 17*2*5 (squarefree number) and (17+2+5)/3 = 8 is an integer.

Links

Crossrefs

Cf. A005117, A078177, A134344, A187073, A229094.

Programs

  • Mathematica
    Select[Range@ 270, And[CompositeQ@ #, SquareFreeQ@ #, IntegerQ@ Mean@ FactorInteger[#][[All, 1]]] &] (* Michael De Vlieger, Jul 25 2016 *)
  • PARI
    sopf(n)= my(f, s=0); f=factor(n); for(i=1, matsize(f)[1], s+=f[i, 1]); s
for(i=2,500,if(issquarefree(i)&&!isprime(i),m=sopf(i)/omega(i);if(m==truncate(m),print1(i,", "))))
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