A134618 -id:A134618 - cp's OEIS frontend

A134605 Composite numbers such that the square root of the sum of squares of their prime factors (with multiplicity) is an integer.

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

Hieronymus Fischer, Nov 11 2007

nonn

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

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

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

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

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

Hieronymus Fischer, Nov 11 2007

nonn

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

Hieronymus Fischer, Table of n, a(n) for n = 1..300

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

Cf. A134600, A134602, A134605, A134611, A134616, A134618, A134620.

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

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

Minor edits by Hieronymus Fischer, Apr 20 2013

A078137 Numbers which can be written as sum of squares>1.

Original entry on oeis.org

4, 8, 9, 12, 13, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82

Offset: 1

Reinhard Zumkeller, Nov 19 2002

A078134(a(n))>0.
Numbers which can be written as a sum of squares of primes. - Hieronymus Fischer, Nov 11 2007
Equivalently, numbers which can be written as a sum of squares of 2 and 3. Proof for numbers m>=24: if m=4*(k+6), k>=0, then m=(k+6)*2^2; if m=4*(k+6)+1 than m=(k+4)*2^2+3^2; if m=4*(k+6)+2 then m=(k+2)*2^2+2*3^2; if m=4*(k+6)+3 then m=k*2^2+3*3^2. Clearly, the numbers a(n)<24 can also be written as sums of squares of 2 and 3. Explicit representation as a sum of squares of 2 and 3 for numbers m>23: m=c*2^2+d*3^2, where c:=(floor(m/4) - 2*(m mod 4))>=0 and d:=m mod 4. - Hieronymus Fischer, Nov 11 2007

Index entries for sequences related to sums of squares
Eric Weisstein's World of Mathematics, Square Number.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Complement of A078135.

Cf. A000290, A078136, A078131, A001597, A025475, A078134, A078135, A078139, A090677, A134600, A134605, A134608, A134612, A134616, A134618, A134620.

Join[{4, 8, 9, 12, 13, 16, 17, 18, 20, 21, 22}, Range[24, 82]] (* Jean-François Alcover, Aug 01 2018 *)

a(n)=if(n>11,n+12,[4,8,9,12,13,16,17,18,20,21,22][n]) \\ Charles R Greathouse IV, Aug 21 2011

a(n)=n + 12 for n >= 12. - Hieronymus Fischer, Nov 11 2007

Edited by N. J. A. Sloane, Oct 17 2009 at the suggestion of R. J. Mathar.

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 *)

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 *)

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

Hieronymus Fischer, Nov 11 2007

nonn

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

			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.

Hieronymus Fischer, Table of n, a(n) for n = 1..250

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

Cf. A134600, A134602, A134605, A134611, A134616, A134618, A134620.

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

Hieronymus Fischer, Nov 11 2007

nonn

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

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

Hieronymus Fischer, Table of n, a(n) for n = 1..1000

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

Cf. A134600, A134602, A134605, A134611, A134616, A134618, A134620.

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 *)

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

Hieronymus Fischer, Nov 11 2007

nonn

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.

			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.

Hieronymus Fischer, Table of n, a(n) for n = 1..50

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

Cf. A134600, A134602, A134605, A134613, A134616, A134618, A134620.

Minor edits and additional terms by the author, Apr 15 2013

cp's OEIS Frontend

A134605 Composite numbers such that the square root of the sum of squares of their prime factors (with multiplicity) is an integer.

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A134608 Composite numbers such that the cube root of the sum of cubes of their prime factors is an integer.

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A078137 Numbers which can be written as sum of squares>1.

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

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

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

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

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Programs

Mathematica

Extensions

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

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