A190882 -id:A190882 - cp's OEIS frontend

A268373 Numbers other than prime powers divisible by the sum of the cubes of their prime divisors.

378, 480, 756, 960, 1134, 1440, 1512, 1920, 2268, 2400, 2548, 2646, 2880, 3024, 3402, 3840, 4320, 4536, 4800, 5096, 5292, 5760, 6048, 6804, 7200, 7680, 7938, 8640, 9072, 9600, 10192, 10206, 10584, 11520, 12000, 12096, 12960, 13608, 14400, 15360, 15876, 17280, 17836, 18144, 18522, 18711

Offset: 1

Michel Marcus, Feb 03 2016

nonn

Koninck & Luca prove that this set is infinite. - Charles R Greathouse IV, Feb 03 2016

			The prime factors of 480 are 2, 3 and 5. The sum of their cubes is 2^3+3^3+5^3=160, and 480 is divisible by 160.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jean-Marie de Koninck, Florian Luca, Integers divisible by sums of powers of their prime factors, Journal of Number Theory, Volume 128, Issue 3 (March 2008), pp. 557-563.

Cf. A066031, A190882.

Select[Range[10^4], Length[(p = FactorInteger[#][[;;,1]])] > 1 && Divisible[#, Total[p^3]] &] (* Amiram Eldar, Sep 05 2019 *)

isok(n) = my(f = factor(n)[,1]) ; (#f>2) && ((n % sum(k=1, #f, f[k]^3)) == 0);

A268417 Numbers other than prime powers divisible by the sum and the sum of squares of their prime divisors.

Original entry on oeis.org

99528, 117040, 143520, 199056, 234080, 287040, 288288, 294216, 298584, 349440, 357357, 383040, 398112, 430560, 468160, 574080, 576576, 585200, 588432, 597168, 631488, 698880, 717600, 766080, 796224, 819280, 861120, 864864, 870870, 882648, 895752, 901824, 936320, 957000

Offset: 1

Michel Marcus, Feb 04 2016

nonn

Intersection of A066031 and A190882.

Prime divisors taken without multiplicity. - Harvey P. Dale, Dec 27 2018

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jean-Marie de Koninck and Florian Luca, Integers divisible by sums of powers of their prime factors, Journal of Number Theory, Volume 128, Issue 3, March 2008, Pages 557-563.

Cf. A066031, A190882.

dssQ[n_]:=Module[{pf=FactorInteger[n][[All,1]]},!PrimePowerQ[ n] && Divisible[ n, Total[pf]]&&Divisible[n,Total[pf^2]]]; Select[ Range[ 960000],dssQ] (* Harvey P. Dale, Dec 27 2018 *)

isok(n) = my(f = factor(n)[,1]); (#f>2) && ((n % vecsum(f)) == 0) && ((n % sum(k=1, #f, f[k]^2)) == 0);

A268418 Numbers other than prime powers divisible by the sum, the sum of squares and the sum of cubes of their prime divisors.

Original entry on oeis.org

12192180, 15724800, 24384360, 31449600, 36576540, 47174400, 48768720, 60960900, 62899200, 73153080, 78624000, 85345260, 94348800, 97537440, 109729620, 110073600, 121921800, 125798400, 134113980, 141523200, 146306160, 157248000, 158498340, 170690520, 182882700, 188697600, 195074880

Offset: 1

Michel Marcus, Feb 04 2016

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..100
Jean-Marie de Koninck, Florian Luca, Integers divisible by sums of powers of their prime factors, Journal of Number Theory, Volume 128, Issue 3, March 2008, Pages 557-563.

Intersection of A268373 and A066031 and A190882.

Intersection of A268373 and A268417.

isok(n) = my(f = factor(n)[,1]); (#f>2) && ((n % vecsum(f)) == 0) && ((n % sum(k=1, #f, f[k]^2)) == 0) && ((n % sum(k=1, #f, f[k]^3)) == 0);

A380902 Integers k with at least 1 proper factorization for which the sum of the squares of the factors equals k.

Original entry on oeis.org

16, 27, 48, 54, 270, 528, 1755, 7216, 7830, 11934, 69168, 81702, 100368, 264654, 340470, 559899, 1397808, 1586340, 1695195, 3837510, 3918420, 8989110, 9815568, 13010448, 15812550, 19468816, 26302590, 75872430, 132825616, 133529580, 180280539, 271165488

Offset: 1

Charles L. Hohn, Feb 07 2025

nonn

It is conjectured that this sequence is infinite, that it does not contain any squarefree terms (A005117), and that with the exception of 16 (2^4) and 27 (3^3) it does not contain any squareful terms (A001694) or examples where the factors are all primes.

It is unknown whether this sequence contains any terms that produce more than one example (improbable if the exponential growth trend holds, but this is also unknown), or whether a more efficient generator algorithm (than the brute-force one given) exists or could be feasible.

			a(1) = 16: 2 * 2 * 2 * 2 = 2^2 + 2^2 + 2^2 + 2^2 = 16.
a(2) = 27: 3 * 3 * 3 = 3^2 + 3^2 + 3^2 = 27.
a(3) = 48: 2 * 2 * 2 * 6 = 2^2 + 2^2 + 2^2 + 6^2 = 48.

Subset of A380760.

Cf. A001694, A005117, A162247, A190882.

a380902_count(x, f=List())={my(r=x/if(#f, vecprod(Vec(f)), 1)); if(r==1, return(if(sum(i=1, #f, f[i]^2)==x, 1, 0))); my(d, c=0); fordiv(r, d, if(d==1 || d==x || (#f && dCharles L. Hohn, Mar 09 2025

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A268373 Numbers other than prime powers divisible by the sum of the cubes of their prime divisors.

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A268417 Numbers other than prime powers divisible by the sum and the sum of squares of their prime divisors.

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A268418 Numbers other than prime powers divisible by the sum, the sum of squares and the sum of cubes of their prime divisors.

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A380902 Integers k with at least 1 proper factorization for which the sum of the squares of the factors equals k.

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PARI