A067688 Composite numbers that are equal to the sum of the r-th powers of its prime factors (counted with multiplicity) for some integer r.

4, 16, 27, 256, 3125, 19683, 65536, 823543, 1096744, 2836295, 4294967296, 4473671462, 23040925705, 285311670611, 7625597484987, 13579716377989, 119429556097859, 302875106592253

Offset

1,1

Comments

Every prime is the sum of the first powers of its prime factors, so only composite numbers have been considered in this sequence.

Every integer of the form p^p^k with p prime and k>0 is in the sequence, since it equals the sum of the (p^k - k)-th powers of its prime factors. The first 8 terms of the sequence are of this form, but 1096744 = 2^3*11^3*103 and 2836295 = 5*7*11*53*139 are not.

4473671462 = 2*13*179*593*1621 is also not a prime power.

a(15) <= 7625597484987. a(16) <= 302875106592253. - Donovan Johnson, May 17 2010

a(16) <= 13579716377989, a(17) <= 119429556097859, a(18) <= 302875106592253. - Jud McCranie, Feb 09 2016

a(19) <= 298023223876953125. - Jud McCranie, Apr 25 2016

By applying Vieta's formulas, I discovered a large number that is not a prime power yet equal the sum of squares of their prime factors: 31980599086523546548147351491272676211458715997231784732063781637489066745716387150725397533911738728841877998280065787261857594370930355385463920685184046649444274178977072736045 = 3 * 3 * 5 * 3975008818510860570063178521896626877423134543956742110665444401955840619438206454686489 * 178787019639056312114272069094606410564986435927077305481693028686338614052391094299501609 = 3^2 + 3^2 + 5^2 + 3975008818510860570063178521896626877423134543956742110665444401955840619438206454686489^2 + 178787019639056312114272069094606410564986435927077305481693028686338614052391094299501609^2. - Yuchen Li, Oct 25 2025

Two more terms (square sum) have been found: 138976859457060202640775760344611999531511108694631847974742443939234326467300434658218470334175459700705971960791869592104 = 2 * 2 * 2 * 23 * 1109123111 * 57766182616657495290612267717977498812931942308391 * 11788844704086155814066994795339207139099517865226893357415731 and 3468964492110178502033312975258483508841840399969823169506562553093783608570218806083200716731395833275441150821349128867534891641545518285925735804347623249366552 = 2 * 2 * 2 * 95783 * 13445403342601809061726519 * 180778871045555786730494902253611483680494135761943 * 1862515635400191345742222416847392668657068433219626455654264772436151243547155029. -Yuchen Li, Oct 26 2025

Links

Table of n, a(n) for n=1..18.

S. P. Hurd and J. S. McCranie, Integers that are Sums of Uniform Powers of all their Prime Factors: the sequence A068916, J. of Int. Seq., vol 22 (2019), article 19.3.4.

Example

The sum of the cubes of the prime factors of 1096744 is 3*2^3 + 3*11^3 + 103^3 = 1096744.

Mathematica

For[n=2, True, n++, If[ !PrimeQ[n], For[r=1; fn=FactorInteger[n]; s=0, s<=n, r++, s=Plus@@((#[[2]]#[[1]]^r)&/@fn); If[s==n, Print[{n, r}]]]]]

Prog

(PARI) is(n)=if(isprime(n)||n<4, return(0)); my(f=factor(n), t=#f~); for(r=1, logint(n\f[t, 2], f[t, 1]), if(sum(i=1, t, f[i, 2]*f[i, 1]^r)==n, return(1))); 0 \\ Charles R Greathouse IV, Jan 30 2016

Crossrefs

Cf. A002808, A068916, A081177 (for values of r), A268036 (for a subsequence).

Sequence in context: A201009 A111260 A381538 * A097374 A257309 A271936

Adjacent sequences: A067685 A067686 A067687 * A067689 A067690 A067691

Keyword

nonn

Author

Joseph L. Pe, Feb 04 2002

Extensions

Edited by Dean Hickerson, Mar 07 2002

More terms from Jud McCranie, Mar 10 2003

a(13)-a(14) from Donovan Johnson, May 17 2010

a(15) confirmed by Jud McCranie, Jan 30 2016

a(16) from Jud McCranie, Feb 13 2016

a(17) from Jud McCranie, Mar 20 2016

a(18) from Jud McCranie, Apr 23 2016

Name edited by Michel Marcus, Oct 26 2025

Status

approved