A068916 -id:A068916 - cp's OEIS frontend

A067688 Composite n such that for some integer r, n equals the sum of the r-th powers of the prime factors of n (counted with multiplicity).

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

Offset: 1

Joseph L. Pe, Feb 04 2002

nonn

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

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

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.

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

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}]]]]]

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

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

A268594 Numbers n of the form p^k - k = q^i - i for primes p < q.

Original entry on oeis.org

2, 12, 58, 238, 3120, 6856, 29788, 50650, 65520, 161046, 262126, 300760, 1295026, 3442948, 9393928, 13997518, 21253930, 49430860, 84604516, 95443990, 237176656, 329939368, 384240580, 487443400, 633839776, 893871732, 904231060, 1284365500, 1605723208, 3183010108, 3301293166, 3588604288, 3936827536

Offset: 1

Jud McCranie, Feb 07 2016

nonn

			50650 = 37^3-3 = 50651^1-1.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
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.

See A268595 for values of p and A268596 for values of q.

Cf. A178251.

is(n)=my(p);sum(e=1,logint(n,2)+1,ispower(n+e,e,&p)&&isprime(p))>1 \\ Charles R Greathouse IV, Feb 08 2016

list(lim)=my(v=List([2]),q,n); for(e=3,logint(1+lim\=1,2), forprime(p=2, sqrtnint(lim+e,e), if(sum(i=1,e-1, n=p^e-e; ispower(n+i,i,&q) && isprime(q)), listput(v,n)))); Set(v) \\ Charles R Greathouse IV, Feb 08 2016

A268036 Cubes of the prime factors (with multiplicity) of n add up to n.

Original entry on oeis.org

1096744, 2836295, 4473671462, 23040925705, 13579716377989, 119429556097859

Offset: 1

Jud McCranie, Jan 24 2016

a(1) = A068916(3).

13579716377989 and 119429556097859 are also terms, but it is not known if they are the next ones. - Jud McCranie, Feb 09 2016

			1096744 = 2^3 * 11^3 * 103 = 2^3+2^3+2^3+11^3+11^3+11^3+103^3.

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.

Cf. A068916, A067688 (a more inclusive sequence), A058041.

isok(n) = {my(f = factor(n)); sum(k=1, #f~, f[k,1]^3*f[k, 2]) == n;} \\ Michel Marcus, Feb 09 2016

a(5) from Jud McCranie, Feb 13 2016

a(6) from Jud McCranie, Mar 20 2016

A318606 Numbers of the form p^k-k for some prime, p, and integer k >= 0.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 10, 12, 16, 18, 22, 23, 24, 27, 28, 30, 36, 40, 42, 46, 47, 52, 58, 60, 66, 70, 72, 77, 78, 82, 88, 96, 100, 102, 106, 108, 112, 119, 121, 122, 126, 130, 136, 138, 148, 150, 156, 162, 166, 167, 172, 178, 180, 190, 192, 196, 198, 210, 222, 226, 228, 232, 238, 240, 248, 250, 256, 262

Offset: 1

Jud McCranie, Aug 29 2018

nonn

			2^7-7=121, so 121 is in the sequence.

Jud McCranie, Table of n, a(n) for n = 1..5000
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.

Cf. A000040, A057897, A068916, A269769.

With[{nn = 265}, Union@ Flatten@ Table[p^k - k, {p, Prime@ Range@ PrimePi@ nn}, {k, Log[p, nn]}]] (* Michael De Vlieger, Jul 16 2019 *)

cp's OEIS Frontend

A067688 Composite n such that for some integer r, n equals the sum of the r-th powers of the prime factors of n (counted with multiplicity).

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A268594 Numbers n of the form p^k - k = q^i - i for primes p < q.

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A268036 Cubes of the prime factors (with multiplicity) of n add up to n.

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A318606 Numbers of the form p^k-k for some prime, p, and integer k >= 0.

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