A296717 Weakly prime-additive numbers: numbers n with at least 2 distinct prime factors that can be represented as n = Sum_{some p|n} p^e_p with e_p > 0.

30, 42, 60, 70, 84, 90, 102, 132, 140, 150, 170, 174, 180, 186, 210, 228, 252, 270, 290, 294, 300, 306, 318, 330, 350, 364, 378, 390, 396, 420, 442, 540, 546, 570, 588, 618, 630, 650, 660, 714, 730, 750, 774, 780, 804, 858, 870, 882, 894, 900, 906, 915, 980

Offset

1,1

Comments

Supersequence of A302753 (prime-additive numbers). Terms that are not prime-additive numbers: 210, 330, 390, 420, 546, 570, 630, 660, 714, 780, 858, 870, ...

Fang & Chen defined shortest weakly prime-additive numbers to be those that can be represented as sum of 3 powers. The weakly prime-additive numbers that are not shortest are 2730, 3570, 5460, 5610, 6090, 6930, 7140, 7854, 8610, 8970, 9030, 9240, 9570, ...

Links

Giovanni Resta, Table of n, a(n) for n = 1..10000

Paul Erdős and Norbert Hegyvári, On prime-additive numbers, Studia Scientiarum Mathematicarum Hungarica, Vol. 27, No. 1-2 (1992), pp. 207-212. Review.

Jin-Hui Fang and Yong-Gao Chen, On the shortest weakly prime-additive numbers, Journal of Number Theory, Vol. 182 (2018), pp. 258-270.

Example

210 = 2 * 3 * 5 * 7 = 2^2 + 3^4 + 5^3, thus 210 is in the sequence. It is not prime-additive number, since there is no power of 7 in the sum.

Mathematica

primes[n_] := First[Transpose[FactorInteger[n]]]; maxPower[p_, n_] := Module[ {k=0, nn=n}, While[nn>1, nn/=p; k++]; k-1 ]; a[n_] := Module[ {ps=primes[n]}, np=Length[ps]; pws=Table[maxPower[ps[[k]], n], {k, 1, np}]; npws = Length[pws]; Coefficient [Product[1+Sum[x^(ps[[k]]^j), {j, 1, pws[[k]]} ], {k, 1, np}], x, n]]; s={}; Do[b=a[n]; If[b>0, AppendTo[s, n]], {n, 1, 2100}]; s

Crossrefs

Cf. A302753, A302755.

Sequence in context: A357685 A091454 A175727 * A302753 A179945 A136152

Adjacent sequences: A296714 A296715 A296716 * A296718 A296719 A296720

Keyword

nonn

Author

Amiram Eldar, Apr 15 2018

Status

approved