A112264 Sum of initial digits of prime factors (with multiplicity) of n.

0, 2, 3, 4, 5, 5, 7, 6, 6, 7, 1, 7, 1, 9, 8, 8, 1, 8, 1, 9, 10, 3, 2, 9, 10, 3, 9, 11, 2, 10, 3, 10, 4, 3, 12, 10, 3, 3, 4, 11, 4, 12, 4, 5, 11, 4, 4, 11, 14, 12, 4, 5, 5, 11, 6, 13, 4, 4, 5, 12, 6, 5, 13, 12, 6, 6, 6, 5, 5, 14, 7, 12, 7, 5, 13, 5, 8, 6, 7, 13, 12, 6, 8, 14, 6, 6, 5, 7, 8, 13, 8, 6, 6, 6

Offset: 1

Author

Jonathan Vos Post, Aug 30 2005

Keywords

• base
• easy
• nonn

Comments

For primes p, elements of A000040, a(p) = A000030(p). The cumulative sum of this sequence is A112265. Primes in the cumulative sum are A112266. This is a base 10 sequence, the base 1 equivalent is A001222(n) = BigOmega(n) = e_1 + e_2 + ... + e_k, the number of prime factors (with multiplicity), where k = A001221(n) = SmallOmega(n). The base 2 equivalent is equal to the base 1 equivalent.

Examples

			a(4) = 4 because 4 = 2*2, so the sum of the initial digits is 2 + 2 = 4.
a(11) = 1 because 11 is prime and its initial digit is 1.
a(22) = 3 because 22 = 2*11, so the sum of the initial digits is 2 + 1 = 3.
a(98) = 16 because 98 = 2 * 7^2, so the sum of the initial digits is 2 + 7 + 7 = 16.
a(100) = 14 because 100 = 2^2 * 5^2, so the sum of the initial digits is 2 + 2 + 5 + 5 = 14.
a(121) = 2 because 121 = 11^2, so the sum of the initial digits is 1 + 1 = 2.
a(361) = 2 because 361 = 19^2, so the sum of the initial digits is 1 + 1 = 2.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Prime Factor
• Eric Weisstein's World of Mathematics, Distinct Prime Factors

Crossrefs

Cf. A000030, A000040, A001221, A001222, A112266.

Programs

Mathematica

f[1] = 0; f[n_] := Plus @@ (#[[2]] First@IntegerDigits[#[[1]]] & /@ FactorInteger[n]); Array[f, 94] (* Giovanni Resta, Jun 17 2016 *)

Formula

a(1) = 0 and given the prime factorization n = (p_1)^(e_1) * (p_2)^(e_2) * ... * (p_k)^(e_k) then a(n) = (e_1)*A000030(p_1) + (e_2)*A000030(p_2) + ... + (e_k)*A000030(p_l).

Extensions

a(6) and a(35) corrected by Giovanni Resta, Jun 17 2016