This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A112264 #19 Feb 16 2025 08:32:59 %S A112264 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, %T A112264 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, %U A112264 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 %N A112264 Sum of initial digits of prime factors (with multiplicity) of n. %C A112264 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. %H A112264 G. C. Greubel, <a href="/A112264/b112264.txt">Table of n, a(n) for n = 1..1000</a> %H A112264 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeFactor.html">Prime Factor</a> %H A112264 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DistinctPrimeFactors.html">Distinct Prime Factors</a> %F A112264 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). %e A112264 a(4) = 4 because 4 = 2*2, so the sum of the initial digits is 2 + 2 = 4. %e A112264 a(11) = 1 because 11 is prime and its initial digit is 1. %e A112264 a(22) = 3 because 22 = 2*11, so the sum of the initial digits is 2 + 1 = 3. %e A112264 a(98) = 16 because 98 = 2 * 7^2, so the sum of the initial digits is 2 + 7 + 7 = 16. %e A112264 a(100) = 14 because 100 = 2^2 * 5^2, so the sum of the initial digits is 2 + 2 + 5 + 5 = 14. %e A112264 a(121) = 2 because 121 = 11^2, so the sum of the initial digits is 1 + 1 = 2. %e A112264 a(361) = 2 because 361 = 19^2, so the sum of the initial digits is 1 + 1 = 2. %t A112264 f[1] = 0; f[n_] := Plus @@ (#[[2]] First@IntegerDigits[#[[1]]] & /@ FactorInteger[n]); Array[f, 94] (* _Giovanni Resta_, Jun 17 2016 *) %Y A112264 Cf. A000030, A000040, A001221, A001222, A112266. %K A112264 base,easy,nonn %O A112264 1,2 %A A112264 _Jonathan Vos Post_, Aug 30 2005 %E A112264 a(6) and a(35) corrected by _Giovanni Resta_, Jun 17 2016