A112264 -id:A112264 - cp's OEIS frontend

A112265 Cumulative sum of sum of initial digits of prime factors (with multiplicity) of n.

0, 2, 5, 9, 14, 19, 26, 32, 38, 45, 46, 53, 54, 63, 71, 79, 80, 88, 89, 98, 108, 111, 113, 122, 132, 135, 144, 155, 157, 167, 170, 180, 184, 187, 199, 209, 212, 215, 219, 230, 234, 246, 250, 255, 266, 270, 274, 285, 299, 311, 315, 320, 325, 336, 342, 355, 359

Offset: 1

Jonathan Vos Post, Aug 30 2005

Primes in this sequence are A112266.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000030, A000040, A001221, A001222, A112264, A112266.

f[1] = 0; f[n_] := Plus @@ (#[[2]] First@IntegerDigits[#[[1]]] & /@ FactorInteger[n]); Accumulate@ Array[f, 90] (* Giovanni Resta, Jun 17 2016 *)

a(1) = 0 and given the prime factorization i = (p_1)^(e_1) * (p_2)^(e_2) * ... * (p_k)^(e_k) then a(n) = Sim_{i=1..n} A112264(i) = Sum_{i=1..n} (e_1)*A000030(p_1) + (e_2)*A000030(p_2) + ... + (e_k)*A000030(p_l).

Corrected and edited by Giovanni Resta, Jun 17 2016

A109834 Startorial numbers: product of initial digits of integers 1 through n.

Original entry on oeis.org

1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 362880, 362880, 362880, 362880, 362880, 362880, 362880, 362880, 362880, 362880, 725760, 1451520, 2903040, 5806080, 11612160, 23224320, 46448640, 92897280, 185794560, 371589120, 1114767360

Offset: 1

Jonathan Vos Post, Aug 31 2005

This is a base 10 sequence; the equivalent exists for other bases. For base 1 and base 2, this gives the all-one sequence (A000012).

			a(30) = 1*2*3*4*5*6*7*8*9*1*1*1*1*1*1*1*1*1*1*2*2*2*2*2*2*2*2*2*2*3 = 1114767360.

G. C. Greubel, Table of n, a(n) for n = 1..1000
A. Cobham, Uniform Tag Sequences, Mathematical Systems Theory, 6 (1972), 164-192.

Cf. A000030, A000040, A001221, A001222, A112264, A112266.

Table[Product[Floor[k/10^(Floor[Log10[k]])], {k,1,n}], {n,1,50}] (* G. C. Greubel, May 16 2017 *)
FoldList[Times,Table[IntegerDigits[n][[1]],{n,30}]] (* Harvey P. Dale, Aug 07 2019 *)

a(n) = Product_{i = 1..n} A000030(i).
a(n+1) = a(n) * InitialDigitOf(n).
a(n) = Product_{i = 1..n} floor(i / 10^(floor(log_10(i)))) where log_10(i) is the logarithm of i in base 10.

cp's OEIS Frontend

A112265 Cumulative sum of sum of initial digits of prime factors (with multiplicity) of n.

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