A112266 -id:A112266 - cp's OEIS frontend

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

Jonathan Vos Post, Aug 30 2005

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.

			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.

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

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

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

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).

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

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

Original entry on oeis.org

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

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

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

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Extensions

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

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Programs

Mathematica

Formula