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A061078
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Sum of the products of the digits of the first n positive even numbers.
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5
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2, 6, 12, 20, 20, 22, 26, 32, 40, 40, 44, 52, 64, 80, 80, 86, 98, 116, 140, 140, 148, 164, 188, 220, 220, 230, 250, 280, 320, 320, 332, 356, 392, 440, 440, 454, 482, 524, 580, 580, 596, 628, 676, 740, 740, 758, 794, 848, 920, 920, 920, 920, 920, 920, 920, 922
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OFFSET
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1,1
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COMMENTS
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For n = (10^r)/2, a(n) is the sum of the r terms of the geometric progression with first term 20 and common ratio 45.
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REFERENCES
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Amarnath Murthy, Smarandache friendly numbers and a few more sequences, Smarandache Notions Journal, Vol. 12, No. 1-2-3, Spring 2001.
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LINKS
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FORMULA
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a(5*10^n-1) = a(5*10^n) = (5/11)*(45^(n+1)-1).
a(n) <= (5/11)*(45^(log((n+1)/5)+1)-1) for all n.
a(n) ~ (4/5)*A061077(n) as n -> infinity.
Conjecture: let a >= 1, b >= 0, where p is not a multiple of 2 nor 5. Then:
a(5^a*2^b*p-1) = a(5^a*2^b*p) = ... = a(5^a*2^b*p + 55...5) where the number of fives is equal to b if a > b, and is equal to a-1 if 1 <= a <= b. (End)
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EXAMPLE
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a(5) = 2 + 4 + 6 + 8 + 1*0 = 20; (a(18)=116, not 114).
a(1199) = a(5^2*2^4*3 - 1) = ... = a(5^2*2^4*3 + 5) = a(1205). In fact, the number of "fives" is exactly equal to 1 = 2-1 (where 2 is the exponent of 5).
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MATHEMATICA
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Accumulate[Times@@@IntegerDigits[Range[2, 120, 2]]] (* Harvey P. Dale, Jun 18 2021 *)
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PROG
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(PARI) pd(n) = my(d = digits(n)); prod(i=1, #d, d[i]);
(PARI) a(n) = sum(k=1, n, vecprod(digits(2*k))); \\ Michel Marcus, Mar 13 2022
(PARI) a(n) = {t=digits(2*n); p=1; d=#t; for(i=1, #t, if(t[i]==0, d=i-1; break));
(5/11) * (45^(#t-1)-1) + (sum(i=1, #t-1, ((prod(j=1, #t-i-1, t[j])) * (t[#t-i]) * (t[#t-i]-1) * 2 * (5^(i))* (9^(i-1)))))+(prod(k=1, #t-1, t[k]))*((((t[#t])^2))/4+(t[#t])/2)} \\ Luca Onnis, Mar 17 2022
(Python)
from math import prod
from itertools import accumulate
def p(n): return prod(map(int, str(n)))
def a(n): return sum(p(2*i) for i in range(1, n+1))
def aupton(nn): return list(accumulate([pd(2*k) for k in range(1, nn+1)]))
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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Incorrect formula removed by Luca Onnis, Mar 13 2022
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STATUS
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approved
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