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A257367 Composite numbers n equal to the sum of prime factors, counted with multiplicity, of the numbers in the interval [n-k,n+k], for some k. 5
4, 75, 186, 531, 627, 5216, 22843, 148050, 1061385, 1490407, 1562485, 9034704, 10422738, 31920786, 76343543, 78824242, 105791155, 111873121, 131515163, 549038887, 1318856915, 1394579379, 1630428366, 1639063828, 3710476544, 3996221763, 4524478925, 6172721935 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Prime numbers are not considered because they are a trivial solution being the sum of their single prime factor (case k = 0).

Composite n such that n = Sum_{i=-k..k} A001414(i+n) for some k.

Values of k are 0, 1, 2, 4, 3, 4, 7, 6, 6, 8, 8, 9, 12, 8, 17, 9, 11, 4, 18, 11, ...

LINKS

Giovanni Resta, Table of n, a(n) for n = 1..40 (terms < 3*10^11)

EXAMPLE

Prime factors of 4 are 2, 2 and 2 + 2 = 4. In this case k = 0.

For 75, k is equal to 1. Let us consider the prime factors of 74, 75 and 76. They are:  2, 37; 3, 5, 5; 2, 2, 19. Their sum is 2 + 37 + 3 + 5 + 5 + 2 + 2 + 19 = 75.

For 186, k is equal to 2. Let us consider the prime factors of 184, 185, 186, 187, 188. They are: 2, 2, 2, 23; 5, 37; 2, 3, 31; 11, 17; 2, 2, 47. Their sum is 2 + 2 + 2 + 23 + 5 + 37 + 2 + 3 + 31 +  11 + 17 + 2 + 2 + 47 = 186.

MAPLE

with(numtheory); P:= proc(q) local a, c, d, j, k, n;

for n from 2 to q do if not isprime(n) then a:=ifactors(n)[2];

k:=0; a:=add(a[j][1]*a[j][2], j=1..nops(a));

while a<n do k:=k+1; c:=ifactors(n-k)[2]; d:=ifactors(n+k)[2];

c:=add(c[j][1]*c[j][2], j=1..nops(c));

d:=add(d[j][1]*d[j][2], j=1..nops(d));

a:=a+c+d; od; if a=n then print(n); fi; fi; od; end: P(10^9);

PROG

(PARI) sopfr(n) = my(f=factor(n)); sum(k=1, #f~, f[k, 1]*f[k, 2]);

isok(n) = {my(s = sopfr(n)); my(k = 1); while (s < n, s += sopfr(n-k) + sopfr(n+k); k++); s == n; }

lista(nn) = {forcomposite(n=2, nn, if (isok(n), print1(n, ", ")); ); } \\ Michel Marcus, May 27 2015

CROSSREFS

Cf. A001414, A257524, A257525.

Sequence in context: A240007 A046057 A280889 * A072373 A006412 A206456

Adjacent sequences:  A257364 A257365 A257366 * A257368 A257369 A257370

KEYWORD

nonn

AUTHOR

Paolo P. Lava, Apr 21 2015

EXTENSIONS

a(21)-a(28) from Giovanni Resta, May 27 2015

STATUS

approved

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Last modified November 20 02:34 EST 2019. Contains 329323 sequences. (Running on oeis4.)