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A003973
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Inverse Möbius transform of A003961; a(n) = sigma(A003961(n)), where A003961 shifts the prime factorization of n one step towards the larger primes.
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85
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1, 4, 6, 13, 8, 24, 12, 40, 31, 32, 14, 78, 18, 48, 48, 121, 20, 124, 24, 104, 72, 56, 30, 240, 57, 72, 156, 156, 32, 192, 38, 364, 84, 80, 96, 403, 42, 96, 108, 320, 44, 288, 48, 182, 248, 120, 54, 726, 133, 228, 120, 234, 60, 624, 112, 480, 144, 128, 62, 624, 68
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OFFSET
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1,2
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COMMENTS
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Sum of the divisors of the prime shifted n, or equally, sum of the prime shifted divisors of n. - Antti Karttunen, Aug 17 2020
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LINKS
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FORMULA
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Multiplicative with a(p^e) = (q^(e+1)-1)/(q-1) where q = nextPrime(p). - David W. Wilson, Sep 01 2001
a(n) is odd if and only if n is a square.
(End)
Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/2) * Product_{p prime} p^3/((p+1)*(p^2-nextprime(p))) = 3.39513795..., where nextprime is A151800. - Amiram Eldar, Dec 08 2022
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MATHEMATICA
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b[1] = 1; b[p_?PrimeQ] := b[p] = Prime[ PrimePi[p] + 1]; b[n_] := b[n] = Times @@ (b[First[#]]^Last[#] &) /@ FactorInteger[n]; a[n_] := Sum[ b[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 70}] (* Jean-François Alcover, Jul 18 2013 *)
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PROG
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(PARI) aPrime(p, e)=my(q=nextprime(p+1)); (q^(e+1)-1)/(q-1)
(PARI) A003973(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); sigma(factorback(f)); }; \\ Antti Karttunen, Aug 06 2020
(Python)
from math import prod
from sympy import factorint, nextprime
def A003973(n): return prod(((q:=nextprime(p))**(e+1)-1)//(q-1) for p, e in factorint(n).items()) # Chai Wah Wu, Jul 05 2022
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CROSSREFS
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Used in the definitions of the following sequences: A326042, A336838, A336841, A336844, A336846, A336847, A336848, A336849, A336850, A336851, A336852, A336856, A336931, A336932.
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KEYWORD
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nonn,easy,mult
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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