A238526 -id:A238526 - cp's OEIS frontend

A238525 n modulo sopfr(n), where sopfr(n) is the sum of the prime factors of n, with multiplicity.

0, 0, 0, 0, 1, 0, 2, 3, 3, 0, 5, 0, 5, 7, 0, 0, 2, 0, 2, 1, 9, 0, 6, 5, 11, 0, 6, 0, 0, 0, 2, 5, 15, 11, 6, 0, 17, 7, 7, 0, 6, 0, 14, 1, 21, 0, 4, 7, 2, 11, 1, 0, 10, 7, 4, 13, 27, 0, 0, 0, 29, 11, 4, 11, 2, 0, 5, 17, 0, 0, 0, 0, 35, 10, 7, 5, 6, 0, 2, 9, 39

Offset: 2

J. Stauduhar, Feb 28 2014

a(A036844(n)) = 0. - Reinhard Zumkeller, Jul 21 2014

			a(6) = 1, because 6 mod sopfr(6) = 6 mod 5 = 1.

Reinhard Zumkeller, Table of n, a(n) for n = 2..10000

Cf. A001414, A238526, A238527, A238528, A238529, A238530.

Cf. A036844.

a238525 n = mod n $ a001414 n  -- Reinhard Zumkeller, Jul 21 2014

Table[Mod[n, Apply[Dot, Transpose[FactorInteger[n]]]], {n, 105}] (* Wouter Meeussen, Mar 01 2014 *)
mms[n_]:=Mod[n,Total[Flatten[Table[#[[1]],#[[2]]]&/@FactorInteger[ n]]]]; Array[mms,90,2] (* Harvey P. Dale, May 25 2016 *)

a(n) = n mod A001414(n).

A238527 Prime values of A238525.

Original entry on oeis.org

2, 3, 3, 5, 5, 7, 2, 2, 5, 11, 2, 5, 11, 17, 7, 7, 7, 2, 11, 7, 13, 29, 11, 11, 2, 5, 17, 7, 5, 2, 19, 41, 23, 3, 11, 11, 23, 5, 2, 2, 2, 31, 7, 3, 17, 3, 23, 11, 59, 19, 5, 2, 37, 3, 41, 23, 71, 11, 2, 11, 47, 11, 29, 13, 13, 2, 31, 5, 5, 53, 17, 19, 13, 61, 23, 101, 19, 29, 41, 23, 107, 67, 11, 3, 3, 47, 73, 2, 3, 17

Offset: 1

J. Stauduhar, Feb 28 2014

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A238525, A238526, A238528.

Select[Table[Mod[n, Plus @@ Times @@@ FactorInteger[n]], {n, 2, 300}], PrimeQ] (* Amiram Eldar, May 17 2021 *)

A238528 Record prime values of A238525.

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, 461, 521, 569, 599, 617, 641, 659, 809, 821, 827, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1289, 1301, 1319, 1427, 1451

Offset: 1

J. Stauduhar, Feb 28 2014

nonn

The prime values in A238526.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A238525, A238526, A238527.

Union @ FoldList[Max, Select[Table[Mod[n, Plus @@ Times @@@ FactorInteger[n]], {n, 2, 3000}], PrimeQ]] (* Amiram Eldar, May 17 2021 *)

A114338 Number of divisors of n!! (double factorial = A006882(n)).

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 10, 8, 16, 16, 36, 32, 66, 64, 144, 120, 192, 240, 340, 480, 570, 864, 1200, 1728, 1656, 2880, 3456, 4320, 5616, 8640, 9072, 17280, 10752, 28800, 22176, 46080, 30240, 92160, 62208, 152064, 84240, 304128, 128000, 608256, 201600

Offset: 0

Giovanni Resta, Feb 07 2006

It appears that a(n+2) = 2*a(n) if n is in A238526. - Michel Lagneau, Dec 07 2015

			a(5) = 4 since 5!! = 15 and the divisors are 1, 3, 5 and 15.
a(6) = 10 because 6!! = A006882(6) = 48 has precisely ten distinct divisors: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. - _Michel Lagneau_, Dec 07 2016

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A000005, A006882, A027423, A238526.

f := proc(n)
numtheory[tau](doublefactorial(n)) ;
end proc: # R. J. Mathar, Dec 14 2015

Mathematica
```
DivisorSigma[0,Range[50]!! ]
```

df(n) = if( n<0, 0, my(E); E = exp(x^2 / 2 + x * O(x^n)); n! * polcoeff( 1 + E * x * (1 + intformal(1 / E)), n)); \\ A006882
vector(100, n, n--; numdiv(df(n))) \\ Altug Alkan, Dec 07 2015

a(n) = sigma_0(n!!) = tau(n!!) = A000005(A006882(n)).

A238621 Position of first occurrence of n in A238525.

Original entry on oeis.org

1, 6, 8, 9, 48, 12, 24, 15, 120, 22, 54, 26, 90, 57, 44, 34, 156, 38, 114, 85, 228, 46, 232, 87, 348, 93, 138, 58, 318, 62, 372, 111, 333, 265, 354, 74, 366, 129, 296, 82, 369, 86, 402, 667, 387, 94, 328, 159, 438, 244, 530, 106, 423, 177, 474, 183, 1416, 118, 498, 122, 742, 201, 590, 415, 534, 134, 610, 219

Offset: 0

Robert G. Wilson v, Mar 01 2014

Robert G. Wilson v, Table of n, a(n) for n = 0..1000

Cf. A238525, A238526, A238527, A238528, A238529.

f[n_] := Mod[n, Dot @@ Transpose@ FactorInteger@ n]; t = Table[0, {1000}]; k = 1; While[k < 100001, a = f[k]; If[a < 1001 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++]; t

cp's OEIS Frontend

A238525 n modulo sopfr(n), where sopfr(n) is the sum of the prime factors of n, with multiplicity.

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A238527 Prime values of A238525.

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A238528 Record prime values of A238525.

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A114338 Number of divisors of n!! (double factorial = A006882(n)).

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A238621 Position of first occurrence of n in A238525.

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