A008472 -id:A008472 - cp's OEIS frontend

A076383 Numbers n such that sum of digits in base 5 is a divisor of sum of prime divisors (A008472).

2, 3, 5, 21, 25, 27, 30, 35, 42, 78, 105, 110, 115, 123, 125, 126, 130, 132, 141, 150, 153, 155, 159, 161, 170, 175, 186, 187, 195, 201, 228, 230, 231, 252, 258, 260, 264, 276, 290, 301, 327, 329, 340, 357, 372, 378, 381, 395, 396, 402, 410, 411, 429, 434

Offset: 1

Floor van Lamoen, Oct 08 2002

Cf. A075657, A076381 - A076387.

A076383 := proc(n) local i,j,t,t1, sod, sopd; t := NULL; for i from 2 to n do t1 := i; sod := 0; while t1 <> 0 do sod := sod + (t1 mod 5); t1 := floor(t1/5); od; sopd := 0; j := 1; while ithprime(j) <= i do if i mod ithprime(j) = 0 then sopd := sopd+ithprime(j); fi; j := j+1; od; if sopd mod sod = 0 then t := t,i; fi; od; t; end;

A076384 Numbers n such that sum of digits in base 6 is a divisor of sum of prime divisors (A008472).

Original entry on oeis.org

2, 3, 5, 6, 25, 30, 36, 38, 39, 42, 60, 78, 84, 90, 106, 114, 120, 122, 126, 130, 150, 152, 156, 171, 178, 180, 183, 186, 187, 194, 198, 216, 217, 218, 219, 221, 222, 228, 230, 240, 244, 252, 255, 258, 259, 260, 262, 264, 270, 287, 294, 297, 299, 300, 303, 321

Offset: 1

Floor van Lamoen, Oct 08 2002

Cf. A075657, A076380 - A076387.

A076384 := proc(n) local i,j,t,t1, sod, sopd; t := NULL; for i from 2 to n do t1 := i; sod := 0; while t1 <> 0 do sod := sod + (t1 mod 6); t1 := floor(t1/6); od; sopd := 0; j := 1; while ithprime(j) <= i do if i mod ithprime(j) = 0 then sopd := sopd+ithprime(j); fi; j := j+1; od; if sopd mod sod = 0 then t := t,i; fi; od; t; end;

Select[Range[2,350],Divisible[Total[Transpose[FactorInteger[#]][[1]]], Total[ IntegerDigits[#,6]]]&] (* Harvey P. Dale, May 26 2013 *)

A076385 Numbers n such that sum of digits in base 7 is a divisor of sum of prime divisors (A008472).

Original entry on oeis.org

2, 3, 5, 7, 8, 9, 42, 49, 78, 84, 105, 114, 115, 126, 130, 154, 156, 161, 168, 170, 186, 228, 235, 252, 258, 294, 305, 336, 343, 350, 357, 366, 371, 372, 378, 402, 410, 425, 429, 430, 434, 442, 444, 455, 456, 460, 474, 504, 516, 520, 555, 558, 574, 588, 616

Offset: 1

Floor van Lamoen, Oct 08 2002

Cf. A075657, A076380 - A076387.

A076385 := proc(n) local i,j,t,t1, sod, sopd; t := NULL; for i from 2 to n do t1 := i; sod := 0; while t1 <> 0 do sod := sod + (t1 mod 7); t1 := floor(t1/7); od; sopd := 0; j := 1; while ithprime(j) <= i do if i mod ithprime(j) = 0 then sopd := sopd+ithprime(j); fi; j := j+1; od; if sopd mod sod = 0 then t := t,i; fi; od; t; end;

A076386 Numbers n such that sum of digits in base 8 is a divisor of sum of prime divisors (A008472).

Original entry on oeis.org

2, 3, 5, 7, 8, 12, 15, 16, 26, 49, 64, 65, 70, 86, 96, 102, 123, 128, 130, 140, 150, 156, 201, 208, 209, 215, 225, 247, 258, 266, 280, 286, 299, 305, 326, 350, 356, 360, 403, 424, 456, 471, 474, 490, 495, 512, 513, 515, 519, 520, 530, 532, 545, 551, 555, 558

Offset: 1

Floor van Lamoen, Oct 08 2002

Cf. A075657, A076380 - A076387.

A076386 := proc(n) local i,j,t,t1, sod, sopd; t := NULL; for i from 2 to n do t1 := i; sod := 0; while t1 <> 0 do sod := sod + (t1 mod 8); t1 := floor(t1/8); od; sopd := 0; j := 1; while ithprime(j) <= i do if i mod ithprime(j) = 0 then sopd := sopd+ithprime(j); fi; j := j+1; od; if sopd mod sod = 0 then t := t,i; fi; od; t; end;

A082881 Least value of A075860(j) when j runs through composite numbers between n-th and (n+1)-th primes. That is, the smallest fixed-point[=prime] reached by iteration of function A008472(=sum of prime factors) initiated with composite values between two consecutive primes.

Original entry on oeis.org

0, 2, 5, 2, 5, 2, 5, 7, 2, 7, 2, 2, 5, 2, 2, 2, 7, 2, 2, 5, 2, 3, 2, 5, 3, 13, 2, 5, 3, 2, 2, 2, 3, 2, 7, 5, 3, 13, 2, 3, 7, 2, 5, 3, 2, 2, 2, 2, 5, 7, 2, 7, 2, 2, 2, 2, 7, 2, 3, 2, 2, 2, 2, 5, 2, 2, 5, 2, 19, 2, 2, 2, 5, 2, 2, 3, 2, 3, 2, 2, 17, 2, 5, 5, 2, 2, 2, 7, 23, 2, 2, 3, 3, 3, 5, 2, 2, 19, 2, 5, 2, 3, 2

Offset: 1

Labos Elemer, Apr 16 2003

nonn

			Between p(23)=83 and p(24)=89, the relevant fixed points are {5,13,2,2,13}, of which the smallest is 2=a(24).

Cf. A075860, A008472, A082087, A082088, A082880, A029908.

ffi[x_] := Flatten[FactorInteger[x]]; lf[x_] := Length[FactorInteger[x]]; ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}]; sopf[x_] := Apply[Plus, ba[x]]; Table[Min[Table[FixedPoint[sopf, w], {w, 1+Prime[n], Prime[n+1]-1}]], {n, 2, 103}]

a(n) = Min_{x=1+prime(n)..prime(n+1)-1} A075860(x).

A082882 Number of distinct values of A075860(j) when j runs through composite numbers between n-th and (n+1)-th primes. That is, the counts of different fixed-points[=prime] reached by iteration of function A008472(=sum of prime factors) initiated with composite values between two consecutive primes.

Original entry on oeis.org

0, 1, 1, 3, 1, 2, 1, 2, 3, 1, 4, 2, 1, 3, 3, 5, 1, 4, 3, 1, 3, 3, 3, 3, 2, 1, 1, 1, 3, 8, 3, 2, 1, 6, 1, 2, 3, 3, 3, 5, 1, 5, 1, 2, 1, 7, 4, 2, 1, 2, 4, 1, 5, 3, 4, 4, 1, 5, 3, 1, 6, 6, 2, 1, 2, 7, 3, 4, 1, 3, 4, 6, 3, 3, 3, 4, 6, 3, 5, 5, 1, 6, 1, 3, 3, 4, 5, 1, 1, 2, 6, 4, 3, 4, 3, 2, 6, 1, 8, 3, 6, 4, 5, 1, 4

Offset: 1

Labos Elemer, Apr 16 2003

nonn

This count is smaller than A001223[n]-1 and albeit not frequently but it can be one even if primes of borders are not twin primes. Such primes are collected into A082883.

			Between p(23)=83 and p(24)=89, the relevant fixed points are
{5,13,2,2,13}, i.e., four are distinct from the 5 values, a(24)=4;
between p(2033)=17707 and p(2034)=170713, the fixed-point set is {5,5,5,5,5}, so a(2033)=1, so a(88)=1.

Cf. A075860, A008472, A082087, A082088, A082880, A082081, A001223.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] sopf[x_] := Apply[Plus, ba[x]] Table[Length[Union[Table[FixedPoint[sopf, w], {w, 1+Prime[n], Prime[n+1]-1}]]], {n, 1, 1000}]

a(n) = Card(Union(A075860(x)); x=1+p(n), ..., -1+p(n+1)).

A139316 An integer k, k>=2, is in the sequence if A001222(k) (the sum of the exponents in the prime factorization of k) divides A008472(k) (the sum of the distinct primes dividing k).

Original entry on oeis.org

2, 3, 4, 5, 7, 11, 13, 15, 17, 19, 21, 23, 27, 28, 29, 31, 33, 35, 37, 39, 41, 42, 43, 47, 48, 51, 52, 53, 55, 57, 59, 61, 65, 67, 69, 71, 72, 73, 76, 77, 78, 79, 83, 84, 85, 87, 89, 91, 93, 95, 97, 98, 101, 103, 105, 107, 108, 109, 110, 111, 113, 114, 115, 119, 120, 123

Offset: 1

Leroy Quet, Jun 07 2008

nonn

			28 has the prime factorization 2^2 * 7^1. The sum of the exponents, 2+1 = 3, divides the sum of the distinct prime divisors, 2+7 = 9. So 28 is in the sequence.

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

Cf. A001222, A008472.

seddpQ[n_]:=Module[{fi=Transpose[FactorInteger[n]]},Divisible[Total[ fi[[1]]],Total[ fi[[2]]]]]; Select[Range[2,150],seddpQ] (* Harvey P. Dale, Apr 13 2015 *)

More terms from D. S. McNeil, Mar 23 2009

A173338 Numbers n such that tau(tau(n)) = sopf(sopf(n)), sopf = A008472.

Original entry on oeis.org

2, 4, 14, 16, 27, 64, 158, 168, 196, 216, 312, 378, 384, 440, 456, 482, 546, 680, 702, 744, 770, 1024, 1026, 1032, 1160, 1454, 1608, 1640, 1674, 1880, 2024, 2058, 2295, 2322, 2472, 2750, 2805, 2944, 3336, 3560, 3608, 3618, 3768, 3828, 3944, 3960, 4040, 4096

Offset: 1

Michel Lagneau, Feb 16 2010

nonn

sopf(n) is the sum of distinct primes dividing n (without repetition, A008472), tau(n) is the number of divisors of n (A000005).

			4 is in the sequence: tau(4) = 3, tau(3) = 2; sopf(4) = 2, sopf(2) = 2.
546 is in the sequence: tau(546) = 16, tau(16) = 5; sopf(546) = 25, sopf(25) = 5.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1919), 75-113.

Cf. A000005, A008472, A010553.

f:=func; g:=func; [k:k in [2..5000]|f(f(k)) eq g(g(k)) ]; // Marius A. Burtea, Nov 14 2019

with(numtheory): for n from 1 to 60000 do : t1:= ifactors(n)[2] : t2 :=sum(t1[i][1], i=1..nops(t1)): tt1:= ifactors(t2)[2] : tt2 :=sum(tt1[i][1], i=1..nops(tt1)):if tau(tau(n))= tt2 then print (n): else fi : od :
# second Maple program:
with(numtheory): sopf:= n-> add(i, i=factorset(n)):
a:= proc(n) option remember; local k;
      for k from 1+ `if`(n=1, 0, a(n-1))
      while tau(tau(k)) <> sopf(sopf(k)) do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 26 2010

Select[Range[4100],DivisorSigma[0,DivisorSigma[0,#]]==Total[ Transpose[ FactorInteger[ Total[Transpose[FactorInteger[#]][[1]]]]][[1]]]&] (* Harvey P. Dale, Aug 05 2013 *)

{ n : A010553(n) = A008472(A008472(n)) }.

Corrected and edited by Michel Lagneau, Apr 25 2010

A175704 Auto-convolution of A008472.

Original entry on oeis.org

0, 0, 4, 12, 17, 32, 54, 78, 95, 102, 149, 188, 213, 254, 360, 408, 463, 488, 617, 636, 769, 784, 1111, 1060, 1231, 1172, 1505, 1408, 1756, 1574, 1990, 2060, 2451, 2096, 3099, 2500, 2729, 2862, 3683, 3368, 4166, 3902, 4523, 4396, 4987, 4424, 6579, 5228

Offset: 1

Michel Lagneau, Aug 10 2010

nonn

A008472 is the sum of the distinct primes dividing n.

A175704 := proc(n)
        add(A008472(k)*A008472(n+1-k),k=1..n) ;
end proc: # R. J. Mathar, Jul 10 2012

a(n) = Sum_{k=1..n} A008472(k)*A008472(n+1-k).

A176147 a(n) = n^sopf(n), where sopf(n) is the sum of the distinct primes dividing n (A008472).

Original entry on oeis.org

1, 4, 27, 16, 3125, 7776, 823543, 64, 729, 10000000, 285311670611, 248832, 302875106592253, 20661046784, 2562890625, 256, 827240261886336764177, 1889568, 1978419655660313589123979, 1280000000, 16679880978201

Offset: 1

Michel Lagneau, Apr 10 2010

nonn

			a(1) = 1^0 = 1;
a(2) = 2^2 = 4;
a(3) = 3^3 = 27;
a(4) = 4^2 = 16.

T. D. Noe, Table of n, a(n) for n = 1..100

Cf. A008472.

Table[n^Total[Transpose[FactorInteger[n]][[1]]], {n, 20}]

a(n) = n^A008472(n).

cp's OEIS Frontend

A076383 Numbers n such that sum of digits in base 5 is a divisor of sum of prime divisors (A008472).

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A076384 Numbers n such that sum of digits in base 6 is a divisor of sum of prime divisors (A008472).

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A076385 Numbers n such that sum of digits in base 7 is a divisor of sum of prime divisors (A008472).

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A076386 Numbers n such that sum of digits in base 8 is a divisor of sum of prime divisors (A008472).

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A082881 Least value of A075860(j) when j runs through composite numbers between n-th and (n+1)-th primes. That is, the smallest fixed-point[=prime] reached by iteration of function A008472(=sum of prime factors) initiated with composite values between two consecutive primes.

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A139316 An integer k, k>=2, is in the sequence if A001222(k) (the sum of the exponents in the prime factorization of k) divides A008472(k) (the sum of the distinct primes dividing k).

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A173338 Numbers n such that tau(tau(n)) = sopf(sopf(n)), sopf = A008472.

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Magma

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A175704 Auto-convolution of A008472.

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Maple