A008472 -id:A008472 - cp's OEIS frontend

A064111 Numbers k such that sopf(k) + 1 = sopf(k+1), where sopf(k) = A008472(k).

2, 8, 120, 168, 175, 247, 860, 1044, 1444, 1659, 1849, 3626, 3834, 4233, 4300, 4345, 4814, 6867, 8240, 14905, 23287, 24476, 28919, 29087, 29464, 30457, 30650, 33725, 34945, 35585, 37214, 49468, 52206, 54900, 58113, 62049, 63440, 65631, 68264

Offset: 1

Jason Earls, Sep 08 2001

nonn

Also k such that z(k) = z(k+1) where z(k) = k - sopf(k).

Prime factors counted without multiplicity. - Harvey P. Dale, Dec 26 2015

			sopf(8) + 1 = 3, sopf(8 + 1) = 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A006145, A008472.

Flatten[Position[Partition[Table[Total[Transpose[FactorInteger[n]] [[1]]], {n, 2,70000}],2,1],?(#[[1]]+1==#[[2]]&),{1},Heads->False]]+1 (* _Harvey P. Dale, Dec 26 2015 *)

sopf(n,s,fac,i)=fac=factor(n); for(i=1,matsize(fac)[1],s=s+fac[i,1]); return(s);
j=[]; for(n=1,100000, if(sopf(n)+1==sopf(n+1), j=concat(j,n))); j

z(n)= { local(f,s=0); f=factor(n); for(i=1, matsize(f)[1], s+=f[i, 1]); return(n - s) }
{ n=0; zm=z(1); for (m=1, 10^9, zp=z(m + 1); if (zm==zp, write("b064111.txt", n++, " ", m); if (n==1000, break)); zm=zp ) } \\ Harry J. Smith, Sep 07 2009

A076381 Numbers n such that sum of digits in base 3 is a divisor of sum of prime divisors (A008472).

Original entry on oeis.org

2, 3, 4, 9, 25, 27, 30, 42, 51, 66, 78, 81, 84, 90, 105, 114, 126, 138, 141, 147, 153, 156, 159, 168, 170, 185, 186, 187, 198, 201, 220, 222, 228, 231, 234, 243, 245, 246, 252, 258, 264, 270, 276, 282, 290, 291, 294, 301, 312, 315, 322, 323, 325, 336, 340, 341

Offset: 1

Floor van Lamoen, Oct 08 2002

Cf. A075657, A076380 - A076387.

A076381 := 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 3); t1 := floor(t1/3); 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[2,400],Divisible[Total[FactorInteger[#][[All,1]]],Total[ IntegerDigits[ #,3]]]&] (* Harvey P. Dale, Jul 09 2018 *)

A081382 a(1) = 1, for n > 1 a(n) = Min{x > n, A008472(x) = A008472(n)}.

Original entry on oeis.org

1, 4, 9, 8, 6, 12, 10, 16, 27, 20, 121, 18, 22, 28, 45, 32, 210, 24, 34, 40, 30, 44, 273, 25, 36, 52, 81, 56, 399, 60, 58, 64, 70, 68, 42, 48, 435, 76, 55, 49, 651, 84, 82, 88, 75, 92, 777, 54, 50, 80, 91, 104, 903, 72, 66, 98, 85, 116, 1645, 63, 118, 124, 90, 128, 77, 117

Offset: 1

Labos Elemer, Mar 26 2003

nonn

Differs from A065642, a(n) <= A065642(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A065642, A008472.

a081382 1 = 1
a081382 n = head [x | let sopf = a008472 n, x <- [n+1..], a008472 x == sopf]
-- Reinhard Zumkeller, Jun 12 2015

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] spf[x_] := Apply[Plus, ba[x]] Table[Min[Flatten[Position[Table[spf[w], {w, n+1, n^2}]-spf[n], 0]]+n], {n, 1, 100}]

A082880 Largest value of A075860(j) when j runs through composite numbers between n-th and (n+1)-th primes. That is, the largest fixed-point[=prime] reached by iteration of A008472(=sum of prime factors) initiated with composite values between two consecutive primes.

Original entry on oeis.org

0, 2, 5, 7, 5, 3, 5, 13, 5, 7, 19, 7, 5, 13, 7, 31, 7, 7, 19, 5, 7, 43, 13, 19, 7, 13, 2, 5, 7, 61, 7, 19, 3, 73, 7, 7, 7, 43, 13, 19, 7, 13, 5, 7, 2, 103, 109, 3, 5, 31, 61, 7, 13, 19, 13, 31, 7, 139, 19, 2, 73, 151, 7, 5, 3, 43, 13, 31, 19, 13, 181, 19, 13, 7, 193, 23, 199, 29, 103, 73

Offset: 1

Labos Elemer, Apr 16 2003

nonn

Cf. A075860, A008472, A082087, A082088, A082881, 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[Max[0, Table[FixedPoint[sopf, w], {w, 1+Prime[n], Prime[n+1]-1}]], {n, 80}]

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

A340677 a(n) = A007947(n) / gcd(A007947(n), A008472(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 10, 1, 6, 1, 14, 15, 1, 1, 6, 1, 10, 21, 22, 1, 6, 1, 26, 1, 14, 1, 3, 1, 1, 33, 34, 35, 6, 1, 38, 39, 10, 1, 7, 1, 22, 15, 46, 1, 6, 1, 10, 51, 26, 1, 6, 55, 14, 57, 58, 1, 3, 1, 62, 21, 1, 65, 33, 1, 34, 69, 5, 1, 6, 1, 74, 15, 38, 77, 13, 1, 10, 1, 82, 1, 7, 85, 86, 87, 22, 1, 3, 91, 46, 93

Offset: 1

Antti Karttunen, Feb 01 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000961 (positions of ones), A006881, A007947, A008472, A099636, A340678.

Cf. also A082344.

A007947(n) = factorback(factorint(n)[, 1]);
A008472(n) = vecsum(factor(n)[, 1]);
A340677(n) = { my(u=A007947(n)); (u/gcd(u, A008472(n))); };

a(n) = A007947(n) / A099636(n) = A007947(n) / gcd(A007947(n), A008472(n)).
a(n) = 1 iff n is power of prime (A000961). - Bernard Schott, Feb 01 2021
a(A006881(n)) = A006881(n). - Bernard Schott and Antti Karttunen, Feb 01 2021

A340678 a(n) = A008472(n) / gcd(A007947(n), A008472(n)).

Original entry on oeis.org

0, 1, 1, 1, 1, 5, 1, 1, 1, 7, 1, 5, 1, 9, 8, 1, 1, 5, 1, 7, 10, 13, 1, 5, 1, 15, 1, 9, 1, 1, 1, 1, 14, 19, 12, 5, 1, 21, 16, 7, 1, 2, 1, 13, 8, 25, 1, 5, 1, 7, 20, 15, 1, 5, 16, 9, 22, 31, 1, 1, 1, 33, 10, 1, 18, 8, 1, 19, 26, 1, 1, 5, 1, 39, 8, 21, 18, 3, 1, 7, 1, 43, 1, 2, 22, 45, 32, 13, 1, 1, 20, 25, 34, 49, 24, 5, 1, 9

Offset: 1

Antti Karttunen, Feb 01 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A007947, A008472, A099636, A340677.

Cf. also A082343.

A007947(n) = factorback(factorint(n)[, 1]);
A008472(n) = vecsum(factor(n)[, 1]);
A340678(n) = { my(v=A008472(n)); (v/gcd(A007947(n), v)); };

a(n) = A008472(n) / A099636(n) = A008472(n) / gcd(A007947(n), A008472(n)).

A365059 a(1) = 2; for n > 2, a(n) is the smallest positive number that has not yet appeared that is a multiple of A008472(a(n-1)), the sum of the distinct primes dividing a(n-1).

Original entry on oeis.org

2, 4, 6, 5, 10, 7, 14, 9, 3, 12, 15, 8, 16, 18, 20, 21, 30, 40, 28, 27, 24, 25, 35, 36, 45, 32, 22, 13, 26, 60, 50, 42, 48, 55, 64, 34, 19, 38, 63, 70, 56, 54, 65, 72, 75, 80, 49, 77, 90, 100, 84, 96, 85, 44, 39, 112, 81, 33, 98, 99, 126, 108, 95, 120, 110, 144, 105, 135, 88, 52, 150, 130, 140

Offset: 1

Scott R. Shannon, Aug 19 2023

nonn

In the first 500000 terms the only fixed points are 38 and 209, although it is likely more exist. In the same range the smallest missing numbers are 311, 313, 337. The sequence is conjectured to be a permutation of the integers >= 2.

			a(3) = 6 as a(2) = 4 and A008472(4) = 2, and 6 is the smallest unused number that is a multiple of 2.
a(11) = 15 as a(10) = 12 and A008472(12) = 5, and 15 is the smallest unused number that is a multiple of 5.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 500000 terms. The green line is a(n) = n.

Cf. A008472, A365060, A300813.

A365060 a(1) = 2; for n > 2, a(n) is the smallest positive number that has not yet appeared that has a common factor with A008472(a(n-1)), the sum of the distinct primes dividing a(n-1).

Original entry on oeis.org

2, 4, 6, 5, 10, 7, 14, 3, 9, 12, 15, 8, 16, 18, 20, 21, 22, 13, 26, 24, 25, 30, 28, 27, 33, 32, 34, 19, 38, 35, 36, 40, 42, 39, 44, 52, 45, 46, 50, 49, 56, 48, 55, 54, 60, 58, 31, 62, 11, 66, 64, 68, 57, 70, 63, 65, 51, 72, 75, 74, 69, 76, 77, 78, 80, 84, 81, 87, 82, 43, 86, 85, 88, 91, 90, 92, 95

Offset: 1

Scott R. Shannon, Aug 19 2023

nonn

In the first 100000 terms the only fixed point is 9, and it is likely no more exist. In the same range the smallest missing numbers are 503, 839, 877. The sequence is conjectured to be a permutation of the integers >= 2.

			a(3) = 6 as a(2) = 4 and A008472(4) = 2, and 6 is the smallest unused number that shares a factor with 2.
a(8) = 3 as a(7) = 14 and A008472(14) = 9, and 3 is the smallest unused number that shares a factor with 9.

Scott R. Shannon, Table of n, a(n) for n = 1..10000.
Scott R. Shannon, Image of the first 100000 terms. The green line is a(n) = n.

Cf. A008472, A365059, A300813.

A058974 a(n) = 0 if n = 1 or a prime, otherwise a(n) = s + a(s) iterated until no change occurs, where s (A008472) is sum of distinct primes dividing n.

Original entry on oeis.org

0, 0, 0, 2, 0, 5, 0, 2, 3, 7, 0, 5, 0, 12, 10, 2, 0, 5, 0, 7, 17, 13, 0, 5, 5, 25, 3, 12, 0, 17, 0, 2, 26, 19, 17, 5, 0, 38, 18, 7, 0, 17, 0, 13, 10, 30, 0, 5, 7, 7, 27, 25, 0, 5, 18, 12, 35, 31, 0, 17, 0, 59, 17, 2, 23, 18, 0, 19, 51, 26, 0, 5, 0, 57, 10, 38, 23, 23, 0, 7, 3, 43, 0, 17, 35, 55, 34, 13, 0

Offset: 1

N. J. A. Sloane, Jan 15 2001

nonn

E. N. Gilbert, An interesting property of 38, unpublished, circa 1992. Shows that 38 is the only solution of a(n) = n.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A008472, A061376.

f := proc(n) option remember; local i,j,k,t1,t2; if n = 1 or isprime(n) then 0 else A008472(n) + f(A008472(n)); fi; end;

f[n_Integer] := If[n == 1 || PrimeQ[n], 0, Plus @@ First[ Transpose[ FactorInteger[n]]]]; Table[Plus @@ Drop[ FixedPointList[f, n], 1], {n, 1, 80}]

A008472(n) = vecsum(factor(n)[, 1]); \\ This function from M. F. Hasler, Jul 18 2015
A058974(n) = if((1==n)||isprime(n),0,A008472(n)+A058974(A008472(n))); \\ Antti Karttunen, Oct 30 2017, after the Maple-program.

More terms from Antti Karttunen, Oct 30 2017

A063968 Numbers k such that sopf(k) = sopf(k+2), where sopf(k) = A008472(k).

Original entry on oeis.org

2, 340, 845, 950, 1340, 3724, 5694, 6102, 7657, 8991, 9331, 9709, 10323, 11388, 11390, 12460, 15870, 18912, 19778, 20882, 21715, 24732, 26978, 29052, 29632, 32428, 33596, 35028, 38178, 42718, 43068, 45750, 46102, 50396, 53251, 61408

Offset: 1

Jason Earls, Sep 05 2001

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A008472, A006145.

sopf(n,s,fac,i)=fac=factor(n); for(i=1,matsize(fac)[1],s=s+fac[i,1]); return(s);
j=[]; for(n=1,100000, if(sopf(n)==sopf(n+2),j=concat(j,n))); j

sopf(n)= { local(f,s=0); f=factor(n); for(i=1, matsize(f)[1], s+=f[i, 1]); return(s) }
{ n=0; r=sopf(1); s=sopf(2); for (m=1, 10^9, t=sopf(m + 2); if(r==t, write("b063968.txt", n++, " ", m); if (n==1000, break)); r=s; s=t ) } \\ Harry J. Smith, Sep 04 2009

cp's OEIS Frontend

A064111 Numbers k such that sopf(k) + 1 = sopf(k+1), where sopf(k) = A008472(k).

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A076381 Numbers n such that sum of digits in base 3 is a divisor of sum of prime divisors (A008472).

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A081382 a(1) = 1, for n > 1 a(n) = Min{x > n, A008472(x) = A008472(n)}.

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A082880 Largest value of A075860(j) when j runs through composite numbers between n-th and (n+1)-th primes. That is, the largest fixed-point[=prime] reached by iteration of A008472(=sum of prime factors) initiated with composite values between two consecutive primes.

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A340677 a(n) = A007947(n) / gcd(A007947(n), A008472(n)).

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A340678 a(n) = A008472(n) / gcd(A007947(n), A008472(n)).

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A365059 a(1) = 2; for n > 2, a(n) is the smallest positive number that has not yet appeared that is a multiple of A008472(a(n-1)), the sum of the distinct primes dividing a(n-1).

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A365060 a(1) = 2; for n > 2, a(n) is the smallest positive number that has not yet appeared that has a common factor with A008472(a(n-1)), the sum of the distinct primes dividing a(n-1).

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A058974 a(n) = 0 if n = 1 or a prime, otherwise a(n) = s + a(s) iterated until no change occurs, where s (A008472) is sum of distinct primes dividing n.

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