A008472 -id:A008472 - cp's OEIS frontend

A182184 a(1)= 0; for n > 1, a(n) = a(n-1) - sopf(n) if that number is positive, otherwise a(n)= a(n-1)+ sopf(n), where sopf(n) is the sum of the distinct primes dividing n (A008472).

0, 2, 5, 3, 8, 3, 10, 8, 5, 12, 1, 6, 19, 10, 2, 0, 17, 12, 31, 24, 14, 1, 24, 19, 14, 29, 26, 17, 46, 36, 5, 3, 17, 36, 24, 19, 56, 35, 19, 12, 53, 41, 84, 71, 63, 38, 85, 80, 73, 66, 46, 31, 84, 79, 63, 54, 32, 1, 60, 50, 111, 78, 68, 66, 48, 32, 99, 80, 54, 40

Offset: 1

Michel Lagneau, Apr 17 2012

nonn

a(n) = 0 for n = 16, 222, 4298, 16652, 37783, 256263, 414054, 530270, ...

Michel Lagneau, Table of n, a(n) for n = 1..10000

Cf. A008472, A185112.

with(numtheory):a0:=0:for n from 2 to 100 do: x:=factorset(n):n1:=nops(x): p:=sum(x[i],i=1..n1):a1:=a0-p:if a1< 0 then a1:=a0+p:else fi: printf(`%d, `, a1): a0:=a1: od:

A242404 a(n) = Sum over i and j with j<=i and i=1.. sopf(n) of binomial(d(i), d(j)), where d(i) are the prime divisors of n and sopf(n) = sum of the distinct primes dividing n (A008472).

Original entry on oeis.org

0, 1, 1, 1, 1, 5, 1, 1, 1, 12, 1, 5, 1, 23, 12, 1, 1, 5, 1, 12, 37, 57, 1, 5, 1, 80, 1, 23, 1, 26, 1, 1, 167, 138, 23, 5, 1, 173, 288, 12, 1, 62, 1, 57, 12, 255, 1, 5, 1, 12, 682, 80, 1, 5, 464, 23, 971, 408, 1, 26, 1, 467, 37, 1, 1289, 226, 1, 138, 1773, 55

Offset: 1

Michel Lagneau, May 13 2014

nonn

This sequence is very rich in properties. We cite a few.
If n = p^m where p is prime, then a(n) = 1 => a(A025475(n)) = 1.
If n = 2^i*3^j, i and j >= 1, then a(n) = 5 => a(A033845(n)) = 5.
We observe a very interesting property with a fractal structure if a(n) = 23.
a(n) = 23 if n belongs to the set E = {n} = {14, 28, 35, 56, 98, 112, 175, 196, 224, 245, 392, 448, 686, ...} and we observe that E1 = {n/7} = {2, 4, 5, 8, 14, 16, 25, 28, 32, 35, 56, 64, 98, 112, 125, 128, 175, 196, 224, 245, 256, ...}  = E2 union E3 union E4 where:
E2 = {2, 4, 8, 16, 32, 64, 128, ...} are powers of 2 (A000079),
E3 = {5, 25, 125, 625, 3125, ...} are powers of 5 (A000351),
E4 = {14, 28, 35, 56, 98, 112, 175, 196, 224, 245, ...} = E, a surprising result: E1 contains E => this shows a fractal structure.
But you can continue with other values of a(n) in order to find similar properties. For example, a(n) = 55 if n belongs to the set F = {n} = {70, 140, 280, 350, 490, 560, 700, 980, 1120, ...} => F1 = {n/70} = { 1, 2, 4, 5, 7, 8, 10, 14, 16, 20, 25, 28, 32, 35, 40, 49, 50, 56, 64, 70, 80, ...} = F2 union F3 union F4 union F5 where:
F2 = {2, 4, 8, 16, 32, 64, 128, ...} are powers of 2 (A000079),
F3 = {5, 25, 125, 625, 3125, ...} are powers of 5 (A000351),
F4 = {1, 7, 49, 343, 2401, ...} are powers of 7 (A000420)
and F5 = {10, 14, 20, 28, 35, 40, 50, 56, 70, 80, 98, 100, 112, 140, ...} contains F but also E.

			a(10) = 12 because 12 = 2^2*3 => sopf(12) = 2+3=5 and a(10) = sum{j<=i=1..5} binomial(d(i),d(j)) = binomial(2,2)+ binomial(5,2)+ binomial(5,5) = 12.

Michel Lagneau, Table of n, a(n) for n = 1..10000

Cf. A000079, A000351, A000420, A008472.

with(numtheory):for n from 1 to 200 do:x:=factorset(n):n1:=nops(x): r:=sum('sum('binomial(x[j],x[i])','j'=i..n1)','i'=1..n1):printf(`%d, `,r):od:

A256122 Number of iterations needed to reach 0 or 1 under the map n-> n-sopf(n), where sopf(n) is the sum of the distinct primes dividing n (A008472).

Original entry on oeis.org

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

Offset: 1

Michel Lagneau, Mar 15 2015

nonn

			a(16) = 3 because:
16 - sopf(16) = 16 - 2 = 14 (first iteration);
14 - sopf(14) = 14 - 9 = 5 (second iteration);
5 - sopf(5)= 5 - 5 = 0 (third iteration and reaching 0).
a(22) = 3 because:
22 - sopf(22) = 22 - 13 = 9 (first iteration);
9 - sopf(9) = 9 - 3 = 6 (second iteration);
6 - sopf(6)= 6 - 5 = 1 (third iteration and reaching 1).

Michel Lagneau, Table of n, a(n) for n = 1..10000

Cf. A008472.

A008472:= n -> add(d, d = select(isprime, numtheory[divisors](n))):
  A:= proc(n)
        a := 0 ;
        x := n ;
        while x>1 do
                x := x-A008472(x) ;
                a := a+1 ;
        end do:
        a ;
end proc:
seq(A(n), n=1..100);

t[n_] := -1 + Length[NestWhileList[#-Total[Transpose[FactorInteger[#]][[1]]]&, n, #>1&]]; Table[t[n], {n, 100}]

A286190 Smallest k such that sopf(k) >= sopf(k+1) >= ... >= sopf(k+n), where sopf = A008472.

Original entry on oeis.org

3, 13, 13, 491, 1516, 12721, 12721, 109453, 1473257, 120797465, 624141002, 4044619541, 136797949237, 315400191511, 1285600699441

Offset: 1

Giovanni Resta, May 04 2017

Here sopf(k) is the sum of the distinct primes dividing k (A008472).

a(16) > 10^13.

			sopf(13) = 13, sopf(14) = 9, sopf(15) = 8, sopf(16) = 2. This is the first run of 4 nonincreasing values, so a(3) = 13.

Cf A008472, A189882.

sopf[n_] := If[n == 1, 0, Total[First /@ FactorInteger@n]]; s = Array[ sopf, 120000]; Table[ SelectFirst[ Range[ Length@s - n], Sort[t = Take[s, {#, # + n}]] == Reverse[t] &], {n, 8}]

A287093 a(0) = 0, a(1) = 2; a(2n) = sopf(a(n)), a(2n+1) = a(n) + a(n+1), where sopf() is the sum of the distinct prime factors (A008472).

Original entry on oeis.org

0, 2, 2, 4, 2, 6, 2, 6, 2, 8, 5, 8, 2, 8, 5, 8, 2, 10, 2, 13, 5, 13, 2, 10, 2, 10, 2, 13, 5, 13, 2, 10, 2, 12, 7, 12, 2, 15, 13, 18, 5, 18, 13, 15, 2, 12, 7, 12, 2, 12, 7, 12, 2, 15, 13, 18, 5, 18, 13, 15, 2, 12, 7, 12, 2, 14, 5, 19, 7, 19, 5, 14, 2, 17, 8, 28, 13, 31, 5, 23, 5, 23, 5, 31, 13, 28, 8, 17, 2, 14, 5

Offset: 0

Ilya Gutkovskiy, May 19 2017

A variation on Stern's diatomic sequence.

			a(0) = 0;
a(1) = 2;
a(2) = a(2*1) = sopf(a(1)) = 2;
a(3) = a(2*1+1) = a(1) + a(2) = 4;
a(4) = a(2*2) = sopf(a(2)) = 2;
a(5) = a(2*2+1) = a(2) + a(3) = 6;
a(6) = a(2*3) = sopf(a(3)) = 2, etc.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Michael Gilleland, Some Self-Similar Integer Sequences
Ilya Gutkovskiy, Extended graphical example
Eric Weisstein's World of Mathematics, Stern's Diatomic Series
Index entries for sequences related to Stern's sequences

Cf. A002487, A008472, A287051.

a[0] = 0; a[1] = 2; a[n_] := If[EvenQ[n], DivisorSum[a[n/2], # &, PrimeQ[#] &], a[(n - 1)/2] + a[(n + 1)/2]]; Table[a[n], {n, 0, 90}]

a(n) = if (n==0, 0, if (n ==1, 2, if (n%2, a((n-1)/2) + a((n+1)/2), vecsum(factor(a(n/2))[,1])))); \\ Michel Marcus, Dec 17 2017

A328174 a(n) is the least integer k such that sigma(k)/(d(k)*sopf(k)) = n where d=A000005, sigma=A000203 and sopf=A008472.

Original entry on oeis.org

20, 140, 54, 189, 378, 1365, 540, 945, 1120, 1755, 1539, 3465, 500, 1815, 4256, 6384, 14645, 5280, 1323, 1029, 864, 23871, 34579, 12903, 1715, 2673, 11934, 5589, 106805, 12285, 5600, 11625, 21070, 41915, 4459, 16905, 61320, 6615, 11178, 5145, 110839, 19656, 109225

Offset: 1

Michel Marcus, Oct 06 2019

nonn

Cf. A000005, A000203, A008472, A134382, A328051, A328052, A328175.

sopf(f) = sum(j=1, #f~, f[j, 1]); \\ A008472
isok(k, n) = my(fk=factor(k)); n*numdiv(fk)*sopf(fk) == sigma(fk);
a(n) = {my(k=1); while (!isok(k, n), k++); k;}

A328175 a(n) is the largest integer k such that sigma(k)/(d(k)*sopf(k)) = n where d=A000005, sigma=A000203 and sopf=A008472.

Original entry on oeis.org

42, 470, 923, 2159, 12924, 3735, 4316, 8786, 23939, 24412, 76502, 26768, 28612, 47849, 145620, 36002, 118204, 189143, 116999, 105657, 109559, 252474, 142687, 236860, 504899, 265682, 388798, 1558808, 154559, 345687, 709564, 544829, 383086, 652049, 361905, 1193075

Offset: 1

Michel Marcus, Oct 06 2019

nonn

Cf. A000005, A000203, A008472, A134382, A328051, A328052, A328174.

sopf(f) = sum(j=1, #f~, f[j, 1]); \\ A008472
lista(nn) = {/* nn should be > 10^7 */ my(nmax = 43, v = vector(nmax, k, List())); for (n=2, nn, my(f=factor(n), q); if (denominator(q=sigma(f)/(numdiv(f)*sopf(f))) == 1, if (q <= nmax, listput(v[q], n)););); for (i=1, nmax, if (#v[i] == 0, break); print1(vecmax(Vec(v[i])), ", "););}

A329463 Carmichael numbers k such that sopf(k) is also a Carmichael number, where sopf(k) is the sum of the distinct primes dividing k (A008472).

Original entry on oeis.org

1618206745, 2265650401, 28645206001, 56969031001, 226244724265, 235389006721, 235771174081, 296423001601, 432133594201, 626086650961, 772165132201, 884500464001, 1167647270401, 4384350028801, 4714081284241, 5438971500481, 5916902791801, 7160462614273, 11458124974801

Offset: 1

Amiram Eldar, Nov 13 2019

nonn

There are 1108 terms below 2^64: 75 have 5 prime factors, 1 have 7 prime factors (307696492063107001), and 1032 have 9 prime factors.

The corresponding values of sopf(a(n)) are 1729, 1105, 1105, 1105, 115921, 2821, 2821, 2821, 15841, 2821, 1729, 10585, 2821, 2821, 75361, 2821, 15841, 2821, 334153, ...

			1618206745 = 5 * 23 * 43 * 229 * 1429 is a Carmichael number, and 5 + 23 + 43 + 229 + 1429 = 1729 is also a Carmichael number.

Amiram Eldar, Table of n, a(n) for n = 1..1108

Cf. A002997, A008472.

carmQ[n_] := CompositeQ[n] && Divisible[n - 1, CarmichaelLambda[n]]; sopf[n_] := Total[FactorInteger[n][[;; , 1]]]; s={}; Do[If[carmQ[n] && carmQ[sopf[n]], AppendTo[s, n]], {n, 2, 3*10^10}]; s

A336657 Numbers k such that 2^k - 1 is divisible by the sum of the distinct primes dividing k (A008472).

Original entry on oeis.org

12, 24, 36, 48, 52, 72, 96, 104, 108, 144, 192, 208, 216, 288, 324, 330, 345, 384, 385, 416, 432, 462, 576, 648, 660, 664, 665, 676, 690, 768, 832, 840, 864, 924, 972, 990, 1035, 1152, 1190, 1296, 1302, 1320, 1328, 1330, 1352, 1380, 1386, 1428, 1430, 1530, 1536

Offset: 1

Amiram Eldar, Jul 28 2020

nonn

Since 2^p == 2 (mod p) for all primes p, all the terms of this sequence are composites. Similar considerations show that there are no semiprimes in this sequence.
The odd terms are relatively rare: 345, 385, 665, 1035, 1725, 1925, ...
If k is a term and d|k then d*k is also a term. In particular, all the numbers of the form 2^i * 3^j, with i > 1 and j > 0, are terms.

			12 = 2^2 * 3 is a term since 2^12 - 1 = 4095 is divisible by 2 + 3 = 5.

Amiram Eldar, Table of n, a(n) for n = 1..10000
William D. Banks and Florian Luca, Sums of prime divisors and Mersenne numbers, Houston J. Math., Vol. 33, No. 2 (2007), pp. 403-413.

Cf. A000225, A008472, A156787.

b[n_] := Total[FactorInteger[n][[;;, 1]]]; Select[Range[2, 1500], PowerMod[2, #, b[#]] == 1 &]

The number of terms not exceeding x is x^(1 - c_1 * log(log(log(x)))/log(log(x))) <= N(x) <= c_2 * x * log(log(x))/log(x) for all sufficiently large values of x, where c_1 and c_2 are positive constants (Banks and Luca, 2007).

A363292 Numbers whose sum of (distinct) prime divisors (A008472) equals 7.

Original entry on oeis.org

7, 10, 20, 40, 49, 50, 80, 100, 160, 200, 250, 320, 343, 400, 500, 640, 800, 1000, 1250, 1280, 1600, 2000, 2401, 2500, 2560, 3200, 4000, 5000, 5120, 6250, 6400, 8000, 10000, 10240, 12500, 12800, 16000, 16807, 20000, 20480, 25000, 25600, 31250, 32000, 40000, 40960

Offset: 1

M. F. Hasler, Jul 20 2023

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A008472 (sopf), A000420 (7^n), A033846 (2^m*5^n), A362948 (A008472 = 5).

seq[max_] := Union[Join[7^Range[Floor[Log[7, max]]], Flatten@ Table[2^i*5^j, {i, 1, Log2[max]}, {j, 1, Log[5, max/2^i]}]]]; seq[40000] (* Amiram Eldar, Jul 27 2023 *)

select( {is_A363292(n)=vecsum(factor(n,0)[,1])==7}, [1..13^4]) \\ alternatively: [n | n<-[1..13^4], A008472(n)==7]

Union of A000420 = {7^k ; k > 0} and A033846 = {2^j*5^k ; j, k > 0}.

Sum_{n>=1} 1/a(n) = 5/12. - Amiram Eldar, Jul 27 2023

cp's OEIS Frontend

A182184 a(1)= 0; for n > 1, a(n) = a(n-1) - sopf(n) if that number is positive, otherwise a(n)= a(n-1)+ sopf(n), where sopf(n) is the sum of the distinct primes dividing n (A008472).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

A242404 a(n) = Sum over i and j with j<=i and i=1.. sopf(n) of binomial(d(i), d(j)), where d(i) are the prime divisors of n and sopf(n) = sum of the distinct primes dividing n (A008472).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A256122 Number of iterations needed to reach 0 or 1 under the map n-> n-sopf(n), where sopf(n) is the sum of the distinct primes dividing n (A008472).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A286190 Smallest k such that sopf(k) >= sopf(k+1) >= ... >= sopf(k+n), where sopf = A008472.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A287093 a(0) = 0, a(1) = 2; a(2*n) = sopf(a(n)), a(2*n+1) = a(n) + a(n+1), where sopf() is the sum of the distinct prime factors (A008472).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A328174 a(n) is the least integer k such that sigma(k)/(d(k)*sopf(k)) = n where d=A000005, sigma=A000203 and sopf=A008472.

Views

Author

Keywords

Crossrefs

Programs

PARI

A328175 a(n) is the largest integer k such that sigma(k)/(d(k)*sopf(k)) = n where d=A000005, sigma=A000203 and sopf=A008472.

Views

Author

Keywords

Crossrefs

Programs

PARI

A329463 Carmichael numbers k such that sopf(k) is also a Carmichael number, where sopf(k) is the sum of the distinct primes dividing k (A008472).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A336657 Numbers k such that 2^k - 1 is divisible by the sum of the distinct primes dividing k (A008472).

Views

Author

Keywords

Comments

Examples

A287093 a(0) = 0, a(1) = 2; a(2n) = sopf(a(n)), a(2n+1) = a(n) + a(n+1), where sopf() is the sum of the distinct prime factors (A008472).