A007947 -id:A007947 - cp's OEIS frontend

A078319 Sum of distinct prime factors of n*rad(n)+1, where rad = A007947 (squarefree kernel).

2, 5, 7, 3, 15, 37, 7, 17, 9, 101, 63, 73, 24, 197, 115, 14, 36, 109, 183, 70, 32, 102, 60, 34, 12, 677, 43, 134, 423, 70, 52, 18, 116, 102, 615, 38, 144, 22, 763, 401, 31, 358, 44, 39, 15, 102, 37, 17, 45, 170, 1303, 55, 288, 18, 108, 162, 20, 678, 1743, 1801, 1863

Offset: 1

Reinhard Zumkeller, Nov 23 2002

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

Cf. A007947, A008472, A078310, A078313, A078318, A078320.

a078319 = a008472 . a078310  -- Reinhard Zumkeller, Jul 23 2013

a[n_] := Plus @@ FactorInteger[1 + n*Times @@ FactorInteger[n][[;; , 1]]][[;; , 1]]; Array[a, 100] (* Amiram Eldar, Apr 10 2025 *)

vecprod(v)=prod(i=1,#v,v[i])
rad(n)=vecprod(factor(n)[,1])
a(n)=vecsum(factor(n*rad(n)+1)[,1]) \\ Charles R Greathouse IV, Jul 09 2013

a(n) = A008472(A078310(n)).

A078320 Sum of all prime factors of n*rad(n)+1, where rad = A007947 (squarefree kernel).

Original entry on oeis.org

2, 5, 7, 6, 15, 37, 12, 17, 11, 101, 63, 73, 24, 197, 115, 14, 36, 109, 183, 70, 32, 102, 60, 34, 15, 677, 43, 134, 423, 70, 52, 18, 116, 102, 615, 38, 144, 39, 763, 401, 60, 358, 49, 39, 30, 102, 37, 34, 49, 170, 1303, 55, 288, 23, 108, 162, 30, 678, 1743, 1801

Offset: 1

Reinhard Zumkeller, Nov 23 2002

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

Cf. A001414, A078310, A078314, A078318, A078319.

a078320 = a001414 . a078310  -- Reinhard Zumkeller, Jul 23 2013

a[n_] := Plus @@ Times @@@ FactorInteger[1 + n * Times @@ FactorInteger[n][[;;, 1]]]; Array[a, 100] (* Amiram Eldar, Apr 10 2025 *)

rad(n)=my(f=factor(n)[,1]); prod(i=1,#f,f[i])
a(n)=my(f=factor(n*rad(n)+1));sum(i=1,#f~,f[i,1]*f[i,2]) \\ Charles R Greathouse IV, Jul 15 2013

a(n) = A001414(A078310(n)).

A078322 a(n) = rad(n*rad(n)+1), where rad = A007947 (squarefree kernel).

Original entry on oeis.org

2, 5, 10, 3, 26, 37, 10, 17, 14, 101, 122, 73, 170, 197, 226, 33, 290, 109, 362, 201, 442, 485, 530, 145, 42, 677, 82, 393, 842, 901, 962, 65, 1090, 1157, 1226, 217, 1370, 85, 1522, 401, 58, 1765, 370, 969, 26, 2117, 2210, 17, 86, 501, 2602, 1353, 2810, 65

Offset: 1

Reinhard Zumkeller, Nov 23 2002

a(n) = A007947(A078310(n)).

			a(25) = rad(25*rad(25)+1) = rad(25*rad(5^2)+1) = rad(25*5+1) = rad(125+1) = rad(126) = rad(2*3*3*7) = 2*3*7 = 42.

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

Cf. A007947, A078310, A078325.

a078322 = a007947 . a078310
-- Reinhard Zumkeller, Oct 19 2011

rad:= n-> mul(i, i=numtheory[factorset](n)):
a:= n-> rad(n*rad(n)+1):
seq(a(n), n=1..70);  # Alois P. Heinz, May 04 2017

rad[n_] := Times @@ FactorInteger[n][[All, 1]]; Table[ rad[n*rad[n] + 1], {n, 1, 54}] (* Jean-François Alcover, Dec 03 2012 *)

rad(n)=vecprod(factor(n)[,1])
a(n)=rad(n*rad(n)+1) \\ Charles R Greathouse IV, Jul 09 2013

A088966 Numbers k such that A007947(k) = A007947(m+1) and A007947(m) = A007947(k+1), where k > m.

Original entry on oeis.org

3, 8, 24, 80, 288, 1088, 4224, 4374, 16640, 66048, 263168, 1050624, 4198400, 16785408, 67125248, 268468224, 1073807360, 4295098368, 17180131328, 68720001024, 274878955520, 1099513724928

Offset: 1

Naohiro Nomoto, Oct 29 2003

For every k >= 0, the sequence includes 4^k + 2^(k+1), with m = 2^k + 1. - David Wasserman, Jan 29 2004
So a(13) <= 4198400. - Michel Marcus, Aug 10 2014
Are there other terms like 4374 that are not of this form? - Michel Marcus, Aug 10 2014

			With n=3 and m=2, rad(3) = rad(3) and rad(2) = rad(4), so 3 is in the sequence.

Christian Hercher, On one of Erdős' Problems - An Efficient Search for Benelux Pairs, arXiv:2506.01099 [math.NT], 2025. See pp. 2, 13.

Cf. A007947 (rad(n)), A087914 (similar sequence), A091697 (the values of m).

rad:= n -> convert(numtheory:-factorset(n),`*`):
count:= 0: lastr:= rad(1):
for n from 2 to 10^7 do
  newr:= rad(n);
  P[lastr,newr]:= n-1;
  if assigned(P[newr,lastr]) then
    count:= count+1; A[count]:= n-1; M[count]:= P[newr,lastr];
  fi;
  lastr:= newr;
od:
seq(A[n],n=1..count); # Robert Israel, Aug 10 2014

(* Recomputation up to a(13), assuming m of the form 2^k+1 *)
rad[n_] := rad[n] = Select[Divisors[n], SquareFreeQ][[-1]];
okQ[n_] := Module[{r = rad[n], r1 = rad[n+1], k, m}, For[k = 0, k < Log[2, n-1], k++, m = 2^k+1; If[r == rad[m+1] && rad[m] == r1, Return[True]]]; False];
Reap[For[n = 1, n <= 5*10^6, n++, If[okQ[n], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Apr 11 2019 *)

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
lista(nn) = {v = vector(nn, i, rad(i)); for (n=1, nn-1, ok = 0; if (n % 2, ma = 2, ma = 1); forstep (m = ma, n-1, 2, if ((v[n] == v[m+1]) && (v[m] == v[n+1]), ok = 1; break);); if (ok, print1(n, ", ")););} \\ Michel Marcus, Aug 10 2014

G.f.: Conjecture: Q(0)/x - 1/x where Q(k)= 1 + 2^k*x/(1 - 2*x/(2*x + 2^k*x/Q(k+1) )); (continued fraction ). - Sergei N. Gladkovskii, Apr 10 2013

More terms from David Wasserman, Jan 29 2004
a(13) confirmed by Robert Israel, Aug 10 2014
a(14)-a(22) from Bill McEachen, Jul 02 2025

A097982 Numbers n such that (phi(n) + sigma(n))/(rad(n))^2 is an integer > 1 (phi=A000010, sigma=A000203, rad=A007947).

Original entry on oeis.org

1, 864, 2430, 7776, 27000, 55296, 69984, 82134, 215622, 432000, 497664, 629856, 675000, 862488, 1499136, 1749600, 2187000, 2667168, 3449952, 3538944, 4287500, 4312440, 4478976, 4563000, 5668704, 6912000, 10800000, 13045131, 13799808, 16875000, 18670176, 19773000

Offset: 1

Lekraj Beedassy, Sep 07 2004

nonn

			For example: 864 is a term since phi(864) = 288, sigma(864) = 2520, 864 = 2^5*3^3, (288+2520)/6^2 = 78.

J.-M. De Koninck and A. Mercier, 1001 Problemes en Theorie Classique Des Nombres, Problem 749, pp. 95, 319, Ellipses, Paris, 2004.

Amiram Eldar, Table of n, a(n) for n = 1..143

Subsequence of A121850.

Cf. A000010, A000203, A007947.

f[n_] := (DivisorSigma[1, n] + EulerPhi[n])/(Times @@ Transpose[FactorInteger[n]][[1]])^2; Do[ If[IntegerQ[f[n] && f[n] != 1], Print[n]], {n, 1, 1000000}] (* Tanya Khovanova, Aug 30 2006 *)
f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := (p - 1)*p^(e - 1); q[1] = True; q[n_] := IntegerQ[(r = (Times @@ f1 @@@ (f = FactorInteger[n]) + Times @@ f2 @@@ f)/ (Times @@ First /@ f)^2)] && r > 1; Select[Range[10^5], q] (* Amiram Eldar, Dec 04 2020 *)

rad(n)=my(f=factor(n)[,1]);prod(i=1,#f,f[i])
is(n)=my(t=(eulerphi(n)+sigma(n))/rad(n)^2);denominator(t)==1 && t>1 \\ Charles R Greathouse IV, Feb 19 2013

More terms from Tanya Khovanova, Aug 30 2006
a(15)-a(29) from Donovan Johnson, Feb 05 2010
a(1)=1 and a(30)-a(32) added by Amiram Eldar, Dec 04 2020

A099984 Bisection of A007947.

Original entry on oeis.org

1, 3, 5, 7, 3, 11, 13, 15, 17, 19, 21, 23, 5, 3, 29, 31, 33, 35, 37, 39, 41, 43, 15, 47, 7, 51, 53, 55, 57, 59, 61, 21, 65, 67, 69, 71, 73, 15, 77, 79, 3, 83, 85, 87, 89, 91, 93, 95, 97, 33, 101, 103, 105, 107, 109, 111, 113, 115, 39, 119, 11, 123, 5, 127, 129, 131, 133, 15, 137

Offset: 1

N. J. A. Sloane, Nov 19 2004

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

Cf. A007947, A065463, A099985.

with(numtheory): A007947 := proc(n) local i,t1,t2; t1 :=ifactors(n)[2]; t2 := mul(t1[i][1],i=1..nops(t1)); end: seq(A007947(2*n-1),n=1..78); # Emeric Deutsch, Dec 15 2004

a[n_] := Times @@ (First /@ FactorInteger[2*n-1]); Array[a, 100]  (* Amiram Eldar, Nov 19 2022*)

a(n) = factorback(factorint(2*n-1)[, 1]); \\ Amiram Eldar, Nov 19 2022

From Amiram Eldar, Nov 19 2022: (Start)
a(n) = A007947(2*n-1).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (6/5) * Product_{p prime} (1 - 1/(p*(p+1))) = (6/5) * A065463 = 0.8453306... . (End)

More terms from Emeric Deutsch, Dec 15 2004

Offset corrected by Amiram Eldar, Nov 19 2022

A147799 Minimal value of A007947(m*(7^n-m)) with m coprime to 7.

Original entry on oeis.org

6, 6, 30, 30, 282, 2262, 17034, 36006, 71070

Offset: 1

M. F. Hasler, Nov 13 2008

The minima are reached for m values given in A147804.
This is related to the abc conjecture.
All terms of this sequence are even, so one could also consider A147799/2 = 3, 3, 15, 15, 141, 1131, 8517, 18003, 35535, ... So far these terms are also multiples of 3, but this might be a coincidence.

Cf. A007947, A147804 (m values); A143702, A147801, A147800 (analog for 2^n, 3^n, 5^n), A147298 (general case).

A147799(n,p=7)={my(m=n=p^n); for(a=1,n\2, a%p || next; m > 2*A007947(a) || next; m > A007947(n-a)*A007947(a) || next; m = A007947(n-a)*A007947(a)); m; }

A147804 Least m coprime to 7 minimizing A007947(m*(7^n-m)).

Original entry on oeis.org

1, 1, 100, 1, 423, 28561, 3072, 124609, 119232

Offset: 1

M. F. Hasler, Nov 13 2008

The minima are given in A147799.

This is related to the abc conjecture: Since m is coprime to 7, it is also coprime to 7^n and thus to 7^n-m. Thus the squarefree kernel A007947(m*(7^n-m)*7^n) = 7*A007947(m(7^n-m)).

Cf. A007947, A147799 (value of minima), A143700, A147802, A147803 (analog for 2^n, 3^n, 5^n), A147300 (analog for any number).

A147804(n,p=7)={my(b, m=3*n=p^n, t); for(a=1, n\2, a%p || next; m>2*(t=A007947(a)) || next; m>(t*=A007947(n-a)) || next; m=t; b=a); b; }

Typo in title corrected by M. F. Hasler, Nov 17 2008

A254791 Nontrivial solutions to n = sigma(a) = sigma(b) (A000203) and rad(a) = rad(b) (A007947) with a != b.

Original entry on oeis.org

4800, 142800, 1909440, 32948784, 210313800, 993938400, 1069286400, 1264808160, 1309463064, 2281635216, 3055104000, 3250790400

Offset: 1

Fred Schneider, Feb 07 2015

nonn

On the term "nontrivial":
If a !=b, sigma(a) = sigma(b) and rad(a) = rad(b) then sigma(a*x) = sigma(b*x) and rad(n*x) = rad(m*x) when gcd(a, b) = gcd(a,x) = gcd(b,x) = 1. So each general solution to the stated problem could generate an infinitude of constructed, "trivial" solutions. So we will limit ourselves to the more interesting "nontrivial" solutions. Precisely, if rad(a) = rad(b) = Product(p(i)), we can write a = Product(p(i)^a(i)), b = Product(p(i)^b(i)) and in this context, a(i) != b(i) for each i in order to have a nontrivial solution.
There is another type of trivial solution, if n can be expressed as the product of two or more smaller solutions, it would be considered a composite solution but still trivial.
The smallest composite solution is below:
210313800: 131576362 = 2 * 17 * 157^3 and 98731648 = 2^7 * 17^3 * 1573250790400: 2196937295 = 5 * 7^3 * 31^3 * 43 and 2156627375 = 5^3 * 7 * 31 * 43^3. Note: the common rads for the two pairs have no factors in common so we have these "trivial" composite solutions below.
sigma(131576362 * 2196937295) = sigma(98731648 * 2156627375) = sigma(131576362 * 2156627375) = sigma(98731648 * 2196937295) = 683686082027520000.

			Sigma => Pair of distinct integers 4800 => 2058 = 2 * 3 * 7^3 and 1512 = 2^3 * 3^3 * 7142800 => 52728 = 2^3 * 3 * 13^3 and 44928 = 2^7 * 3^3 * 131909440 => 1038230 = 2 * 5 * 47^3 and 752000 = 2^7 * 5^3 * 4732948784 => 10825650 = 2 * 3^9 * 5^2 * 11 and 8624880 = 2^4 * 3^4 * 5 * 11^3210313800 => 131576362 = 2 * 17 * 157^3 and 98731648 = 2^7 * 17^3 * 157993938400 => 336110688 = 2^5 * 3^3 * 73^3 and 326965248 = 2^11 * 3^7 * 73.
The pairs that contribute to the solution each have the same rad or squarefree kernel and they are "nontrivial" because within a pair for the same prime, none of the exponents match.

Cf. A000203, A007947.

Subsequence of A254035. Cf. also A255334, A255425, A255426.

A255335 Numbers n for which there exists k < n such that A000203(k) = A000203(n) and A007947(k) = A007947(n), where A000203 gives the sum of divisors, and A007947 gives the squarefree kernel of n.

Original entry on oeis.org

2058, 10290, 22638, 26754, 34986, 39102, 47334, 51450, 52728, 59682, 63798, 76146, 84378, 88494, 96726, 109074, 113190, 121422, 125538, 133770, 137886, 146118, 150234, 162582, 170814, 174930, 183162, 195510, 199626, 207858, 211974, 220206, 224322, 232554, 236670, 249018, 257250, 261366, 263640, 269598, 281946, 286062, 294294

Offset: 1

Antti Karttunen, Mar 23 2015, suggested by Michel Marcus, Feb 23 2015

nonn

Sequence A255423 sorted into ascending order.
Note that both for u = a(17) = 113190 and v = a(22) = 146118, A000203(u) = A000203(v) = 345600.
Also, both for w = a(20) = 133770 and x = a(25) = 170814, A000203(w) = A000203(x) = 403200.
Question: Does this have any common terms with A255334 ?

Antti Karttunen, Table of n, a(n) for n = 1..2434

Subsequence of A013929.
Cf. A000203, A007947.
Cf. also A255334, A255423, A254035.

allocatemem(234567890);
A007947(n) = factorback(factorint(n)[, 1]); \\ Andrew Lelechenko, May 09 2014
upto = (2^24)-4;
bigvec = vector(upto);
i=0; for(n=1, upto, bigvec[n] = Set([]); my(r=A007947(n), s=sigma(n)); if(setsearch(bigvec[r],s), i++; write("b255335.txt", i, " ", n), bigvec[r] = setunion(Set([s]),bigvec[r])));

;; With Antti Karttunen's IntSeq-library. Quite naive implementation.
(define A255335 (MATCHING-POS 1 1 isA255335?))
(define (isA255335? n) (let ((sig_n (A000203 n)) (rad_n (A007947 n))) (let loop ((try (- n rad_n))) (cond ((< try rad_n) #f) ((and (= sig_n (A000203 try)) (= rad_n (A007947 try))) #t) (else (loop (- try rad_n)))))))

cp's OEIS Frontend

A078319 Sum of distinct prime factors of n*rad(n)+1, where rad = A007947 (squarefree kernel).

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A078320 Sum of all prime factors of n*rad(n)+1, where rad = A007947 (squarefree kernel).

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A078322 a(n) = rad(n*rad(n)+1), where rad = A007947 (squarefree kernel).

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A088966 Numbers k such that A007947(k) = A007947(m+1) and A007947(m) = A007947(k+1), where k > m.

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A097982 Numbers n such that (phi(n) + sigma(n))/(rad(n))^2 is an integer > 1 (phi=A000010, sigma=A000203, rad=A007947).

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A099984 Bisection of A007947.

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A147799 Minimal value of A007947(m*(7^n-m)) with m coprime to 7.

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