A007850 -id:A007850 - cp's OEIS frontend

A346555 7-Sondow numbers: numbers k such that p^s divides k/p + 7 for every prime power divisor p^s of k.

1, 2, 6, 15, 78, 294, 12642, 539026980558

Offset: 1

José María Grau Ribas, Feb 04 2022

Numbers k such that A235137(k) == 7 (mod k).
A positive integer k is a 7-Sondow number if satisfies any of the following equivalent properties:
1) p^s divides k/p + 7 for every prime power divisor p^s of k.
2) 7/k + Sum_{prime p|k} 1/p is an integer.
3) 7 + Sum_{prime p|k} k/p == 0 (mod k).
4) Sum_{i=1..k} i^phi(k) == 7 (mod k).

Github, Jonathan Sondow (1943 - 2020)
J. M. Grau, A. M. Oller-Marcén and D. Sadornil, On µ-Sondow Numbers, arXiv:2111.14211 [math.NT], 2021.
J. M. Grau, A. M. Oller-Marcen and J. Sondow, On the congruence 1^n + 2^n + ... + n^n = d (mod n), where d divides n, arXiv:1309.7941 [math.NT], 2013-2014.

Cf. A054377, A007850, A235137, A348058, A348059.
(-1) and (-2) -Sondow numbers: A326715, A330069.
1-Sondow to 9-Sondow numbers: A349193, A330068, A346551, A346552, A346553, A346554, this sequence, A346556, A346557.

Sondow[mu_][n_]:=Sondow[mu][n]=Module[{fa=FactorInteger[n]},IntegerQ[mu/n+Sum[1/fa[[i,1]],{i,Length[fa]}]]]
Select[Range[10000000],Sondow[7][#]&]

a(8) from Martin Ehrenstein, Feb 04 2022

A346556 8-Sondow numbers: numbers k such that p^s divides k/p + 8 for every prime power divisor p^s of k.

Original entry on oeis.org

1, 3, 16, 48, 336, 14448, 376464, 17716019376, 419963172816

Offset: 1

José María Grau Ribas, Feb 04 2022

Numbers k such that A235137(k) == 8 (mod k).
A positive integer k is a 8-Sondow number if satisfies any of the following equivalent properties:
1) p^s divides k/p + 8 for every prime power divisor p^s of k.
2) 8/k + Sum_{prime p|k} 1/p is an integer.
3) 8 + Sum_{prime p|k} k/p == 0 (mod k).
4) Sum_{i=1..k} i^phi(k) == 8 (mod k).
Other numbers in the  sequence: 17716019376, 419963172816, 67923372668477507285654170088688

Github, Jonathan Sondow (1943 - 2020)
J. M. Grau, A. M. Oller-Marcén and D. Sadornil, On µ-Sondow Numbers, arXiv:2111.14211 [math.NT], 2021.
J. M. Grau, A. M. Oller-Marcen and J. Sondow, On the congruence 1^n + 2^n + ... + n^n = d (mod n), where d divides n, arXiv:1309.7941 [math.NT], 2013-2014.

Cf. A054377, A007850, A235137, A348058, A348059.
(-1) and (-2) -Sondow numbers: A326715, A330069.
1-Sondow to 9-Sondow numbers: A349193, A330068, A346551, A346552, A346553, A346554, A346555, this sequence, A346557.

Sondow[mu_][n_]:=Sondow[mu][n]=Module[{fa=FactorInteger[n]},IntegerQ[mu/n+Sum[1/fa[[i,1]],{i,Length[fa]}]]]
Select[Range[400000],Sondow[8][#]&]

a(8)-a(9) verified by Martin Ehrenstein, Feb 04 2022

A346557 9-Sondow numbers: numbers k such that p^s divides k/p + 9 for every prime power divisor p^s of k.

Original entry on oeis.org

1, 2, 5, 22, 54, 378, 16254, 423522, 19930521798, 472458569418

Offset: 1

José María Grau Ribas, Feb 04 2022

Numbers k such that A235137(k) == 9 (mod k).
A positive integer k is a 9-Sondow number if satisfies any of the following equivalent properties:
1) p^s divides k/p + 9 for every prime power divisor p^s of k.
2) 9/k + Sum_{prime p|k} 1/p is an integer.
3) 9 + Sum_{prime p|k} k/p == 0 (mod k).
4) Sum_{i=1..k} i^phi(k) == 9 (mod k).
Other numbers in the sequence: 19930521798, 472458569418, 76413794252037195696360941349774

Github, Jonathan Sondow (1943 - 2020)
J. M. Grau, A. M. Oller-Marcén and D. Sadornil, On µ-Sondow Numbers, arXiv:2111.14211 [math.NT], 2021.
J. M. Grau, A. M. Oller-Marcen and J. Sondow, On the congruence 1^n + 2^n + ... + n^n = d (mod n), where d divides n, arXiv:1309.7941 [math.NT], 2013-2014.

Cf. A054377, A007850, A235137, A348058, A348059.
(-1) and (-2) -Sondow numbers: A326715, A330069.
1-Sondow to 9-Sondow numbers: A349193, A330068, A346551, A346552, A346553, A346554, A346555, A346556, this sequence.

Sondow[mu_][n_]:=Sondow[mu][n]=Module[{fa=FactorInteger[n]},IntegerQ[mu/n+Sum[1/fa[[i,1]],{i,Length[fa]}]]]
Select[Range[1000000],Sondow[9][#]&]

a(9)-a(10) verified by Martin Ehrenstein, Feb 04 2022

A349193 1-Sondow numbers: numbers j such that p divides j/p + 1 for every prime divisor p of j.

Original entry on oeis.org

1, 2, 6, 42, 1806, 47058, 2214502422, 52495396602, 8490421583559688410706771261086

Offset: 1

José María Grau Ribas, Nov 10 2021

nonn

These are the weak primary pseudoperfect numbers mentioned in Grau-Oller-Sondow (2013).
Includes the primary pseudoperfect numbers (A054377). Any weak primary pseudoperfect number which is not a primary pseudoperfect number must have more than 58 distinct prime factors, and therefore must be greater than 10^110; none are known.
A positive integer j is a k-Sondow number if satisfies any of the following equivalent properties:
1) p^s divides j/p + k for every prime power divisor p^s of j.
2) k/j + Sum_{prime p|j} 1/p is an integer.
3) k + Sum_{prime p|j} j/p == 0 (mod j).
4) Sum_{i=1..j} i^A000010(j) == k (mod j).
Numbers m such that A235137(m) == 1 (mod m).

Github, Jonathan Sondow (1943 - 2020)
J. M. Grau, A. M. Oller-Marcén, and D. Sadornil, On µ-Sondow Numbers, arXiv:2111.14211 [math.NT], 2021.
J. M. Grau, A. M. Oller-Marcen and J. Sondow, On the congruence 1^n + 2^n + ... + n^n = d (mod n), where d divides n, arXiv:1309.7941 [math.NT], 2013.

Cf. A000010, A054377, A007850, A235137, A348058, A348059.
(-1) and (-2)-Sondow numbers: A326715, A330069.
2-Sondow to 9-Sondow numbers: A330068, A346551, A346552, A346553, A346554, A346555, A346556, A346557.

Sondow[mu_][n_]:= Sondow[mu][n]= Module[{fa=FactorInteger[n]},IntegerQ[mu/n+Sum[1/fa[[i,1]],{i,Length[fa]}]]];
Select[Range[100000],Sondow[1][#]&]

A199767 Numbers m for which Sum_{i=1..k} (1+1/p_i) + Product_{i=1..k} (1+1/p_i) is an integer, where p_i are the k prime factors of m (with multiplicity).

Original entry on oeis.org

21, 45, 432, 740, 1728, 3456, 3888, 5616, 12096, 23760, 46656, 52164, 131328, 152064, 186624, 195656, 233280, 311472, 606528, 618192, 746496, 926208, 933120, 979776, 1273536, 1403136, 2985984, 3221456, 3732480, 5178816, 5412096, 5971968, 9704448, 13651200

Offset: 1

Paolo P. Lava, Nov 22 2011

The numbers of the sequence are the solution of the differential equation m’=(a-k)*m-b, which can also be written as A003415(m)=(a-k)*m-A003958(m), where k is the number of prime factors of m, and a is the integer Sum_{i=1..k} (1+1/p_i) + Product_{1=1..k} (1+1/p_i).

The numbers of the sequence satisfy also Sum_{i=1..k} (1-1/p_i) - Product_{i=1..k} (1+1/p_i) = some integer.

			740 has prime factors 2, 2, 5, 37. 1 + 1/2 + 1 + 1/2 + 1 + 1/5 + 1 + 1/37 = 967/185 is the sum over 1+1/p_i. (1+1/2) * (1+1/2) * (1+1/5) * (1+1/37) = 513/185 is the product over 1+1/p_i. 967/185 + 513/185 = 8 is an integer.

J. M. Borwein and E. Wong, A survey of results relating to Giuga’s conjecture on primality, May 8, 1995.
Romeo Mestrovic, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.

Cf. A003415, A003958, A007850, A198391.

isA199767 := proc(n)
    p := ifactors(n)[2] ;
    add(op(2,d)+op(2,d)/op(1,d),d=p) + mul((1+1/op(1,d))^op(2,d),d=p) ;
    type(%,'integer') ;
end proc:
for n from 20 do
    if isA199767(n) then
           printf("%d,\n",n);
    end if;
end do: # R. J. Mathar, Nov 23 2011

A326689 Numerator of the fraction (Sum_{prime p | n} 1/p - 1/n).

Original entry on oeis.org

-1, 0, 0, 1, 0, 2, 0, 3, 2, 3, 0, 3, 0, 4, 7, 7, 0, 7, 0, 13, 3, 6, 0, 19, 4, 7, 8, 17, 0, 1, 0, 15, 13, 9, 11, 29, 0, 10, 5, 27, 0, 20, 0, 25, 23, 12, 0, 13, 6, 17, 19, 29, 0, 22, 3, 5, 7, 15, 0, 61, 0, 16, 29, 31, 17, 10, 0, 37, 25, 29, 0, 59, 0, 19, 13, 41

Offset: 1

Jonathan Sondow, Jul 18 2019

See Comments on denominators in A326690.

			-1/1, 0/1, 0/1, 1/4, 0/1, 2/3, 0/1, 3/8, 2/9, 3/5, 0/1, 3/4, 0/1, 4/7, 7/15, 7/16, 0/1, 7/9, 0/1, 13/20, 3/7, 6/11, 0/1, 19/24, 4/25, 7/13, 8/27, 17/28, 0/1, 1/1

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

Denominators are A326690. Cf. also A007850, A309132, A309235, A309378.

Cf. A028235.

PrimeFactors[n_] := Select[Divisors[n], PrimeQ];
g[n_] := Numerator[Sum[1/p, {p, PrimeFactors[n]}] - 1/n];
Table[ g[n], {n, 100}]

a(n) = numerator(sumdiv(n, d, isprime(d)/d) - 1/n); \\ Michel Marcus, Jul 19 2019

a(p) = 0 if p is a prime.

a(g) = 1 if g is a known Giuga number (see my 2nd comment in A007850).

A189710 Numbers n such that n'' = n'-1 where n' and n'' are respectively the first and the second arithmetic derivative of n (A003415).

Original entry on oeis.org

2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 185, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263

Offset: 1

Giorgio Balzarotti, Apr 25 2011

nonn

The composite numbers in the sequences are: 9, 185, 341, 377, 437, 9005, 30413, 33953, 41009, 51533, 82673, 92909,....

			9' = 6, 9''= 6'= 5, 9" = 9'- 1 -> 9 is in the sequence.

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

Cf. A003415, A007850, A054377, A189639.

import Data.List (elemIndices)
a189710 n = a189710_list !! (n-1)
a189710_list = elemIndices 0 $
   zipWith (-) (map a003415 a003415_list) (map pred a003415_list)
-- Reinhard Zumkeller, May 09 2011

#using Michael B. Porter's code from A003415
der:=n->n*add(op(2,p)/op(1,p),p=ifactors(n)[2])
for i from 1 to n do a:=der(der(i))-der(i)+1; if a=0 then j:=j+1; B[j]:=i; end if od

A190273 Numbers n such that n' = m+1, with n and m semiprimes and gcd(m,n)>1, where n' is the arithmetic derivative of n.

Original entry on oeis.org

6, 10, 21, 26, 39, 55, 57, 74, 93, 111, 122, 146, 155, 201, 203, 253, 301, 305, 314, 327, 381, 386, 417, 471, 497, 543, 554, 597, 626, 633, 689, 737, 755, 791, 794, 842, 889, 905, 914, 921, 1011, 1027, 1055, 1081, 1082, 1137, 1226, 1227, 1322, 1346, 1379, 1461, 1466, 1477, 1497, 1514, 1623, 1655, 1703, 1711, 1713, 1731, 1751, 1754, 1893, 1967, 1994

Offset: 1

Giorgio Balzarotti, May 07 2011

nonn

The sequence is related to the Rassias Conjecture ("for any prime p there are two primes p1 and p2 such that p*p1=p1+p2+1, p>2, p2>p1", see A190272-A190275), because n = p1*p2, m=p1*p -> p1*p = p1+p2-1. The sequence includes the cases with p=p1 (or p2). Generalization can be achieved by removing semiprimarity condition or accepting gcd(n,m)=1. The differential equation in its general form n'=m+1 includes Giuga Numbers, i.e., n'=b*n+1, or n'=n+1 (A007850).

These are semiprimes n = p*q such that 1/p + 1/q - 1/n = P/Q, where P <> Q are primes. Cf. A326690. - Amiram Eldar and Thomas Ordowski, Jul 25 2019

			n=6, 6'=5, m=5+1=6, gcd(6,6)=6 -> a(1)=6

For Rassias conjecture: Preda Mihăilescu, Review of Problem Solving and Selected Topics in Number Theory, Newsletter of the European Mathematical Society, March 2011, p. 46.

Cf. A001358 (semiprimes), A003415 (arithmetic derivative), A007850 (Giuga numbers), A190272 (n'=m-1), A190273, A190274, A190275.

der:=n->n*add(op(2,p)/op(1,p),p=ifactors(n)[2]);
seq(`if`(bigomega(i)=2 and bigomega(der(i)-1)=2 and gcd(i,der(i)-1)>1,i,NULL),i=1..2000);

A235138 a(n) = Sum_{k=1..n} k^phi(n) (mod n) where phi(n) = A000010(n).

Original entry on oeis.org

0, 1, 2, 2, 4, 1, 6, 4, 6, 3, 10, 2, 12, 5, 7, 8, 16, 3, 18, 6, 11, 9, 22, 4, 20, 11, 18, 10, 28, 29, 30, 16, 19, 15, 23, 6, 36, 17, 23, 12, 40, 1, 42, 18, 21, 21, 46, 8, 42, 15, 31, 22, 52, 9, 39, 20, 35, 27, 58, 58, 60, 29, 33, 32, 47, 5, 66, 30, 43, 11, 70, 12, 72, 35, 35, 34, 59, 7, 78, 24, 54, 39, 82, 2, 63, 41, 55, 36, 88, 87, 71, 42, 59, 45, 71, 16, 96, 35, 57, 30

Offset: 1

Jonathan Sondow and Emmanuel Tsukerman, Jan 03 2014

a(n) = n-1 if and only if n is prime or is a Giuga number A007850.
a(n) = 1 if (and probably only if) n is a primary pseudoperfect number A054377.
a(2^k*p) = 2^(k-1)*p-2^k if p is an odd prime. - Robert Israel, Apr 25 2017

			a(4) = 30 (mod 4) = 2 since 1^(phi(4)) + 2^(phi(4)) + 3^(phi(4)) + 4^(phi(4))= 1^2 + 2^2 + 3^2 + 4^2 = 1 + 4 + 9 + 16 = 30.

Robert Israel, Table of n, a(n) for n = 1..10000
Jonathan Sondow and K. MacMillan, Reducing the Erdos-Moser equation 1^n + 2^n + . . . + k^n = (k+1)^n modulo k and k^2, Integers 11 (2011), #A34.
Jonathan Sondow and E. Tsukerman, The p-adic order of power sums, the Erdos-Moser equation, and Bernoulli numbers, arXiv:1401.0322 [math.NT], 2014; see section 4.
Wikipedia, Giuga number
Wikipedia, Primary pseudoperfect number

Cf. A007850, A054377, A235137.

f:= proc(n) local q; q:= numtheory:-phi(n);
   add(k&^q, k=1..n) mod n
end proc:
map(f, [$1..100]); # Robert Israel, Apr 25 2017

a[n_] := Mod[Sum[PowerMod[i, EulerPhi@n, n], {i, n}], n]

a(n)=my(p=eulerphi(n));sum(k=1,n,k^p) \\ Charles R Greathouse IV, Jan 04 2014

a(n) = A235137(n) (mod n).

Conjecture : a(n) = Sum_{d|n} phi(n/d)*d^phi(n) (mod n). - Ridouane Oudra, Feb 17 2024

A326691 a(n) = n/denominator(Sum_{prime p | n} 1/p - 1/n).

Original entry on oeis.org

1, 2, 3, 1, 5, 2, 7, 1, 1, 2, 11, 3, 13, 2, 1, 1, 17, 2, 19, 1, 3, 2, 23, 1, 1, 2, 1, 1, 29, 30, 31, 1, 1, 2, 1, 1, 37, 2, 3, 1, 41, 2, 43, 1, 1, 2, 47, 3, 1, 2, 1, 1, 53, 2, 5, 7, 3, 2, 59, 1, 61, 2, 1, 1, 1, 6, 67, 1, 1, 2, 71, 1, 73, 2, 3, 1, 1, 2, 79

Offset: 1

Jonathan Sondow, Jul 20 2019

nonn

Denominator(Sum_{prime p | n} 1/p - 1/n) is a factor of n, since all primes in the sum divide n. So a(n) is an integer.

			a(18) = 18/denominator(Sum_{prime p | 18} 1/p - 1/18) = 18/denominator(1/2 + 1/3 - 1/18) = 18/denominator(7/9) = 18/9 = 2.
a(30) = 30/denominator(Sum_{prime p | 30} 1/p - 1/30) = 30/denominator(1/2 + 1/3 + 1/5 - 1/30) = 30/denominator(1/1) = 30/1 = 30, and 30 is a Giuga number.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Christian Krause, et al, LODA, an assembly language, a computational model and a tool for mining integer sequences
Wikipedia, Giuga number

Cf. A003415, A007850, A326689, A326690, A326692, A326715.

PrimeFactors[n_] := Select[Divisors[n], PrimeQ];
f[n_] := Denominator[Sum[1/p, {p, PrimeFactors[n]}] - 1/n];
Table[n/f[n], {n, 79}]

A326691(n) = (n/A326690(n)); \\ Antti Karttunen, Mar 15 2021

a(n) = n/A326690(n).
a(n) = n > 1 iff n is either a prime or a Giuga number A007850.
a(n) = gcd(n, 1+((n-1)*A003415(n))). [Conjectured, after an empirical formula found by LODA miner. This holds at least up to n=2^27] - Antti Karttunen, Mar 15 2021

cp's OEIS Frontend

A346555 7-Sondow numbers: numbers k such that p^s divides k/p + 7 for every prime power divisor p^s of k.

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A346556 8-Sondow numbers: numbers k such that p^s divides k/p + 8 for every prime power divisor p^s of k.

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A346557 9-Sondow numbers: numbers k such that p^s divides k/p + 9 for every prime power divisor p^s of k.

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A349193 1-Sondow numbers: numbers j such that p divides j/p + 1 for every prime divisor p of j.

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A199767 Numbers m for which Sum_{i=1..k} (1+1/p_i) + Product_{i=1..k} (1+1/p_i) is an integer, where p_i are the k prime factors of m (with multiplicity).

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A326689 Numerator of the fraction (Sum_{prime p | n} 1/p - 1/n).

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A189710 Numbers n such that n'' = n'-1 where n' and n'' are respectively the first and the second arithmetic derivative of n (A003415).

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A190273 Numbers n such that n' = m+1, with n and m semiprimes and gcd(m,n)>1, where n' is the arithmetic derivative of n.

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