A125137 -id:A125137 - cp's OEIS frontend

A362957 a(n) is the least prime p such that the number of distinct prime factors of p^n + 1 sets a new record.

2, 3, 5, 43, 17, 47, 151, 1697, 59, 2153, 521, 13183, 30089, 20753, 3769

Offset: 1

Hugo Pfoertner, Jun 11 2023

a(16) > 2.3*10^6; to see if there has been any progress see also A280005(16).

smallf(q,nmax) = {my(qq=q,j=0); forprime (p=2, nmax, my(k=0); while (qq%p==0, k++; qq/=p); if (k>0, j++;)); [j,qq]};
a362957(upto) = {my(nfmax=0); for (n=1, upto, forprime (p=2, oo, my(f=p^n+1, s=smallf(f,p)); if (s[1]nfmax, print1(p,", "); nfmax=nf; break)))};
a362957(12)

A238981 Sum of n-th powers of unitary divisors of n (d such that d divides n, gcd(d, n/d) = 1).

Original entry on oeis.org

1, 5, 28, 257, 3126, 47450, 823544, 16777217, 387420490, 10009766650, 285311670612, 8916117756914, 302875106592254, 11112685048647250, 437893920912786408, 18446744073709551617, 827240261886336764178, 39346558169931834836690, 1978419655660313589123980

Offset: 1

José María Grau Ribas, Mar 07 2014

nonn

Amiram Eldar, Table of n, a(n) for n = 1..387

Cf. A034448, A238982, A238983.

a[n_, k_] := Sum[If[GCD[i, n] == i && GCD[i, n/i] == 1, i^k, 0], {i, n}]; Table[a[n, n], {n, 1, 24}]
a[1] = 1; a[n_] := Times @@ (1 + First[#]^(n * Last[#]) &/@ FactorInteger[n]);  Array[a, 19] (* Amiram Eldar, Aug 10 2019 *)

a(n) = sumdiv(n, d, d^n*(gcd(d, n/d) == 1)); \\ Michel Marcus, Mar 19 2014

For prime p, a(p) = p^p + 1; A125137 is a subsequence. - Michel Marcus, Nov 20 2015
a(n) = n^n+1 (A014566) if n is a prime power (A246655). - Michel Marcus, Nov 21 2015
a(n) = Sum_{d|n, gcd(d,n/d)=1} d^n. - Wesley Ivan Hurt, Apr 28 2023

A115973 Number of distinct prime factors of p^p + 1 where p is prime(n).

Original entry on oeis.org

1, 2, 3, 3, 6, 5, 3, 5, 8, 7, 7, 7, 6, 8, 10, 8, 9, 8, 7, 8, 4, 7, 12, 6, 8, 7, 12, 11, 6, 9, 7, 17, 9, 9

Offset: 1

Parthasarathy Nambi, Mar 14 2006

a(35) >= 9. See link to factordb.com - Hugo Pfoertner, Aug 07 2019

			If p=29 then (29^29 + 1) contains 7 distinct prime factors.

Chris Caldwell, The First 1000 Primes.
Dario Alejandro Alpern, Factorization using the Elliptic Curve Method
factordb.com, Status of prime(n)^prime(n)+1.

Cf. A125137.

f[n_] := Length@ FactorInteger[Prime[n]^Prime[n] + 1]; Array[f, 20] (* Robert G. Wilson v, Apr 06 2006 *)

{ for(n=1,20, p = prime(n); d = factor(p^p+1); dec=matsize(d); print1(dec[1],","); ); } \\ R. J. Mathar, Mar 29 2006

8 more terms from R. J. Mathar, Mar 29 2006
a(19)-a(25) from Robert G. Wilson v, Apr 06 2006
a(26)-a(32) from Sean A. Irvine, Oct 20 2011
a(33)-a(34) from Hugo Pfoertner, Aug 07 2019
a(28) corrected by Sean A. Irvine, Aug 04 2023

A376432 a(n) is the least odd prime factor of prime(n)^prime(n)+1.

Original entry on oeis.org

5, 7, 3, 113, 3, 7, 3, 5, 3, 3, 373, 19, 3, 11, 3, 3, 3, 31, 17, 3, 37, 5, 3, 3, 7, 3, 13, 3, 5, 3, 921259, 3, 3, 5, 3, 19, 79, 41, 3, 3, 3, 7, 3, 97, 3, 5, 53, 7, 3, 5, 3, 3, 11, 3, 3, 3, 3, 17, 139, 3, 71, 3, 7, 3, 157, 3, 83, 13, 3, 5, 3, 3, 23, 11, 5, 3, 3, 199

Offset: 1

Hugo Pfoertner, Sep 27 2024

nonn

			While almost all terms lie in the range between 3 and (prime(n)+1)/2, there are some notable outliers: a(31) = 921259 with prime(31)=127 (127^127+1=2^7*a(31)*C268), and a(1028)=1528928750837 with prime(1028)=8191 (8191^8191+1=2^13*a(1028)*C32039), Cx being composite with x decimal digits.

Hugo Pfoertner, Table of n, a(n) for n = 1..12250

Cf. A078701, A125137, A376431.

a376432(n) = my(pp=prime(n)^prime(n)+1); forprime (p=3, oo, if(pp%p==0, return(p)))

a(n) = A078701(A125137(n)). - Michel Marcus, Sep 27 2024

A345263 a(n) = Sum_{d|n} d^rad(d).

Original entry on oeis.org

1, 5, 28, 21, 3126, 46688, 823544, 85, 757, 10000003130, 285311670612, 3032688, 302875106592254, 11112006826381564, 437893890380862528, 341, 827240261886336764178, 34059641, 1978419655660313589123980, 10250000003146, 5842587018385982521381947992, 341427877364219557681958394200

Offset: 1

Wesley Ivan Hurt, Jun 12 2021

nonn

If p is prime, a(p) = Sum_{d|p} d^rad(d) = 1^1 + p^p = p^p + 1.

Inverse Möbius transform of n^rad(n). - Wesley Ivan Hurt, Mar 31 2025

			a(4) = Sum_{d|4} d^rad(d) = 1^1 + 2^2 + 4^2 = 21.
a(6) = Sum_{d|6} d^rad(d) = 1^1 + 2^2 + 3^3 + 6^6 = 46688.

Cf. A007947 (rad), A125137, A345261.

Table[Sum[(1 - Ceiling[n/i] + Floor[n/i]) i^Product[k^((PrimePi[k] - PrimePi[k - 1]) (1 - Ceiling[i/k] + Floor[i/k])), {k, i}], {i, n}], {n, 30}]

rad(n) = factorback(factorint(n)[, 1]);
a(n) = sumdiv(n, d, d^rad(d)); \\ Michel Marcus, Jun 12 2021

a(prime(n)) = A125137(n). - Michel Marcus, Jun 12 2021

A277208 Numbers m such that m-1 = (tau(m-1)-1)^k for some k>=0, where tau(m) is the number of divisors of m (A000005).

Original entry on oeis.org

2, 5, 17, 28, 3126, 3376, 65537, 823544, 3748097, 52521876

Offset: 1

Jaroslav Krizek, Oct 10 2016

Corresponding pairs of numbers (tau(m-1)-1, k): (0, 0); (2, 2); (4, 2); (3, 3); (5, 5); (15, 3); (16, 4); (7, 7); ...
Numbers from A125137 (numbers of the form p^p + 1 where p = prime) are terms: 285311670612, 302875106592254, 827240261886336764178, 1978419655660313589123980, 20880467999847912034355032910568, ...
Prime terms are in A258429: 2, 5, 17, 65537, ...
A Fermat prime from A019434 of the form F(n) = 2^(2^n) + 1 is a term if k = 2^n * log(2) / log(2^n) is an integer.
a(11), if it exists, is > 10^10. - Lars Blomberg, Nov 14 2016

			3376 is in the sequence because 3375 = (tau(3375)-1)^3 = 15^3.

Cf. A000005, A019434, A125137, A258429.

Set(Sort([n: n in[2..1000000], k in [0..20] |  (n-1) eq (NumberOfDivisors(n-1)-1)^k]));

isok(n) = {if (n==2, return(1)); my(dd = numdiv(n-1) - 1); if (dd > 1, my(k = 1); while(dd^k < n-1, k++); dd^k == n-1;);} \\ Michel Marcus, Oct 11 2016

a(9)-a(10) from Michel Marcus, Oct 11 2016

A348393 a(n) = Sum_{d|n} (n^d)', where ' is the arithmetic derivative.

Original entry on oeis.org

0, 5, 28, 1060, 3126, 233885, 823544, 201351372, 2324524398, 70000350147, 285311670612, 142657631177872, 302875106592254, 100008061904383173, 3503151123048905408, 590295810427425653792, 827240261886336764178, 826274569583310299739525, 1978419655660313589123980, 2516582400000122880019968984

Offset: 1

Wesley Ivan Hurt, Oct 16 2021

nonn

			a(4) = 1060; a(4) = (4^1)' + (4^2)' + (4^4)' = 4' + 16' + 256' = 4 + 32 + 1024 = 1060.

Cf. A003415, A125137.

d[0] = d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); a[n_] := DivisorSum[n, d[n^#] &]; Array[a, 20] (* Amiram Eldar, Oct 16 2021 *)

ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
a(n) = sumdiv(n, d, ad(n^d)); \\ Michel Marcus, Oct 18 2021

If p is prime, a(p) = p^p + 1. See A125137. - Bernard Schott, Oct 18 2021

cp's OEIS Frontend

A362957 a(n) is the least prime p such that the number of distinct prime factors of p^n + 1 sets a new record.

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A238981 Sum of n-th powers of unitary divisors of n (d such that d divides n, gcd(d, n/d) = 1).

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A115973 Number of distinct prime factors of p^p + 1 where p is prime(n).

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A376432 a(n) is the least odd prime factor of prime(n)^prime(n)+1.

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A345263 a(n) = Sum_{d|n} d^rad(d).

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A277208 Numbers m such that m-1 = (tau(m-1)-1)^k for some k>=0, where tau(m) is the number of divisors of m (A000005).

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A348393 a(n) = Sum_{d|n} (n^d)', where ' is the arithmetic derivative.

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