A078701 -id:A078701 - cp's OEIS frontend

A079432 a(n) = A079431(A079431(n)).

4, 8, 9, 16, 20, 12, 28, 32, 15, 25, 44, 18, 52, 49, 21, 64, 68, 24, 76, 35, 27, 88, 92, 30, 40, 104, 33, 56, 116, 36, 124, 128, 39, 136, 50, 42, 148, 152, 45, 55, 164, 48, 172, 121, 51, 184, 188, 54, 77, 65, 57, 169, 212, 60, 70, 91, 63, 232, 236, 66, 244, 248, 69, 256

Offset: 1

Reinhard Zumkeller, Jan 09 2003

nonn

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

Cf. A078701, A079431.

f[n_] := FactorInteger[n/2^IntegerExponent[n, 2]][[1, 1]]; A079431[n_] := Module[{p = f[n], m}, m = n+p; While[f[m] != p, m+=p]; m]; a[n_] := A079431[A079431[n]]; Array[a, 100] (* Amiram Eldar, Mar 26 2025 *)

A078701(a(n)) = A078701(A079431(n)) = A078701(n).

A080215 Binomial(greatest prime factor of n, smallest odd prime factor of n).

Original entry on oeis.org

1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 10, 2, 1, 1, 1, 1, 35, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 2, 165, 1, 21, 1, 1, 1, 286, 1, 1, 35, 1, 1, 10, 1, 1, 1, 1, 1, 680, 1, 1, 1, 462, 1, 969, 1, 1, 10, 1, 1, 35, 2, 1287, 165, 1, 1, 1771, 21, 1, 1, 1, 1, 10, 1, 330, 286, 1, 1, 1, 1, 1, 35

Offset: 1

Reinhard Zumkeller, Feb 06 2003

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A006530, A078701, A080214, A080212.

a[n_] := With[{f = Select[FactorInteger[n], #[[1]] != 2&]}, If[f == {}, 2, Binomial[f[[-1, 1]], f[[1, 1]]]]];
Array[a, 100] (* Jean-François Alcover, Dec 02 2021 *)

A080215(n) = if(1==n,n,my(f = factor(n)[, 1], v = select(x->((x%2)==1), f)); binomial(f[#f], if(#v, vecmin(v), 1))); \\ Antti Karttunen, Dec 23 2018

a(n) = binomial(A006530(n), A078701(n)). - Antti Karttunen, Dec 23 2018

A108738 a(n) = n/(smallest odd prime divisor of n), if any.

Original entry on oeis.org

1, 2, 1, 4, 1, 2, 1, 8, 3, 2, 1, 4, 1, 2, 5, 16, 1, 6, 1, 4, 7, 2, 1, 8, 5, 2, 9, 4, 1, 10, 1, 32, 11, 2, 7, 12, 1, 2, 13, 8, 1, 14, 1, 4, 15, 2, 1, 16, 7, 10, 17, 4, 1, 18, 11, 8, 19, 2, 1, 20, 1, 2, 21, 64, 13, 22, 1, 4, 23, 14, 1, 24, 1, 2, 25, 4, 11, 26, 1, 16, 27, 2, 1, 28, 17, 2, 29, 8, 1

Offset: 1

S. Muthukrishnan (muthu(AT)research.att.com), Jun 23 2005

a(n) = n if n has no odd prime divisor, i.e. for n = 2^k (k>=0).

			a(21) = 21/3 = 7.

David A. Corneth, Table of n, a(n) for n = 1..10000
Z. Nedev and S. Muthukrishnan, The Nagger-Mover Game, DIMACS Tech. Report 2005-22.

Cf. A078701, A108514.

with(numtheory): a:=proc(n) local nn: nn:=factorset(n): if n=1 then 1 elif nn={2} then n elif nn[1]=2 then n/nn[2] else n/nn[1] fi end: seq(a(n),n=1..100); # Emeric Deutsch, Jun 24 2005

f[n_] := If[IntegerQ@Log[2, n], n, pf = First /@ FactorInteger@n; If[ EvenQ@n, n/pf[[2]], n/pf[[1]] ]]; Array[f, 89] (* Robert G. Wilson v, Sep 02 2006 *)

a(n) = my(v = select(x->((x%2)==1), factor(n)[,1]));  n/if (#v, vecmin(v), 1); \\ Michel Marcus, Oct 25 2017

first(n) = {my(res = vector(n, i, i)); forprime(p = 3, n, for(k = 1, n\p, if(res[k*p] == k*p, res[k*p]\=p))); res} \\ David A. Corneth, Oct 25 2017

a(n) = n/A078701(n).

More terms from Emeric Deutsch and Reinhard Zumkeller, Jun 24 2005

A112545 Least odd number k greater than 1 such that the sum of the predecessor and successor primes of the n-th prime is divisible by k or if no such odd k exists then 2.

Original entry on oeis.org

7, 5, 2, 5, 7, 2, 5, 3, 3, 3, 3, 5, 11, 3, 53, 3, 3, 3, 5, 3, 3, 3, 3, 5, 5, 13, 53, 5, 59, 61, 3, 3, 11, 5, 3, 157, 3, 3, 173, 3, 5, 11, 97, 7, 3, 211, 3, 113, 5, 3, 3, 5, 3, 257, 263, 3, 3, 3, 5, 7, 5, 151, 5, 157, 7, 3, 3, 7, 5, 3, 3, 3, 373, 3, 3, 3, 5, 13, 5, 5, 5, 7, 3, 3, 3, 3, 5, 5, 29, 3, 3

Offset: 2

Robert G. Wilson v, Jan 11 2006

nonn

From Robert Israel, Apr 20 2017: (Start)
a(n) = A078701(prime(n-1)+prime(n+1)) unless that is 1, in which case a(n)=2.
a(n) = 2 if and only if for some m, A007053(m) = n or n-1 with prime(n-1)+prime(n+1) = 2^(m+1). The first m for which this occurs are 3,4,9,379,593, corresponding to n = 4,7,97 and approximately 3*10^116 and 1*10^181.  Are there infinitely many? (End)

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A000040, A112686, A112681, A112794, A112731, A112789, A112795, A112796, A112804, A112847, A112859, A113155, A113156, A113157, A113158.

Cf. A007053, A078701.

f:= proc(n) local t; t:= min(numtheory:-factorset(ithprime(n-1)+ithprime(n+1)) minus {2}); if t::integer then t else 2 fi end proc:
map(f, [$2..200]); # Robert Israel, Apr 20 2017

f[n_] := Block[{k = 3, s = Prime[n - 1] + Prime[n + 1]}, While[Mod[s, k] != 0 && k <= s, k += 2]; If[k > s, 2, k]]; Table[ f[n], {n, 2, 92}]

a(n) = {p = prime(n); s = precprime(p-1) + nextprime(p+1); f = factor(s); if (#f~ > 1, f[2,1], f[1,1]);} \\ Michel Marcus, Apr 22 2017

A293958 Smallest odd prime divisor of (2n+1)^2 + 1.

Original entry on oeis.org

5, 13, 5, 41, 61, 5, 113, 5, 181, 13, 5, 313, 5, 421, 13, 5, 613, 5, 761, 29, 5, 1013, 5, 1201, 1301, 5, 17, 5, 1741, 1861, 5, 2113, 5, 2381, 2521, 5, 29, 5, 3121, 17, 5, 3613, 5, 17, 41, 5, 4513, 5, 13, 5101, 5, 37, 5, 13, 61, 5, 17, 5, 73, 7321, 5, 13, 5, 53, 8581, 5, 13, 5, 9661, 9941, 5

Offset: 1

N. J. A. Sloane, Nov 04 2017, following a suggestion from Zoran Sunic

nonn

If the map "x -> smallest odd prime divisor of n^2+1" is iterated, does it always terminate in the 2-cycle (5 <-> 13)? - Zoran Sunic, Oct 25 2017

A027862 is a subsequence. - David A. Corneth, Nov 04 2017

David A. Corneth, Table of n, a(n) for n = 1..10000

A bisection of A125256. Cf. A027862, A069894, A078701, A256970.

sod[n_]:=With[{fi=FactorInteger[n]},If[fi[[1,1]]==2,fi[[2,1]],fi[1,1]]]; sod/@(Range[3,151,2]^2+1) (* Harvey P. Dale, Dec 23 2023 *)

a(n) = factor((2*n+1)^2 + 1)[2,1]; \\ Michel Marcus, Nov 04 2017

a(n) = A078701(A069894(n)). - Michel Marcus, Nov 04 2017

A336650 a(n) = p^e, where p is the smallest odd prime factor of n, and e is its exponent, with a(n) = 1 when n is a power of two.

Original entry on oeis.org

1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, 3, 13, 7, 3, 1, 17, 9, 19, 5, 3, 11, 23, 3, 25, 13, 27, 7, 29, 3, 31, 1, 3, 17, 5, 9, 37, 19, 3, 5, 41, 3, 43, 11, 9, 23, 47, 3, 49, 25, 3, 13, 53, 27, 5, 7, 3, 29, 59, 3, 61, 31, 9, 1, 5, 3, 67, 17, 3, 5, 71, 9, 73, 37, 3, 19, 7, 3, 79, 5, 81, 41, 83, 3, 5, 43, 3, 11, 89, 9, 7, 23, 3

Offset: 1

Antti Karttunen, Jul 30 2020

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. A000265, A028233, A078701, A336362.

A336650(n) = if(!bitand(n,n-1),1,my(f=factor(n>>valuation(n,2))); f[1, 1]^f[1, 2]);

a(n) = A028233(A000265(n)).

A376431 a(n) is the least odd prime factor of prime(n)^prime(n)-1.

Original entry on oeis.org

3, 13, 11, 3, 5, 3, 10949, 3, 11, 7, 3, 3, 5, 3, 23, 13, 29, 3, 3, 5, 3, 3, 41, 11, 3, 5, 3, 53, 3, 7, 3, 5, 17, 3, 37, 3, 3, 3, 83, 43, 89, 3, 5, 3, 7, 3, 3, 3, 113, 3, 29, 7, 3, 5, 21589, 131, 67, 3, 3, 5, 3, 73, 3, 5, 3, 79, 3, 3, 173, 3, 11, 179, 3, 3, 3, 191

Offset: 1

Hugo Pfoertner, Sep 27 2024

nonn

Hugo Pfoertner, Table of n, a(n) for n = 1..6542

Cf. A078701, A088730, A376432.

a376431(n) = my(pp=prime(n)^prime(n)-1); forprime (p=3, oo, if(pp%p==0, return(p)))

a(n) = A078701(A088730(n)). - Michel Marcus, Sep 27 2024

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

A023505 Least odd prime divisor of prime(n) - 1, or 1 if prime(n) - 1 is a power of 2.

Original entry on oeis.org

1, 1, 1, 3, 5, 3, 1, 3, 11, 7, 3, 3, 5, 3, 23, 13, 29, 3, 3, 5, 3, 3, 41, 11, 3, 5, 3, 53, 3, 7, 3, 5, 17, 3, 37, 3, 3, 3, 83, 43, 89, 3, 5, 3, 7, 3, 3, 3, 113, 3, 29, 7, 3, 5, 1, 131, 67, 3, 3, 5, 3, 73, 3, 5, 3, 79, 3, 3, 173, 3, 11, 179, 3, 3, 3, 191, 97, 3, 5, 3, 11, 3, 5, 3, 3

Offset: 1

Clark Kimberling

nonn

Note that a(n)=1 for n= 1,2,3,7,55,6543, ... . - Michel Marcus, Oct 01 2013

Cf. A006093, A078701.

a(n) = my(p = prime(n) - 1, v = p/(2^valuation(p, 2))) ; if (v == 1, 1, factor(v)[1, 1]); \\ Michel Marcus, Oct 01 2013

a(n) = A078701(A006093(n)). - R. J. Mathar, Feb 06 2019

A023578 Least odd prime divisor of prime(n)+3, or 1 if prime(n)+3 is a power of 2.

Original entry on oeis.org

5, 3, 1, 5, 7, 1, 5, 11, 13, 1, 17, 5, 11, 23, 5, 7, 31, 1, 5, 37, 19, 41, 43, 23, 5, 13, 53, 5, 7, 29, 5, 67, 5, 71, 19, 7, 5, 83, 5, 11, 7, 23, 97, 7, 5, 101, 107, 113, 5, 29, 59, 11, 61, 127, 5, 7, 17, 137, 5, 71, 11, 37, 5, 157, 79, 5, 167, 5, 5, 11, 89, 181, 5, 47, 191

Offset: 1

Clark Kimberling

nonn

Cf. A023505, A078701, A113935.

a(n) = my(p = prime(n)+3, v = p/(2^valuation(p, 2))) ; if (v == 1, 1, factor(v)[1, 1]); \\ Michel Marcus, Aug 05 2021

from sympy import factorint, prime
def A023578(n): return min((p for p in factorint(prime(n)+3) if p > 2), default=1) # Chai Wah Wu, Feb 03 2022

a(n) = A078701(A113935(n)). - Michel Marcus, Aug 05 2021

cp's OEIS Frontend

A079432 a(n) = A079431(A079431(n)).

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A080215 Binomial(greatest prime factor of n, smallest odd prime factor of n).

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A108738 a(n) = n/(smallest odd prime divisor of n), if any.

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A112545 Least odd number k greater than 1 such that the sum of the predecessor and successor primes of the n-th prime is divisible by k or if no such odd k exists then 2.

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A293958 Smallest odd prime divisor of (2n+1)^2 + 1.

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A336650 a(n) = p^e, where p is the smallest odd prime factor of n, and e is its exponent, with a(n) = 1 when n is a power of two.

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A376431 a(n) is the least odd prime factor of prime(n)^prime(n)-1.

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

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