A037276 -id:A037276 - cp's OEIS frontend

A037274 Home primes: for n >= 2, a(n) = the prime that is finally reached when you start with n, concatenate its prime factors (A037276) and repeat until a prime is reached (a(n) = -1 if no prime is ever reached).

1, 2, 3, 211, 5, 23, 7, 3331113965338635107, 311, 773, 11, 223, 13, 13367, 1129, 31636373, 17, 233, 19, 3318308475676071413, 37, 211, 23, 331319, 773, 3251, 13367, 227, 29, 547, 31, 241271, 311, 31397, 1129, 71129, 37, 373, 313, 3314192745739, 41, 379, 43, 22815088913, 3411949, 223, 47, 6161791591356884791277

Offset: 1

N. J. A. Sloane, Jeff Burch

The initial 1 could have been omitted.
Probabilistic arguments give exactly zero for the chance that the sequence of integers starting at n contains no prime, the expected number of primes being given by a divergent sequence. - J. H. Conway
After over 100 iterations, a(49) is still composite - see A056938 for the latest information.
More terms:
a(50) to a(60) are 3517, 317, 2213, 53, 2333, 773, 37463, 1129, 229, 59, 35149;
a(61) to a(65) are 61, 31237, 337, 1272505013723, 1381321118321175157763339900357651;
a(66) to a(76) are 2311, 67, 3739, 33191, 257, 71, 1119179, 73, 379, 571, 333271.
This is different from A195264. Here 8 = 2^3 -> 222 -> ... -> 3331113965338635107 (a prime), whereas in A195264 8 = 2^3 -> 23 (a prime). - N. J. A. Sloane, Oct 12 2014

			9 = 3*3 -> 33 = 3*11 -> 311, prime, so a(9) = 311.
The trajectory of 8 is more interesting:
8 ->
2 * 2 * 2 ->
2 * 3 * 37 ->
3 * 19 * 41 ->
3 * 3 * 3 * 7 * 13 * 13 ->
3 * 11123771 ->
7 * 149 * 317 * 941 ->
229 * 31219729 ->
11 * 2084656339 ->
3 * 347 * 911 * 118189 ->
11 * 613 * 496501723 ->
97 * 130517 * 917327 ->
53 * 1832651281459 ->
3 * 3 * 3 * 11 * 139 * 653 * 3863 * 5107
and 3331113965338635107 is prime, so a(8) = 3331113965338635107.

Jeffrey Heleen, Family Numbers: Mathemagical Black Holes, Recreational and Educational Computing, 5:5, pp. 6, 1990.
Jeffrey Heleen, Family numbers: Constructing Primes by Prime Factor Splicing, J. Recreational Math., Vol. 28 #2, 1996-97, pp. 116-119.

Christian N. K. Anderson, Table of known values of n, # of steps to reach a(n), and a(n) or NA if a(n) has 30 digits or more. Also, the trajectory, with factors separated by a |, terminated by either "(end)" or "-> ?" if a(n) has 30 digits or more.
Patrick De Geest, Home Primes < 100 and Beyond
M. Herman and J. Schiffman, Investigating home primes and their families, Math. Teacher, 107 (No. 8, 2014), 606-614.
N. J. A. Sloane, Confessions of a Sequence Addict (AofA2017), slides of invited talk given at AofA 2017, Jun 19 2017, Princeton. Mentions this sequence.
N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, Part I, Part 2, Slides. (Mentions this sequence)
Eric Weisstein's World of Mathematics, Home Prime.
Wikipedia, Home prime

Cf. A006919, A037271, A037272, A037273, A037275, A037276, A037919-A037941, A048986, A056938.
Cf. A195264 (use exponents instead of repeating primes).
Cf. A084318 (use only one copy of each prime), A248713 (Fermi-Dirac analog: use unique representation of n>1 as a product of distinct terms of A050376).
Cf. also A120716 and related sequences.

b:= n-> parse(cat(sort(map(i-> i[1]$i[2], ifactors(n)[2]))[])):
a:= n-> `if`(isprime(n) or n=1, n, a(b(n))):
seq(a(n), n=1..48);  # Alois P. Heinz, Jan 09 2021

f[n_] := FromDigits@ Flatten[ IntegerDigits@ Table[ #[[1]], { #[[2]] }] & /@ FactorInteger@n, 2]; g[n_] := NestWhile[ f@# &, n, !PrimeQ@# &]; g[1] = 1; Array[g, 41] (* Robert G. Wilson v, Sep 22 2007 *)

step(n)=my(f=factor(n),s="");for(i=1,#f~,for(j=1,f[i,2],s=Str(s,f[i,1]))); eval(s)
a(n)=if(n<4,return(n)); while(!isprime(n), n=step(n)); n \\ Charles R Greathouse IV, May 14 2015

from sympy import factorint, isprime
def f(n): return int("".join(str(p)*e for p, e in factorint(n).items()))
def a(n):
    if n == 1: return 1
    fn = n
    while not isprime(fn): fn = f(fn)
    return fn
print([a(n) for n in range(1, 40)]) # Michael S. Branicky, Jul 11 2022

def digitLen(x,n):
    r=0
    while(x>0):
        x//=n
        r+=1
    return r
def concatPf(x,n):
    r=0
    f=list(factor(x))
    for c in range(len(f)):
        for d in range(f[c][1]):
            r*=(n**digitLen(f[c][0],n))
            r+=f[c][0]
    return r
def hp(x,n):
    x1=concatPf(x,n)
    while(x1!=x):
        x=x1
        x1=concatPf(x1,n)
    return x
#example: prints the home prime of 8 in base 10
print(hp(8,10))

Corrected and extended by Karl W. Heuer, Sep 30 2003

A340633 Primes of the form k + A037276(k).

Original entry on oeis.org

2, 29, 41, 233, 239, 251, 257, 269, 293, 311, 359, 383, 401, 419, 449, 467, 491, 2269, 2309, 2339, 2377, 2381, 2393, 2411, 2417, 2447, 2473, 2503, 2543, 2579, 2591, 2621, 2633, 2671, 2687, 2699, 2713, 2753, 2789, 2797, 2819, 2843, 2879, 2939, 3011, 3041, 3067, 3083, 3119, 3137, 3167, 3191, 3203

Offset: 1

J. M. Bergot and Robert Israel, Jan 13 2021

All terms have first digit 2, 3 or 4.

			a(1) =   2 =  1 + A037276(1)  =  1 +   1;
a(2) =  29 =  6 + A037276(6)  =  6 +  23;
a(3) =  41 = 14 + A037276(14) = 14 +  27;
a(4) = 233 = 22 + A037276(22) = 22 + 211;
a(5) = 251 = 18 + A037276(18) = 18 + 233
           = 34 + A037276(34) = 34 + 217.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A037276, A340634, A340636.

N:= 5000: # for terms <= N
dcat:= proc(L) local i, x;
  x:= L[-1];
  for i from nops(L)-1 to 1 by -1 do
    x:= 10^(1+ilog10(x))*L[i]+x
  od;
  x
end proc:
A037276:= proc(n) local F;
  F:= sort(ifactors(n)[2], (a, b) -> a[1] < b[1]);
  dcat(map(t -> t[1]$t[2], F));
end proc:
A037276(1):= 1:
R:= NULL:
for n from 1 to N/2 do
  v:= n + A037276(n);
  if v < N and isprime(v) then R:= R, v fi;
od:
sort(convert({R},list));

A340634 Numbers k such that k + A037276(k) is prime.

Original entry on oeis.org

1, 6, 14, 18, 22, 26, 34, 36, 38, 46, 48, 62, 66, 74, 106, 108, 110, 122, 134, 146, 156, 166, 170, 174, 178, 194, 196, 198, 206, 226, 230, 254, 262, 274, 278, 290, 294, 298, 306, 318, 354, 362, 374, 386, 392, 394, 416, 420, 422, 426, 458, 466, 468, 490, 502, 504, 516, 526, 528, 530, 532, 544, 562

Offset: 1

J. M. Bergot and Robert Israel, Jan 14 2021

All terms except 1 are even.

			a(3) = 14 is a term because 14 + A037276(14) = 14 + 27 = 41 is prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A037276, A340633, A340636.

dcat:= proc(L) local i, x;
  x:= L[-1];
  for i from nops(L)-1 to 1 by -1 do
    x:= 10^(1+ilog10(x))*L[i]+x
  od;
  x
end proc:
A037276:= proc(n) local F;
  F:= sort(ifactors(n)[2], (a, b) -> a[1] < b[1]);
  dcat(map(t -> t[1]$t[2], F));
end proc:
A037276(1):= 1:
select(t -> isprime(t + A037276(t)), [1,seq(i,i=3..1000)]);

A340636 Primes of the form k + A037276(k) in more than one way.

Original entry on oeis.org

251, 2671, 2687, 2753, 23327, 23561, 27827, 28499, 28789, 28817, 29411, 34757, 223441, 226001, 227537, 230849, 231359, 232217, 232259, 232367, 232643, 232919, 233591, 234791, 236129, 236609, 236867, 237857, 238141, 239023, 239873, 240899, 241169, 241343, 241687, 241691, 242447, 242747, 245299

Offset: 1

J. M. Bergot and Robert Israel, Jan 14 2021

			a(3) = 2687 = 170 + A037276(170) = 170 + 2517
            = 458 + A037276(458) = 458 + 2229.
The first term that occurs in more than two ways is
a(163) = 2255299 =  4180 + A037276(4180)  =  4180 + 2251119
                 = 21156 + A037276(21156) = 21156 + 2234143
                 = 29560 + A037276(29560) = 29560 + 2225739.

Robert Israel, Table of n, a(n) for n = 1..7500

Cf. A037276, A340633, A340634.

N:= 5*10^5: # for terms <= N
dcat:= proc(L) local i, x;
  x:= L[-1];
  for i from nops(L)-1 to 1 by -1 do
    x:= 10^(1+ilog10(x))*L[i]+x
  od;
  x
end proc:
A037276:= proc(n) local F;
  F:= sort(ifactors(n)[2], (a, b) -> a[1] < b[1]);
  dcat(map(t -> t[1]$t[2], F));
end proc:
A037276(1):= 1:
R:= NULL:
for n from 1 to N/2 do
  v:= n + A037276(n);
  if v < N and isprime(v) then R:= R, v fi;
od:
S:= {R}:
select(s -> numboccur(s,[R])>1, S);

A287883 Partial sums of A037276.

Original entry on oeis.org

1, 3, 6, 28, 33, 56, 63, 285, 318, 343, 354, 577, 590, 617, 652, 2874, 2891, 3124, 3143, 3368, 3405, 3616, 3639, 5862, 5917, 6130, 6463, 6690, 6719, 6954, 6985, 29207, 29518, 29735, 29792, 32025, 32062, 32281, 32594, 34819, 34860, 35097, 35140, 37351, 37686, 37909, 37956, 60179, 60256, 60511

Offset: 1

N. J. A. Sloane, Jun 20 2017

N. J. A. Sloane, Table of n, a(n) for n = 1..20000

Cf. A037276.

co[n_, k_] := Nest[Flatten[IntegerDigits[{#, n}]] &, n, k - 1]; A037276 =
Table[FromDigits[Flatten[IntegerDigits[co @@@ FactorInteger[n]]]], {n, 50}]; Table[Sum[A037276[[k]], {k, 1, n}], {n, 1, 25}] (* G. C. Greubel, Jun 23 2017 *)

from sympy import factorint
def a037276(n):
    f=factorint(n)
    l=sorted([i for i in f])
    return 1 if n==1 else int("".join(str(i)*f[i] for i in l))
l=[0, 1]
for n in range(2, 101): l.append(l[n - 1] + a037276(n))
print(l[1:]) # Indranil Ghosh, Jun 23 2017

A289660 a(n) = A037276(n) - n.

Original entry on oeis.org

0, 0, 0, 18, 0, 17, 0, 214, 24, 15, 0, 211, 0, 13, 20, 2206, 0, 215, 0, 205, 16, 189, 0, 2199, 30, 187, 306, 199, 0, 205, 0, 22190, 278, 183, 22, 2197, 0, 181, 274, 2185, 0, 195, 0, 2167, 290, 177, 0, 22175, 28, 205, 266, 2161, 0, 2279, 456, 2171, 262, 171, 0, 2175, 0, 169, 274, 222158, 448, 2245, 0

Offset: 1

N. J. A. Sloane, Jul 24 2017

The fact that a(n)>0 if n is composite shows that (unlike A080670), A037276 has no nontrivial fixed points (nor any cycles).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A037274, A037276, A080670.

FromDigits@Flatten@IntegerDigits[Table[#1,{#2}]&@@@FactorInteger@#]-#&/@Range@54 (* Vincenzo Librandi, Jul 25 2017 *)

from sympy import factorint
def A289660(n):
    return 0 if n == 1 else int(''.join(map(lambda x: str(x[0])*x[1],sorted(factorint(n).items())))) - n # Chai Wah Wu, Jul 25 2017

A351975 Numbers k such that A037276(k) == -1 (mod k).

Original entry on oeis.org

1, 6, 14, 18, 48, 124, 134, 284, 3135, 4221, 9594, 16468, 34825, 557096, 711676, 746464, 1333334, 2676977, 6514063, 11280468, 16081252, 35401658, 53879547, 133333334, 198485452, 223856659, 1333333334, 2514095219, 2956260256, 3100811124, 10912946218, 19780160858

Offset: 1

J. M. Bergot and Robert Israel, Feb 26 2022

Numbers k such that the concatenation of prime factors of k is 1 less than a multiple of k.
Contains 2*m for m in A093170.
Terms k where k-1 is prime include 6, 14, 18, 48 and 284. Are there others?

			a(4) = 48 is a term because 48=2*2*2*2*3 and 22223 == -1 (mod 48).

Martin Ehrenstein, Table of n, a(n) for n = 1..38

Cf. A037276, A093170.

tcat:= proc(x,y) x*10^(1+ilog10(y))+y end proc:
filter:= proc(n) local F,t,i;
F:= map(t -> t[1]$t[2], sort(ifactors(n)[2],(a,b)->a[1]

from sympy import factorint
def A037276(n):
    if n == 1: return 1
    return int("".join(str(p)*e for p, e in sorted(factorint(n).items())))
def afind(limit, startk=1):
    for k in range(startk, limit+1):
        if (A037276(k) + 1)%k == 0:
            print(k, end=", ")
afind(10**6) # Michael S. Branicky, Feb 27 2022
# adapted and corrected by Martin Ehrenstein, Mar 06 2022

from itertools import count, islice
from sympy import factorint
def A351975_gen(startvalue=1): # generator of terms >= startvalue
    for k in count(max(startvalue,1)):
        c = 0
        for d in sorted(factorint(k,multiple=True)):
            c = (c*10**len(str(d)) + d) % k
        if c == k-1:
            yield k
A351975_list = list(islice(A351975_gen(),10)) # Chai Wah Wu, Feb 28 2022

a(24)-a(25) from Michael S. Branicky, Feb 27 2022

Prepended 1 and more terms from Martin Ehrenstein, Feb 28 2022

A080670 Literal reading of the prime factorization of n.

Original entry on oeis.org

1, 2, 3, 22, 5, 23, 7, 23, 32, 25, 11, 223, 13, 27, 35, 24, 17, 232, 19, 225, 37, 211, 23, 233, 52, 213, 33, 227, 29, 235, 31, 25, 311, 217, 57, 2232, 37, 219, 313, 235, 41, 237, 43, 2211, 325, 223, 47, 243, 72, 252, 317, 2213, 53, 233, 511, 237, 319, 229, 59, 2235

Offset: 1

Jon Perry, Mar 02 2003

Exponents equal to 1 are omitted and therefore this sequence differs from A067599.
Here the first duplicate (ambiguous) term appears already with a(8)=23=a(6), in A067599 this happens only much later. - M. F. Hasler, Oct 18 2014
The number n = 13532385396179 = 13·53^2·3853·96179 = a(n) is (maybe the first?) nontrivial fixed point of this sequence, making it the first known index of a -1 in A195264. - M. F. Hasler, Jun 06 2017

			8=2^3, which reads 23, hence a(8)=23; 12=2^2*3, which reads 223, hence a(12)=223.

T. D. Noe, Table of n, a(n) for n = 1..10000
Tony Padilla and Brady Haran, 13532385396179, Numberphile Video, 2017
N. J. A. Sloane, Confessions of a Sequence Addict (AofA2017), slides of invited talk given at AofA 2017, Jun 19 2017, Princeton. Mentions this sequence.
N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, Part I, Part 2, Slides. (Mentions this sequence)

Cf. A037276, A067599, A230305, A230625, A027748, A124010, A288818.
See A195330, A195331 for those n for which a(n) is a contraction.
See also home primes, A037271.
See A195264 for what happens when k -> a(k) is repeatedly applied to n.
Partial sums: A287881, A287882.

import Data.Function (on)
a080670 1 = 1
a080670 n = read $ foldl1 (++) $
zipWith (c `on` show) (a027748_row n) (a124010_row n) :: Integer
where c ps es = if es == "1" then ps else ps ++ es
-- Reinhard Zumkeller, Oct 27 2013

ifsSorted := proc(n)
        local fs,L,p ;
        fs := sort(convert(numtheory[factorset](n),list)) ;
        L := [] ;
        for p in fs do
                L := [op(L),[p,padic[ordp](n,p)]] ;
        end do;
        L ;
end proc:
A080670 := proc(n)
        local a,p ;
        if n = 1 then
                return 1;
        end if;
        a := 0 ;
        for p in ifsSorted(n) do
                a := digcat2(a,op(1,p)) ;
                if op(2,p) > 1 then
                        a := digcat2(a,op(2,p)) ;
                end if;
        end do:
        a ;
end proc: # R. J. Mathar, Oct 02 2011
# second Maple program:
a:= proc(n) option remember; `if`(n=1, 1, (l->
      parse(cat(seq(`if`(l[i, 2]=1, l[i, 1], [l[i, 1],
      l[i, 2]][]), i=1..nops(l)))))(sort(ifactors(n)[2])))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Mar 17 2020

f[n_] := FromDigits[ Flatten@ IntegerDigits[ Flatten[ FactorInteger@ n /. {1 -> {}}]]]; f[1] = 1; Array[ f, 60] (* Robert G. Wilson v, Mar 02 2003 and modified Jul 22 2014 *)

A080670(n)=if(n>1, my(f=factor(n),s=""); for(i=1,#f~,s=Str(s,f[i,1],if(f[i,2]>1, f[i,2],""))); eval(s),1) \\ Charles R Greathouse IV, Oct 27 2013; case n=1 added by M. F. Hasler, Oct 18 2014

A080670(n)=if(n>1,eval(concat(apply(f->Str(f[1],if(f[2]>1,f[2],"")),Vec(factor(n)~)))),1) \\ M. F. Hasler, Oct 18 2014

import sympy
[int(''.join([str(y) for x in sorted(sympy.ntheory.factorint(n).items()) for y in x if y != 1])) for n in range(2,100)] # compute a(n) for n > 1
# Chai Wah Wu, Jul 15 2014

Edited and extended by Robert G. Wilson v, Mar 02 2003

A037271 Number of steps to reach a prime under "replace n with concatenation of its prime factors" when applied to n-th composite number, or -1 if no such number exists.

Original entry on oeis.org

2, 1, 13, 2, 4, 1, 5, 4, 4, 1, 15, 1, 1, 2, 3, 4, 4, 1, 2, 2, 1, 5, 3, 2, 2, 1, 9, 2, 9, 6, 1, 15

Offset: 1

Jeff Burch

a(33) is presently unknown: starting with 49, no prime has been reached after 110 steps. See A037274 for the latest information.

			Starting with 14 (the seventh composite number) we get 14=2*7, 27=3*3*3, 333=3*3*37, 3337=47*71, 4771=13*367, 13367 is prime; so a(7)=5.

Patrick De Geest, Home Primes
M. Herman and J. Schiffman, Investigating home primes and their families, Math. Teacher, 107 (No. 8, 2014), 606-614.
Eric Weisstein's World of Mathematics, Home Prime

Cf. A037272, A037273, A037274, A056938, A002808, A037276, A027746, A230305.

a037271 = length . takeWhile ((== 0) . a010051'') .
                             iterate a037276 . a002808
-- Reinhard Zumkeller, Apr 03 2012

maxComposite = 49; maxIter = 40; concat[n_] := FromDigits[ Flatten[ IntegerDigits /@ Flatten[ Apply[ Table, {#[[1]], {#[[2]]}} & /@ FactorInteger[n], {1}]]]]; composites = Select[ Range[2, maxComposite], ! PrimeQ[#] &]; a[n_] := ( lst = NestWhileList[ concat, composites[[n]], ! PrimeQ[#] &, 1, maxIter]; If[PrimeQ[ Last[lst]], Length[lst] - 1, - 1]); Table[a[n], {n, 1, Length[composites]}] (* Jean-François Alcover, Jul 10 2012 *)

A067599 Decimal encoding of the prime factorization of n: concatenation of prime factors and exponents.

Original entry on oeis.org

21, 31, 22, 51, 2131, 71, 23, 32, 2151, 111, 2231, 131, 2171, 3151, 24, 171, 2132, 191, 2251, 3171, 21111, 231, 2331, 52, 21131, 33, 2271, 291, 213151, 311, 25, 31111, 21171, 5171, 2232, 371, 21191, 31131, 2351, 411, 213171, 431, 22111, 3251, 21231

Offset: 2

Joseph L. Pe, Jan 31 2002

If n has prime factorization p_1^e_1 * ... * p_r^e_r with p_1 < ... < p_r, then its decimal encoding is p_1 e_1...p_r e_r. For example, 15 = 3^1 * 5^1, so has decimal encoding 3151.
Sequence A068633 is a duplicate, up to a conventional initial term a(1)=11.
a(31) = a(177147) = 311. Is there any solution to a(n) = n? - Franklin T. Adams-Watters, Dec 18 2006
The earliest duplicate is a(223) = 2231 = a(12). There is no fixed point below 3*10^6. - M. F. Hasler, Oct 06 2013

			The prime factorization of 24 = 2^3 * 3^1 has corresponding encoding 2331. So a(24) = 2331.
a(42) = 213171 since 42 = 2^1*3^1*7^1. - _Amarnath Murthy_, Feb 27 2002

Reinhard Zumkeller, Table of n, a(n) for n = 2..10000

Cf. A037276, A080670, A112375.

Cf. A027748, A124010.

import Data.Function (on)
a067599 n = read $ foldl1 (++) $
   zipWith ((++) `on` show) (a027748_row n) (a124010_row n) :: Integer
-- Reinhard Zumkeller, Oct 27 2013

with(ListTools): with(MmaTranslator[Mma]): seq(FromDigits(FlattenOnce(ifactors(n)[2])), n=2..46); # Wolfdieter Lang, Aug 16 2014
# second Maple program:
a:= n-> parse(cat(map(i-> i[], sort(ifactors(n)[2]))[])):
seq(a(n), n=2..60);  # Alois P. Heinz, Mar 16 2018

f[n_] := FromDigits[ Flatten[ IntegerDigits[ FactorInteger[ n]]]]; Table[ f[n], {n, 2, 50} ]

A067599(n)=eval(concat(concat([""],concat(Vec(factor(n)~))~))) \\ - M. F. Hasler, Oct 06 2013

Edited by Robert G. Wilson v, Feb 02 2002

Merged contributions from A068633 to here, and minor edits by M. F. Hasler, Oct 06 2013

cp's OEIS Frontend

A037274 Home primes: for n >= 2, a(n) = the prime that is finally reached when you start with n, concatenate its prime factors (A037276) and repeat until a prime is reached (a(n) = -1 if no prime is ever reached).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

SageMath

Extensions

A340633 Primes of the form k + A037276(k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A340634 Numbers k such that k + A037276(k) is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A340636 Primes of the form k + A037276(k) in more than one way.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

A287883 Partial sums of A037276.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Python

A289660 a(n) = A037276(n) - n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

A351975 Numbers k such that A037276(k) == -1 (mod k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Python

Python

Extensions

A080670 Literal reading of the prime factorization of n.

Views