A234575 -id:A234575 - cp's OEIS frontend

A357190 a(n) is the least prime p such that A234575(p, A007953(p)) is the n-th power of a prime.

17, 13, 131, 107, 383, 613, 43607, 1021, 334403, 26099, 40637, 138967, 212867, 360049, 502210997, 2227399, 5682166613, 7339303, 13630913, 35650627, 92273957, 142605709, 4424729404133, 671087119, 42364430471219, 2684353351, 404156666702231, 10737417109, 4872756792902003

Offset: 1

J. M. Bergot and Robert Israel, Oct 25 2022

a(n) is the least prime p such that the sum of the quotient and remainder on division of p by the sum of digits of p is the n-th power of an integer.

			a(3) = 131 because 131 is prime, has sum of digits 5, 131 = 26*5 + 1 and 26 + 1 = 27 = 3^3 where 3 is prime; and 131 is the least prime that works.

Robert Israel, Table of n, a(n) for n = 1..100

Cf. A007953, A234575.

g:= proc(t,M) local s,q,r,n;
  for s from 2 to 9*M do
    for r from s-1 to 1 by -1 do
      q:= t-r;
      n:= q*s+r;
      if convert(convert(n,base,10),`+`) = s and isprime(n) then return n fi;
      if n >= 10^M then return -1 fi;
   od od;
   -1
end proc:
G:= proc(m) local i,M,found,v,r;
  found:= false; r:= infinity;
  for M from 3 while not found do
    for i from 1 while ithprime(i)^m < 10^M do
      v:= g(ithprime(i)^m, M);
      if v > 0 then found:= true; r:= min(v,r) fi
  od od:
  r
end proc:
map(G, [$1..30]);

A358034 Numbers k such that A234575(k,s) = s^2 where s = A007953(k).

Original entry on oeis.org

1, 113, 313, 331, 512, 1271, 2065, 2137, 2173, 2705, 3291, 3931, 4066, 4913, 5832, 6535, 6553, 6571, 6607, 6625, 6643, 6661, 6715, 6733, 6751, 6805, 6823, 6841, 7715, 13479, 13686, 15289, 15577, 17576, 19449, 19683, 21898, 23969, 49789, 49897, 49969

Offset: 1

J. M. Bergot and Robert Israel, Oct 25 2022

Numbers k such that, if the sum of digits of k is s, the quotient and remainder on division of k by s sum to s^2.

			a(3) = 313 is a term because the sum of digits of 313 is 7, 313 = 44*7+5, and 44+5 = 49 = 7^2.

Cf. A007953, A234575.

filter:= proc(n) local s,q,r;
  s:= convert(convert(n,base,10),`+`);
r:= n mod s;
  q:= (n-r)/s;
q+r = s^2
end proc:
select(filter, [$1..10^6]);

from itertools import count, islice
def A358034_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:(s:=sum(int(d) for d in str(n)))**2 == sum(divmod(n,s)),count(max(startvalue,1)))
A358034_list = list(islice(A358034_gen(),30)) # Chai Wah Wu, Oct 26 2022

A372523 Triangle read by rows: T(n, k) is equal to n/k if k | n, else to the concatenation of A003988(n, k) = floor(n/k) and A051127(k, n) = n mod k.

Original entry on oeis.org

1, 2, 1, 3, 11, 1, 4, 2, 11, 1, 5, 21, 12, 11, 1, 6, 3, 2, 12, 11, 1, 7, 31, 21, 13, 12, 11, 1, 8, 4, 22, 2, 13, 12, 11, 1, 9, 41, 3, 21, 14, 13, 12, 11, 1, 10, 5, 31, 22, 2, 14, 13, 12, 11, 1, 11, 51, 32, 23, 21, 15, 14, 13, 12, 11, 1, 12, 6, 4, 3, 22, 2, 15, 14, 13, 12, 11, 1

Offset: 1

Stefano Spezia, May 04 2024

			The triangle begins:
  1;
  2,  1;
  3, 11,  1;
  4,  2, 11,  1;
  5, 21, 12, 11,  1;
  6,  3,  2, 12, 11,  1;
  7, 31, 21, 13, 12, 11, 1;
  ...

Stefano Spezia, First 150 rows of the triangle, flattened

Cf. A003988, A004216, A051127, A055642.
Cf. A000012 (right diagonal), A000027 (1st column).
Cf. A048158, A051126, A051127, A234575.

T[n_,k_]:=If[Divisible[n,k],n/k,FromDigits[Join[IntegerDigits[Floor[n/k]],IntegerDigits[Mod[n,k]]]]]; Table[T[n,k],{n,12},{k,n}]//Flatten (* or *)
T[n_,k_]:=Floor[n/k]10^IntegerLength[Mod[n,k]]+Mod[n,k]; Table[T[n,k],{n,12},{k,n}]//Flatten (* or *)
T[n_, k_]:=SeriesCoefficient[x^k(1+Sum[(i + 10^(1+Floor[Log10[Mod[n,k]]]))*x^i, {i, k-1}] - Sum[i*x^(k+i), {i, k-1}])/(1-x^k)^2, {x, 0, n}]; Table[T[n, k], {n, 12}, {k, n}]//Flatten

T(n, k) = floor(n/k)*10^(1+floor(log10(n mod k))) + (n mod k) if n is not divisible by k.
T(n, n) = 1.
T(n, 1) = n.
T(n, k) = 2*T(n-k, k) - T(n-2*k, k) for n >= 3*k.
T(n, k) = [x^n] x^k*(1 + (Sum_{i=1..k-1} (i + 10^(1+floor(log10(n mod k))))*x^i) - (Sum_{i=1..k-1} i*x^(k+i)))/(1 - x^k)^2.

A358052 Triangular array read by rows. For T(n,k) where 1 <= k <= n, start with x = k and repeat the map x -> floor(n/x) + (n mod x) until an x occurs that has already appeared. The number of applications of the map is T(n,k).

Original entry on oeis.org

1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 3, 3, 2, 2, 2, 1, 1, 2, 3, 2, 2, 2, 3, 2, 3, 4, 3, 2, 2, 2, 1, 2, 1, 3, 2, 3, 2, 2, 2, 2, 1, 2, 3, 2, 3, 3, 2, 2, 2, 2, 3, 2, 1, 3, 4, 3, 3, 2, 2, 2, 2, 2, 3, 2, 3, 4, 3, 3, 3, 2, 2, 2, 2, 1, 1, 3, 1, 4, 2, 2, 3, 3, 2, 2, 2, 4, 3, 3, 3, 2, 3

Offset: 1

J. M. Bergot and Robert Israel, Oct 27 2022

T(n,k) = 1 if k^2 >= n > k and k-1 divides n-k.

If A234575(n,k) = j then either T(n,k) = T(n,j)+1 and A357554(n,k) = A357554(n,j), or T(n,k) = T(n,j), A357554(n,k) = k and A357554(n,j) = j.

			For T(13,2) we have 2 -> floor(13/2) + (13 mod 2) = 7 -> floor(13/7) + (13 mod 7) = 7.  That is two applications of the map so T(13,2) = 2.
Triangle starts:
  1;
  2, 2;
  2, 1, 2;
  2, 1, 2, 2;
  2, 2, 1, 3, 2;
  2, 2, 2, 3, 3, 2;
  2, 2, 1, 1, 2, 3, 2;
  2, 2, 3, 2, 3, 4, 3, 2;
  2, 2, 1, 2, 1, 3, 2, 3, 2;
  2, 2, 2, 1, 2, 3, 2, 3, 3, 2;

Robert Israel, Table of n, a(n) for n = 1..10011(rows 1 to 141, flattened)

Cf. A234575, A357554.

f:= proc(n, k) local x, S,count;
  S:= {k};
  x:= k;
  for count from 1 do
     x:= iquo(n, x) + irem(n, x);
     if member(x, S) then return count fi;
     S:= S union {x};
  od
end proc:
for n from 1 to 20 do seq(f(n, k), k=1..n) od;

f[n_, k_] := Module[{x, S, count}, S = {k}; x = k; For[count = 1, True, count++, x = Quotient[n, x] + Mod[n, x]; If[MemberQ[S, x], Return[count]]; S = S~Union~{x}]];
Table[f[n, k], {n, 1, 20}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 29 2023, after Maple code *)

A234541 Least k such that floor(n/k) + (n mod k) is a prime, or 0 if no such k exists.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 4, 1, 2, 3, 8, 1, 6, 1, 4, 2, 2, 1, 8, 2, 2, 5, 4, 1, 6, 1, 6, 2, 2, 3, 12, 1, 2, 3, 8, 1, 6, 1, 4, 2, 2, 1, 16, 3, 10, 3, 4, 1, 18, 3, 6, 2, 2, 1, 8, 1, 2, 6, 18, 3, 6, 1, 4, 3, 4, 1, 14, 1, 2, 9, 4, 5, 6, 1, 16, 2, 2, 1, 12, 2, 2

Offset: 1

Alex Ratushnyak, Dec 27 2013

a(n) = 1 only if n is a prime.
a(2m) <= m, because with k=m, floor(2m/m)+(2m mod m) = 2.
a(2m+1) <= 2m: floor((2m+1)/2m) + ((2m+1) mod 2m) = 1 + 1 = 2.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000040, A234575.

primes = [2, 3]
primFlg = [0]*100000
primFlg[2] = primFlg[3] = 1
def appPrime(k):
  for p in primes:
    if k%p==0:  return
    if p*p > k:  break
  primes.append(k)
  primFlg[k] = 1
for n in range(5, 100000, 6):
  appPrime(n)
  appPrime(n+2)
for n in range(1, 100000):
  a = 0
  for k in range(1, n):
    c = n//k + n%k
    if primFlg[c]:  # if c in primes:
      a = k
      break
  print(str(a), end=', ')

;; MIT/GNU Scheme, with Aubrey Jaffer's SLIB Scheme library and function A234575bi as defined in A234575
(require 'factor) ;; For predicate prime? from SLIB-library.
(define (A234541 n) (let loop ((k 1)) (cond ((prime? (A234575bi n k)) k) ((> k n) 0) (else (loop (+ 1 k))))))
;; Antti Karttunen, Dec 29 2013

A358055 a(n) is the least m such that A358052(m,k) = n for some k.

Original entry on oeis.org

1, 2, 5, 8, 14, 20, 32, 38, 59, 59, 63, 116, 122, 158, 158, 218, 278, 278, 402, 548, 642, 642, 642, 642, 642, 1062, 1062, 1668, 2474, 2690, 2690, 2690, 2690, 2690, 3170, 3170, 3170, 3170, 3170, 3170, 3170, 9260, 9260, 9260, 9788, 9788, 11772, 11942, 11942, 11942, 11942, 11942

Offset: 1

J. M. Bergot and Robert Israel, Oct 27 2022

nonn

a(n) is the least m such that iteration of the map x -> floor(m/x) + (m mod x), starting at some k in [1,m], produces n distinct values before repeating.

			a(4) = 8 because A358052(8,6) = 4 and this is the first appearance of 4 in A358052.
Thus the map x -> floor(8/x) + (8 mod x) starting at 6 produces 4 distinct values before repeating: 6 -> 3 -> 4 -> 2 -> 4.

Cf. A234575, A358052.

f:= proc(n, k) local x, S, count;
  S:= {k};
  x:= k;
  for count from 1 do
     x:= iquo(n, x) + irem(n, x);
     if member(x, S) then return count fi;
     S:= S union {x};
  od
end proc:
V:= Vector(50): count:= 0:
for n from 1 while count < 50 do
  for k from 1 to n do
    v:= f(n,k);
    if v <= 50 and V[v] = 0 then
       V[v]:= n; count:= count+1;
    fi
od od:
convert(V,list);

f[n_, k_] := Module[{x, S, count}, S = {k}; x = k; For[count = 1, True, count++, x = Quotient[n, x] + Mod[n, x]; If[MemberQ[S, x], Return@count]; S = S~Union~{x}]];
V = Table[0, {vmax = 40}]; count = 0;
For[n = 1, count < vmax, n++, For[k = 1, k <= n, k++, v = f[n, k]; If[v <= vmax && V[[v]] == 0, Print[n]; V[[v]] = n; count++]]];
V (* Jean-François Alcover, Mar 12 2024, after Maple code *)

cp's OEIS Frontend

A357190 a(n) is the least prime p such that A234575(p, A007953(p)) is the n-th power of a prime.

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A358034 Numbers k such that A234575(k,s) = s^2 where s = A007953(k).

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A372523 Triangle read by rows: T(n, k) is equal to n/k if k | n, else to the concatenation of A003988(n, k) = floor(n/k) and A051127(k, n) = n mod k.

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A358052 Triangular array read by rows. For T(n,k) where 1 <= k <= n, start with x = k and repeat the map x -> floor(n/x) + (n mod x) until an x occurs that has already appeared. The number of applications of the map is T(n,k).

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A234541 Least k such that floor(n/k) + (n mod k) is a prime, or 0 if no such k exists.

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Scheme

A358055 a(n) is the least m such that A358052(m,k) = n for some k.

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