A061762 -id:A061762 - cp's OEIS frontend

A061763 Numbers k such that k is divisible by A061762(k) and the product of digits of k (A007954(k)) is not zero.

19, 29, 39, 42, 49, 59, 69, 79, 89, 99, 126, 132, 285, 312, 522, 594, 1134, 1144, 1159, 1211, 1275, 1323, 1365, 1573, 1632, 1634, 1674, 1715, 1813, 1815, 1911, 1919, 1932, 1944, 2133, 2139, 2516, 2793, 3132, 3135, 3161, 3211, 3213, 3216, 3321, 3363, 3393

Offset: 1

Amarnath Murthy, May 20 2001

Intersection of A038366 and A052382 (zeroless numbers). - Michel Marcus, Oct 29 2019

			42 is a term as 4+2 + 2*4 = 14 and 42 = 14*3.

S. Parmeswaran, S+P numbers, Mathematics Informatics Quarterly, Vol. 9, No. 3 Sept. 1999, Bulgaria.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1001 terms from Harry J. Smith)

Cf. A061762, A038366, A052382.

Select[Range[3400], (y = Times @@ (x = IntegerDigits[#])) != 0 && Divisible[#, Plus @@ x + y] &] (* Jayanta Basu, Jul 14 2013 *)

isok(k) = my(d=digits(k)); vecmin(d) && ((k % (vecprod(d) + vecsum(d))) == 0);

Corrected and extended by Larry Reeves (larryr(AT)acm.org), May 23 2001

Offset corrected by Giovanni Resta, Oct 29 2019

A130858 Smallest k such that A061762(k) = n.

Original entry on oeis.org

0, 10, 1, 11, 2, 12, 3, 13, 4, 14, 5, 15, 6, 16, 7, 17, 8, 18, 9, 19, 26, 126, 1126, 27, 44, 144, 28, 36, 136, 29, 129, 37, 137, 234, 46, 38, 138, 1138, 11138, 39, 139, 56, 156, 1156, 48, 148, 334, 57, 66, 49, 149, 245, 1245, 58, 158, 67, 167, 1167, 11167, 59

Offset: 0

Zak Seidov and Giovanni Resta, Aug 22 2007

nonn

Giovanni Resta, Table of n, a(n) for n = 0..1000

A344021 Numbers k such that A061762(k) and k+A061762(k) are both prime.

Original entry on oeis.org

1, 12, 16, 32, 34, 54, 56, 78, 104, 106, 160, 232, 236, 250, 252, 298, 302, 304, 326, 328, 340, 362, 382, 388, 474, 490, 502, 508, 526, 560, 580, 610, 650, 656, 670, 676, 706, 740, 760, 838, 850, 890, 898, 928, 940, 980, 1004, 1006, 1024, 1028, 1042, 1048, 1082, 1084, 1152, 1190, 1192, 1246, 1248

Offset: 1

J. M. Bergot and Robert Israel, May 06 2021

			a(3) = 16 is a term because A061762(16) = 1*6+1+6=13 is prime and 16+13=29 is prime.

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

Cf. A061762, A185300.

f:= proc(n) local L;
L:= convert(n,base,10);
convert(L,`*`)+convert(L,`+`);
end proc:
filter:= proc(n) local t; t:= f(n); isprime(t) and isprime(n+t) end proc:
select(filter, [1,seq(i,i=2..10000,2)]);

A344032 a(n) is the least prime that begins a sequence of at least n distinct primes under iteration of A061762.

Original entry on oeis.org

2, 11, 23, 53, 12451, 36779999

Offset: 1

J. M. Bergot and Robert Israel, May 07 2021

			12451 is prime and A061762(12451) = 1*2*4*5*1+1+2+4+5+1 = 53.
53 is prime and A061762(53) = 5*3+5+3 = 23.
23 is prime and A061762(23) = 2*3+2+3 = 11.
11 is prime and A061762(11) = 1*1+1+1 = 3.
3 is prime and A061762(3) = 3+3 = 6 is not prime.
Thus 12451 begins a sequence of 5 distinct primes under the iteration of A061762.  Since 12451 is the least such prime, a(5) = 12451.

Cf. A061762, A214629.

f:= proc(n) local L;
   L:= convert(n,base,10);
   convert(L,`+`)+convert(L,`*`)
end proc:
g:= proc(n) local S,v;
  S:= {n}:
  v:= n;
  do
    v:= f(v);
    if member(v,S) or not isprime(v) then return nops(S) fi;
    S:= S union {v}
  od
end proc:
R:= NULL: p:= 1: m:= 0:
while m < 5 do
  p:= nextprime(p);
  v:= g(p);
  if v > m then R:= R, p$(v-m); m:= v fi
od:
R;

A038366 n is divisible by (product of digits) + (sum of digits).

Original entry on oeis.org

10, 19, 20, 29, 30, 39, 40, 42, 49, 50, 59, 60, 69, 70, 79, 80, 89, 90, 99, 100, 102, 108, 110, 120, 126, 132, 140, 150, 180, 190, 200, 201, 204, 207, 209, 210, 220, 230, 240, 270, 280, 285, 300, 306, 308, 312, 320, 330, 360, 370, 400, 402, 405, 407, 408, 410

Offset: 1

Felice Russo

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A061762, A061763.

Select[Range[410], Divisible[#, Plus @@ (x = IntegerDigits[#]) + Times @@ x] &] (* Jayanta Basu, Jul 14 2013 *)

isok(m) = my(d=digits(m)); (m % (vecprod(d) + vecsum(d))) == 0; \\ Michel Marcus, Oct 29 2019

from math import prod
def ok(n): d = list(map(int, str(n))); return n and n%(sum(d)+prod(d)) == 0
print([k for k in range(411) if ok(k)]) # Michael S. Branicky, Oct 19 2021

A036120 a(n) = 2^n mod 19.

Original entry on oeis.org

1, 2, 4, 8, 16, 13, 7, 14, 9, 18, 17, 15, 11, 3, 6, 12, 5, 10, 1, 2, 4, 8, 16, 13, 7, 14, 9, 18, 17, 15, 11, 3, 6, 12, 5, 10, 1, 2, 4, 8, 16, 13, 7, 14, 9, 18, 17, 15, 11, 3, 6, 12, 5, 10, 1, 2, 4, 8, 16, 13, 7, 14, 9, 18, 17, 15

Offset: 0

N. J. A. Sloane

The sequence can be generated via a(n) = A061762(a(n-1)). Apparently any other choice of the first element leads also to periodic sequences, with fixed points of A061762 as special cases. - Zak Seidov, Aug 22 2007

I. M. Vinogradov, Elements of Number Theory, pp. 220 ff.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, -1, 1).

CF. A000079 (2^n).

List([0..60],n->PowerMod(2,n,19)); # Muniru A Asiru, Oct 17 2018

[Modexp(2, n, 19): n in [0..100]]; // G. C. Greubel, Oct 17 2018

with(numtheory) ; i := pi(19) ; [ seq(primroot(ithprime(i))^j mod ithprime(i), j=0..100) ];

PowerMod[2, Range[0, 100], 19] (* G. C. Greubel, Oct 17 2018 *)

a(n)=lift(Mod(2,19)^n) \\ Charles R Greathouse IV, Mar 22 2016

for n in range(0, 100): print(int(pow(2, n, 19)), end=' ') # Stefano Spezia, Oct 17 2018

[power_mod(2,n,19) for n in range(0,66)] # Zerinvary Lajos, Nov 03 2009

a(n)= +a(n-1) -a(n-9) +a(n-10). - R. J. Mathar, Apr 13 2010
G.f.: (1+x+2*x^2+4*x^3+8*x^4-3*x^5-6*x^6+7*x^7-5*x^8+10*x^9)/ ((1-x) * (1+x) * (x^2- x+1) * (x^6-x^3+1)). - R. J. Mathar, Apr 13 2010
a(n) = a(n+18). - Vincenzo Librandi, Sep 09 2011

A214629 Primes p such that the sum of the digits plus the product of the digits is a prime.

Original entry on oeis.org

11, 13, 19, 23, 29, 31, 37, 43, 53, 59, 61, 73, 79, 89, 97, 101, 223, 263, 283, 401, 409, 443, 601, 607, 809, 823, 829, 883, 1013, 1019, 1031, 1033, 1039, 1051, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1163, 1171, 1181, 1187, 1193, 1213, 1231, 1259

Offset: 1

Vicente Izquierdo Gomez, Aug 13 2012

			11 is in the sequence because A061762(11) = 3 is prime.

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

Cf. A061762, A344032. Primes in A185300.

f:= proc(n) local L;
   L:= convert(n,base,10);
   convert(L,`+`)+convert(L,`*`)
end proc:
select(p -> isprime(f(p)), [seq(ithprime(i),i=1..1000)]); # Robert Israel, May 07 2021

f[n_] := Module[{in = IntegerDigits[n]}, Times @@ in + Plus @@ in];Select[Prime[Range[300]], PrimeQ[f[#]] &]

{p in A000040: A061762(p) in A000040}. - R. J. Mathar, Aug 13 2012

A171772 Number of steps needed to reach a prime when the map S(n)+M(n) is applied to n, or -1 if a prime is never reached. Here S(n) and M(N) mean the sum and the product of the digits of n in base 10.

Original entry on oeis.org

1, 0, 0, 3, 0, 2, 0, 2, 2, 2, 0, 1, 0, 3, 1, 1, 0, 1, 0, 1, 1, 3, 0, 4, 1, 2, 1, 3, 0, 1, 0, 1, 2, 1, 1, 2, 0, 2, -1, 4, 0, 4, 0, 5, 1, 2, 0, 6, -1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 0, 3, 0, 2, 2, 2, 1, 7, 0, 3, -1, 1, 0, 1, 0, -1, 1, 3, 3, 1, 0, 3

Offset: 1

R. J. Mathar, Oct 12 2010

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

			a(4)=3 because 4->8->16->13 is prime.
a(39)=-1 because 39 -> 39 ->39 ... never reaches a prime.
a(49)=-1 because 49 -> 49 ->49 ... never reaches a prime.
a(69)=-1 because 69 -> 69 ->69 ... never reaches a prime.
a(74)=-1 because 74 -> 39 ->39 ... never reaches a prime.
a(28)=3 because 28 ->26 ->20 ->2.

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

A variant of A074871.

Cf. A007954, A053837, A061762.

f:= proc(n) local L;
  L:= convert(n,base,10);
  convert(L,`+`)+convert(L,`*`);
end proc:
g:= proc(n) option remember; local v,w;
     if n::prime then return 0 fi;
     v:= f(n);
     if v = n then return -1 fi;
     w:= procname(v);
     if w = -1 then -1 else w+1 fi
end proc:
map(g, [$1..100]); # Robert Israel, Nov 03 2019

A249121 a(n) = n - (sum of digits of n) - (product of digits of n).

Original entry on oeis.org

0, -1, -2, -3, -4, -5, -6, -7, -8, -9, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 16, 14, 12, 10, 8, 6, 4, 2, 0, 27, 24, 21, 18, 15, 12, 9, 6, 3, 0, 36, 32, 28, 24, 20, 16, 12, 8, 4, 0, 45, 40, 35, 30, 25, 20, 15, 10, 5, 0, 54, 48, 42, 36, 30, 24, 18, 12, 6, 0, 63, 56, 49, 42, 35, 28, 21, 14, 7, 0, 72, 64, 56

Offset: 0

Zak Seidov, Oct 21 2014

In the graph, the upper bound is the line:

a(10*m) = 9m: a(10) = 9, a(20) = 18, a(30) = 27,...

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Zak Seidov, Color graph with line connecting the consecutive points.

Cf. A001477, A007953, A007954, A061762.

a:= n-> (l-> n-add(i, i=l)-mul(i, i=l))(convert(n, base, 10)):
seq(a(n), n=0..120); # Alois P. Heinz, Oct 22 2014

Table[id = IntegerDigits[n]; n - (Plus @@ id) - (Times @@ id), {n, 0, 200}]

a(n)=d=digits(n);if(!n,return(0));n-sumdigits(n)-prod(i=1,#d,d[i])
vector(100,n,a(n-1)) \\ Derek Orr, Oct 21 2014

a(n) = n - A007953(n) - A007954(n) = A001477(n) - A061762(n).

A185400 Numbers with property that the digital sum plus the product of the digits is a power of 2.

Original entry on oeis.org

1, 2, 4, 8, 10, 20, 22, 40, 80, 100, 101, 103, 107, 110, 111, 113, 117, 130, 131, 133, 137, 170, 171, 173, 177, 200, 202, 206, 220, 260, 301, 305, 310, 311, 313, 317, 331, 350, 371, 400, 404, 440, 503, 530, 602, 620, 701, 709, 710, 711, 713, 717, 731, 771, 790, 800, 808, 880, 907, 970, 1000, 1001, 1003, 1007, 1010, 1012, 1016

Offset: 1

Michel Lagneau, Feb 03 2011

			371 is in the sequence because (3+7+1) + (3*7*1) = 11 + 21 = 32 = 2^5.
116291 is in the sequence because (1+1+6+2+9+1) + (1*1*6*2*9*1) = 20 + 108 = 128 = 2^7.

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

Cf. A061762.

A007953 := proc(n) add(d,d=convert(n,base,10)) ; end proc:
A007954 := proc(n) mul(d,d=convert(n,base,10)) ; end proc:
A061762 := proc(n) A007953(n)+A007954(n) ; end proc:
isA000079 := proc(n) if n < 1 then false; elif n = 1 then true; else if type(n,'even') then is( nops(numtheory[factorset](n)) = 1) ; else false; end if; end if; end proc:
isA185400 := proc(n) isA000079(A061762(n)) ; end proc:
for n from 1 to 1300 do if isA185400(n) then printf("%a,",n) ; end if; end do: # R. J. Mathar, Feb 08 2011

pwrs2Q[n_]:=Module[{idn=IntegerDigits[n],x,y},x=Total[idn]+Times@@idn;y=Round[Log[x]/Log[2]];2^y==x]
Select[Range[1100],pwrs2Q]  (* Harvey P. Dale, Feb 16 2011 *)

cp's OEIS Frontend

A061763 Numbers k such that k is divisible by A061762(k) and the product of digits of k (A007954(k)) is not zero.

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A130858 Smallest k such that A061762(k) = n.

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A344021 Numbers k such that A061762(k) and k+A061762(k) are both prime.

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A344032 a(n) is the least prime that begins a sequence of at least n distinct primes under iteration of A061762.

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A038366 n is divisible by (product of digits) + (sum of digits).

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A036120 a(n) = 2^n mod 19.

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A214629 Primes p such that the sum of the digits plus the product of the digits is a prime.

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A171772 Number of steps needed to reach a prime when the map S(n)+M(n) is applied to n, or -1 if a prime is never reached. Here S(n) and M(N) mean the sum and the product of the digits of n in base 10.

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