Peter Weiss - cp's OEIS frontend

A292568 a(n) = a(n-1) + sum of base-1000 digits of a(n-1), a(0)=1.

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 1049, 1099, 1199, 1399, 1799, 2599, 3200, 3403, 3809, 4621, 5246, 5497, 5999, 7003, 7013, 7033, 7073, 7153, 7313, 7633, 8273, 8554, 9116, 9241, 9491, 9991, 10991, 11992, 12995, 14002, 14018, 14050, 14114, 14242, 14498

Offset: 0

Peter Weiss, Sep 19 2017

In Germany you just write Q3 for the base-1000 digit sum (see book: "Taschenbuch der Mathematik" by Bronstein, Semendjajew, Musiol, Mühlig, p. 332) and you need it for the so-called "Teilbarkeitskriterium" for the number 37. If you add Q3 to a number you can also find this rule for the number 37.
Sum of base-1000 digits of m can also be described as "break the digit-string of m into triples starting at the right, and add these 3-digit numbers". For example, 1234567 -> 567 + 234 + (00)1 = 802.
None of the numbers of this sequence is divisible by 3 or 37.
The general form of this sequence is n + sum of base-(10^m) digits of n.
m=1: 1, 2, 4, 8, 16, 23, 28, 38, 49, 62, 70, 77, 91, 101, 103, ...  (Cf. A004207.)
m=2: 1, 2, 4, 8, 16, 32, 64, 128, 157, 215, 232, 266, 334, 371, ...  (Cf. A286660.)
m=3: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 1049, 1099, ...  (this sequence)

			a(16) = 2599 = 2 * 1000^1 + 599 * 1000^0. The sum of digits of a(17 - 1) = 2599 in base 1000 is therefore 2 + 599 = 601. a(17) = a(16) + the sum of digits of a(60) in base 1000 is therefore 2599 + 601 = 3200.

Cf. A004207, A286660.

NestList[Total[IntegerDigits[#,1000]]+#&,1,50] (* Harvey P. Dale, Dec 12 2018 *)

a(n) = if (n==0, 1, prev = a(n-1); prev + sumdigits(prev, 1000)); \\ Michel Marcus, Sep 20 2017

A292566 Primes that can be reached with their prime-index, if you start with the prime-index and use iterations of A062028.

Original entry on oeis.org

2, 11, 107, 191, 587, 719, 1061, 1171, 1181, 1259, 1327, 1487, 1597, 1619, 1933, 1949, 2011, 2141, 2269, 2477, 2803, 2999, 3041, 3049, 3079, 3169, 3229, 3259, 3617, 3733, 4493, 4799, 5009, 5023, 5171, 5261, 5581, 5657, 6131, 6211, 6301, 6311, 6421, 6451, 6529

Offset: 1

Peter Weiss, Sep 19 2017

If p is in the sequence, its index A000720(p) is not divisible by 3. - Robert Israel, Sep 19 2017

			The prime-index of 11 is 5: 5+5=10, 10+1+0=11 -> after two iterations you reach 11, so 11 is in the sequence.

Cf. A000720, A062028, A007618, A004207.

f:= proc(n) local t, p;
  p:= ithprime(n);
  t:= n;
  do
    t:= t + convert(convert(t,base,10),`+`);
    if t > p then return NULL
    elif t = p then return p
    fi
  od;
end proc:
map(f, [$1..1000]); # Robert Israel, Sep 19 2017

ok[p_] := Block[{n = PrimePi@ p}, While [n < p, n += Total@ IntegerDigits@ n]; n == p]; Select[Prime@ Range@ 600, ok] (* Giovanni Resta, Sep 19 2017 *)

is(n) = my(x=primepi(n)); while(1, x=x+sumdigits(x); if(x==n, return(1), if(x > n, return(0))))
forprime(p=1, 7000, if(is(p), print1(p, ", "))) \\ Felix Fröhlich, Sep 19 2017

More terms from Felix Fröhlich, Sep 19 2017

A292512 Sequence A: Start with n, add the sum of digits of n (A062028) and repeat. Sequence B: Start with n, add the sum of base-100 digits of n and repeat. a(n) is the smallest common number > n.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 221, 341, 24, 109, 218, 30, 1171, 173, 36, 406, 80, 84, 88, 851, 96, 163, 104, 54, 218, 346, 120, 628, 1171, 231, 173, 181, 72, 197, 406, 213, 538, 260, 237, 1003, 1705, 90, 184, 719, 1041, 1015, 365, 111, 320, 127, 117, 418, 488, 114, 1487, 137, 120, 122, 199, 126, 1171, 298, 231, 134, 677

Offset: 1

Peter Weiss, Sep 18 2017

If you start with n=1 and take a third sequence C (n + sum of base-1000 digits of n), the first common numbers of the three sequences are 2, 4, 8, 16 and 1027975.
The common numbers for the first ten primes are:
                    2 -> 4, 8, 16, 1027975, ...
                    3 -> 24, 96, 60342, ...
                    5 -> 10, 469534, ...
                    7 -> 14, 131558, ...
                   11 -> 923428, ...
                   13 -> 668495, ...
                   17 -> 81820, ...
                   19 -> 2061797, ...
                   23 -> 2227118, ...
                   29 -> 12278, ...

			n=10: Sequence A: 10, 11, 13, 17, 25, 32, 37, 47, 58, 71, 79, 95, 109, 119, 130, 134, 142, 149, 163, 173, 184, 197, 214, 221, ...
Sequence B: 10, 20, 40, 80, 160, 221, ...
-> 221 is the first common number > 10, so a(n)=221.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A286660, A062028, A007618, A006507, A016096.

With[{m = 10^3}, Table[With[{A = Rest@ NestList[# + Total@ IntegerDigits@ # &, n, m]}, NestWhile[# + Total@ IntegerDigits[#, 100] &, n, FreeQ[A, #] &, 1, m]], {n, 68}]] (* Michael De Vlieger, Sep 23 2017 *)

a(n) = my (A=n + sumdigits(n), B=n + sumdigits(n,100)); while (1, if (A==B, return (A), ARémy Sigrist, Sep 23 2017

A292218 The n-th iteration of (k -> k + sum of base-100 digits of k) applied to n.

Original entry on oeis.org

2, 8, 24, 64, 160, 285, 203, 266, 450, 538, 550, 642, 566, 488, 678, 958, 1120, 945, 686, 1241, 1032, 1144, 1339, 1671, 1229, 1499, 1377, 1492, 1945, 1515, 1745, 2108, 1650, 2326, 2110, 2106, 2558, 2105, 2049, 2548, 2590, 2247, 2786, 2750

Offset: 1

Peter Weiss, Sep 11 2017

Main diagonal of square array A(n, k), where A(n, k) gives the result of k applications of the map x -> x + sum of base-100 digits of x, starting at n. - Felix Fröhlich, Sep 12 2017

			n=6:  6+6=12, 12+12=24, 24+24=48, 48+48=96, 96+96=192, 192+92+(0)1=285. After 6 iterations you get 285, so a(6)=285.

Cf. A286660, A292202.

Table[Nest[# + Total@ IntegerDigits[#, 100] &, n, n], {n, 44}] (* Michael De Vlieger, Sep 11 2017 *)

a(n) = my(x=n); for (k=1, n, x += vecsum(digits(x, 100))); x; \\ Michel Marcus, Sep 12 2017

A292202 The n-th iteration of A062028 starting with n.

Original entry on oeis.org

2, 8, 15, 28, 25, 33, 59, 77, 90, 79, 109, 111, 134, 137, 129, 148, 184, 189, 218, 197, 204, 226, 250, 231, 284, 284, 297, 313, 325, 309, 341, 398, 354, 418, 418, 414, 488, 440, 438, 478, 529, 465, 545, 554, 531, 628, 658, 561, 620, 677, 624, 697, 697, 657, 785, 818, 735, 865, 835, 762, 851, 851

Offset: 1

Peter Weiss, Sep 11 2017

a(n) == n*2^n (mod 9). This has a period of 18. - Robert Israel, Sep 11 2017

			n=5: 5+5=10, 10+1+0=11, 11+1+1=13, 13+1+3=17, 17+1+7=25. After 5 iterations you get 25, so a(5)=25.

Robert Israel, Table of n, a(n) for n = 1..10000
Robert Israel, Plot of a(n)/n for n = 1 .. 4000

Cf. A007618, A006507, A062028, A016096.

A062028:= proc(t) option remember; t + convert(convert(t,base,10),`+`):end proc:
seq((A062028@@n)(n), n=1..100); # Robert Israel, Sep 11 2017

Table[Nest[# + Total@ IntegerDigits@ # &, n, n], {n, 62}] (* Michael De Vlieger, Sep 11 2017 *)

a(n) = my(x=n); for (k=1, n, x += sumdigits(x)); x; \\ Michel Marcus, Sep 12 2017

A290574 Self numbers that are the product of two self numbers greater than one.

Original entry on oeis.org

9, 378, 400, 525, 602, 1155, 1188, 1862, 2055, 2200, 2325, 2415, 2492, 2560, 2907, 3045, 3348, 3392, 3460, 3515, 3717, 3752, 3965, 4180, 4360, 4382, 4415, 4865, 4920, 5115, 5418, 5517, 5719, 6138, 6228, 6900, 7038, 7060, 7396, 7532, 7565, 7609, 7947, 8162, 8342, 8465, 8520, 8700, 8757, 8869, 8970, 9152, 9365, 9387, 9409, 9420, 9422, 9499, 9870, 9925

Offset: 1

Peter Weiss, Aug 06 2017

			The product of the self numbers 31 and 75 is the self number 2325, so 2325 is in the sequence.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A003052, A290426.

Block[{nn = 10^4, s}, s = Rest@ Complement[Range@ nn, Union[Table[n + Total@ IntegerDigits@ n, {n, nn}]]]; Select[Range@ nn, Function[n, And[MemberQ[s, n], AnyTrue[Map[{#, n/#} &, Rest@ TakeWhile[Divisors@ n, # <= Sqrt@ n &]], AllTrue[#, MemberQ[s, #] &] &]]]]] (* or *)
Block[{nn = 5000, s}, s = Rest@ Complement[Range@ nn, Union@ Table[n + Total@ IntegerDigits@ n, {n, nn}]]; Select[Union@ Sort@ Map[Times @@ # &@ # &, Tuples[s, {2}]], MemberQ[s, #] &]] (* Michael De Vlieger, Aug 23 2017, after T. D. Noe at A003052 *)

is(n)=if(!is_A003052(n), return(0)); fordiv(n,d, if(d==1, next); if(d^2>n, break); if(is_A003052(d) && is_A003052(n/d), return(1))); 0 \\ Charles R Greathouse IV, Aug 23 2017

is_A290574(n)={is_A003052(n) && fordiv(n,d, d^2>n && break; d>1 && is_A003052(d) && is_A003052(n/d) && return(1))} \\ M. F. Hasler, Nov 09 2018

Corrected by Charles R Greathouse IV, Aug 23 2017

A290426 Products of two self numbers (A003052).

Original entry on oeis.org

1, 3, 5, 7, 9, 15, 20, 21, 25, 27, 31, 35, 42, 45, 49, 53, 60, 63, 64, 75, 81, 86, 93, 97, 100, 108, 110, 121, 126, 132, 140, 143, 154, 155, 159, 165, 176, 180, 187, 192, 198, 209, 210, 211, 217, 222, 225, 233, 244, 255, 258, 265, 266, 277, 279, 288, 291, 294, 299, 310, 312, 320, 323, 324, 330, 334, 345, 356, 363, 367, 371

Offset: 1

Peter Weiss, Jul 31 2017

			The product of the self numbers 3 and 86 is 258, so 258 is in the sequence.

Cf. A003052.

A288469 a(n) = n if n is a nonprime, otherwise take the prime index of n and repeat until you get a nonprime which is then a(n).

Original entry on oeis.org

1, 1, 1, 4, 1, 6, 4, 8, 9, 10, 1, 12, 6, 14, 15, 16, 4, 18, 8, 20, 21, 22, 9, 24, 25, 26, 27, 28, 10, 30, 1, 32, 33, 34, 35, 36, 12, 38, 39, 40, 6, 42, 14, 44, 45, 46, 15, 48, 49, 50, 51, 52, 16, 54, 55, 56, 57, 58, 4, 60, 18, 62, 63, 64, 65, 66, 8, 68, 69, 70, 20, 72, 21, 74, 75, 76, 77, 78, 22, 80, 81, 82, 9, 84, 85, 86

Offset: 1

Peter Weiss, Jun 09 2017

nonn

a(n) = 1 for n in A007097. - Robert Israel, Jun 09 2017

			For n = 17:  17 is a prime, so you take the prime index of 17 which is 7. 7 is a prime, so you take the prime index of 7 which is 4. 4 is a nonprime, so a(17) = 4.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000720, A007097, A010051, A114537.

f:= proc(n) option remember; if isprime(n) then procname(numtheory:-pi(n)) else n fi end proc:
map(f, [$1..100]); # Robert Israel, Jun 09 2017

Table[If[!PrimeQ@ n, n, NestWhile[PrimePi, n, PrimeQ]], {n, 86}] (* Michael De Vlieger, Jun 09 2017 *)

a(n)=while(isprime(n), n=primepi(n)); n \\ Charles R Greathouse IV, Jun 09 2017

From Robert Israel, Jun 09 2017: (Start)
a(n) = n + A010051(n)*(a(A000720(n))-n).
a(A114537(n,k)) = A114537(n,1). (End)

A286660 a(n) = a(n-1) + sum of base-100 digits of a(n-1), a(0) = 1.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 157, 215, 232, 266, 334, 371, 445, 494, 592, 689, 784, 875, 958, 1025, 1060, 1130, 1171, 1253, 1318, 1349, 1411, 1436, 1486, 1586, 1687, 1790, 1897, 2012, 2044, 2108, 2137, 2195, 2311, 2345, 2413, 2450, 2524, 2573, 2671, 2768, 2863, 2954, 3037, 3104, 3139

Offset: 0

Peter Weiss, May 12 2017

			a(7) = 128 = 1 * 100^1 + 28 * 100^0. The sum of digits of a(8 - 1) = 128 in base 100 is therefore 1 + 28 = 29. a(8) = a(7) + the sum of digits of a(7) in base 100 is therefore 128 + 29 = 157.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A004207, A010062.

g:= n -> n+convert(convert(n,base,100),`+`):
A[0]:= 1:
for n from 1 to 100 do A[n]:= g(A[n-1]) od:
seq(A[i],i=0..100); # Robert Israel, May 22 2017

a[0] = 1; a[n_] := a[n] = a[n-1] + Total[IntegerDigits[a[n-1], 100]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, May 21 2017 *)
NestList[#+Total[IntegerDigits[#,100]]&,1,60] (* Harvey P. Dale, May 26 2019 *)

a(n) = if(n < 8, return(1<<(n-1))); my(r = cr = 128); for(i=8, n, while(cr > 0, r += cr % 100; cr \= 100); cr = r); r \\ David A. Corneth, May 15 2017

A281592 Products of three distinct primes p1, p2 and p3 (sphenic numbers) with p1

Original entry on oeis.org

138, 777, 4642, 10258, 10263, 12207, 13282, 16167, 19762, 30783, 37407, 38482, 46978, 48927, 56127, 60145, 63543, 73767, 81687, 89823, 95367, 95627, 103863, 110905, 115527, 128545, 202705, 208879, 223643, 284119, 324947, 325793, 360151, 395003, 477538, 541163, 558322, 585538, 672199, 673693, 780082, 914551, 1016643

Offset: 1

Peter Weiss, Apr 14 2017

			10258 is in the sequence because 10258 = 2*23*223 and 223 is the concatenation of 2 with 23.

Cf. A007304, A133980 (the p3 primes).

c[x_, y_] := x 10^IntegerLength[y] + y; upto[mx_] := Sort@ Reap[Block[{p=2, q=3, v=1}, While[v <= mx, While[p < q && (v = p q (r = c[p, q])) <= mx, If[PrimeQ@r, Sow@v]; p = NextPrime[p]]; p=2; q = NextPrime[q]; v = p q c[p, q]]]][[2, 1]]; upto[10^6] (* Giovanni Resta, Apr 14 2017 *)

isok(n) = f = factor(n); ((#f~ == 3) && (vecmax(f[,2]) == 1) && (f[3,1] == fromdigits(concat(digits(f[1,1]), digits(f[2,1]))))); \\ Michel Marcus, Apr 14 2017

cp's OEIS Frontend

User: Peter Weiss

A292568 a(n) = a(n-1) + sum of base-1000 digits of a(n-1), a(0)=1.

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A292566 Primes that can be reached with their prime-index, if you start with the prime-index and use iterations of A062028.

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A292512 Sequence A: Start with n, add the sum of digits of n (A062028) and repeat. Sequence B: Start with n, add the sum of base-100 digits of n and repeat. a(n) is the smallest common number > n.

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A292218 The n-th iteration of (k -> k + sum of base-100 digits of k) applied to n.

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A292202 The n-th iteration of A062028 starting with n.

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A290574 Self numbers that are the product of two self numbers greater than one.

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A290426 Products of two self numbers (A003052).

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A288469 a(n) = n if n is a nonprime, otherwise take the prime index of n and repeat until you get a nonprime which is then a(n).

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