A057147 -id:A057147 - cp's OEIS frontend

A229549 Numbers k such that k*(sum of digits of k) is a palindrome.

0, 1, 2, 3, 11, 22, 42, 53, 56, 101, 111, 113, 121, 124, 182, 187, 202, 272, 353, 434, 515, 572, 616, 683, 739, 829, 888, 1001, 1111, 1357, 1507, 1508, 1624, 1717, 2002, 2074, 2852, 3049, 3146, 3185, 3326, 3342, 3687, 3747, 4058, 4066, 4391, 4719, 4724, 5038, 7579, 8569, 9391, 9471

Offset: 1

Derek Orr, Sep 26 2013

			829*(8+2+9) = 15751 (palindrome), so 829 is a term of this sequence.

Delbert L. Johnson, Table of n, a(n) for n = 1..20000

Cf. A057147.

palQ[n_] := Block[{d = IntegerDigits@ n}, d == Reverse@ d]; Select[Range@ 10000, palQ[# Plus @@ IntegerDigits@ #] &] (* Michael De Vlieger, Apr 12 2015 *)
Select[Range[0,10000],PalindromeQ[# Total[IntegerDigits[#]]]&] (* Harvey P. Dale, Jun 30 2025 *)

ispal(n)=d=digits(n);d==Vecrev(d)
for(n=0,10^4,s=sumdigits(n);if(ispal(n*s),print1(n,", "))) \\ Derek Orr, Apr 10 2015

def ispal(n):
    r = ''
    for i in str(n):
        r = i + r
    return n == int(r)
def DS(n):
    s = 0
    for i in str(n):
        s += int(i)
    return s
{print(n, end=', ') for n in range(10**4) if ispal(n*DS(n))}
## Simplified by Derek Orr, Apr 10 2015

More terms from Derek Orr, Apr 10 2015

A037478 Number of positive solutions to "numbers that are n times sum of their digits".

Original entry on oeis.org

9, 1, 1, 4, 1, 1, 4, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 11, 1, 1, 3, 1, 1, 3, 2, 2, 12, 1, 1, 3, 1, 1, 4, 1, 2, 15, 2, 1, 4, 1, 1, 3, 1, 1, 13, 2, 2, 3, 1, 1, 4, 1, 1, 13, 1, 1, 2, 1, 1, 3, 0, 0, 7, 0, 1, 4, 1, 1, 4, 1, 1, 8, 1, 0, 3, 1, 1, 4, 1, 1, 10, 1, 0, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 0, 1, 3, 1, 1, 9, 1

Offset: 1

Henry Bottomley, Sep 12 2000

It appears that the largest terms occur when n=1 mod 9 and moderately large terms when n=4 or 7 mod 9.

			a(13)=3 since the only three solutions are 117=9*13, 156=12*13 and 195=15*13.

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

Cf. A003634, A003635, A005349, A052489, A052490, A057147, A058913.

read("transforms"):
A037478 := proc(n)
    local a,x,k;
    a := 0 ;
    for k from 1 do
        x := n*k ;
        if digsum(x)*n = x then
            a := a+1 ;
        end if;
        # may stop if x/digsum(x)>n, so if x/#digits(x) > 9*n
        if x/A055642(x) > 9*n then
            break;
        end if;
    end do:
    a ;
end proc:
seq(A037478(n),n=1..101) ; # R. J. Mathar, May 11 2016

A337816 Numbers that can be written as (m * sum of digits of m) for some m.

Original entry on oeis.org

0, 1, 4, 9, 10, 16, 22, 25, 36, 40, 49, 52, 63, 64, 70, 81, 88, 90, 100, 112, 115, 124, 136, 144, 160, 162, 175, 190, 198, 202, 205, 208, 220, 238, 243, 250, 252, 280, 301, 306, 319, 324, 333, 352, 360, 364, 370, 400, 405, 412, 418, 424, 427, 448, 460, 468, 484, 486, 490

Offset: 1

Bernard Schott, Sep 23 2020

If 3 divides a(n), then 9 divides a(n).

			10 = 10 * (1+0);
22 = 11 * (1+1).

Range of A057147 and of A117570.
Similar sequences: A176995 (m + sum of digits of m), A336826 (m * product of digits of m), A337718 (m + product of digits of m).
Cf. A337817.
Some subsequences: A011557, A052268, A093141.

m = 500; Select[Union @ Table[k * Plus @@ IntegerDigits[k], {k, 0, m}], # <= m &] (* Amiram Eldar, Sep 23 2020 *)

is(k)={if(k==0, return(1)); fordiv(k, d, if(d*sumdigits(d)==k, return(1))); 0} \\ Andrew Howroyd, Sep 23 2020

A117570 Numbers of the form k * (sum of digits of k) listed sorted with multiplicity.

Original entry on oeis.org

0, 1, 4, 9, 10, 16, 22, 25, 36, 36, 40, 49, 52, 63, 64, 70, 81, 88, 90, 90, 100, 112, 115, 124, 136, 144, 160, 160, 162, 175, 190, 198, 202, 205, 208, 220, 238, 243, 250, 252, 280, 280, 301, 306, 306, 319, 324, 333, 352, 360, 360, 364, 370, 400, 405, 412, 418, 424

Offset: 1

Ernesto Estrada (estrada66(AT)yahoo.com), Jan 02 2008

			2008 is a term since 2008 = 251*(2+5+1).

David A. Corneth, Table of n, a(n) for n = 1..10000
Ömer Eğecioğlu and Bünyamin Şahin, On twin EP numbers, Transact. Comb. (2025) Vol. 14, Iss. 4, Art. No. 4, 261-270. See p. 262.
Ernesto Estrada, Puri Pereira-Ramos, Spatial 'Artistic' Networks: From Deconstructing Integer-Functions to Visual Arts, Complexity, Vol. 2018 (2018), Article ID 9893867.

Sorted terms of A057147.

Terms without duplicates are given by A337816.

With[{nn = 400}, TakeWhile[#, # <= nn &] &@ Union@ Array[# Total@ IntegerDigits[#] &, nn + 1, 0]] (* Michael De Vlieger, Apr 19 2018 *)

A338976 Primes p such that p*A007953(p)+1 is prime.

Original entry on oeis.org

2, 11, 13, 17, 19, 59, 71, 97, 107, 109, 149, 167, 181, 239, 271, 419, 431, 499, 509, 523, 547, 563, 613, 631, 691, 727, 811, 853, 859, 983, 1009, 1063, 1087, 1117, 1151, 1193, 1229, 1409, 1427, 1487, 1559, 1579, 1601, 1759, 1823, 1913, 1973, 2039, 2099, 2161, 2237, 2251, 2309, 2411, 2437, 2473

Offset: 1

J. M. Bergot and Robert Israel, Dec 18 2020

			a(3) = 13 is a term because 13 and 13*(1+3)+1 = 53 are prime.

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

Cf. A007953, A057147.

Subsequence of A119449.

select(t -> isprime(t) and isprime(t*convert(convert(t,base,10),`+`)+1), [$2..10^4]);

isok(p) = isprime(p) && isprime(p*sumdigits(p)+1); \\ Michel Marcus, Dec 18 2020

A089227 Numbers k such that 1 + k*ds(k) is prime, where ds(k) is the sum of digits of k.

Original entry on oeis.org

1, 2, 4, 6, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 22, 28, 33, 34, 35, 38, 44, 46, 48, 50, 51, 54, 56, 59, 64, 68, 70, 71, 78, 80, 82, 84, 88, 90, 91, 92, 93, 94, 97, 98, 99, 100, 102, 104, 105, 106, 107, 109, 112, 116, 118, 123, 128, 129, 130, 136, 138, 140, 144, 145

Offset: 1

Yalcin Aktar, Dec 10 2003

			10 is in the sequence because A007953(10) = 1 and 1 + 10*1 = 11 is prime.

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

Cf. A007953, A057147.

[k:k in [1..145] | IsPrime(1+k*(&+Intseq(k,10)))]; // Marius A. Burtea, Jun 21 2019

ds:= n -> convert(convert(n,base,10),`+`):
filter:= n -> isprime(1+n*ds(n)):
select(filter, [$1..1000]); # Robert Israel, Jun 20 2019

Do[k = Plus @@ IntegerDigits[n]; If[PrimeQ[n*k + 1], Print[n]], {n, 1, 100}] (* Ryan Propper *)
Select[Range[150],PrimeQ[#*Total[IntegerDigits[#]]+1]&] (* Harvey P. Dale, May 25 2024 *)

isok(k) = isprime(1+k*sumdigits(k)); \\ Michel Marcus, Jun 20 2019

More terms from David Wasserman, Aug 31 2005

A213630 a(t_1 t_2...t_n) = (t_1 + t_2)t_1 t_2 + ... + (t_n-1 + t_n)t_n-1 t_n.

Original entry on oeis.org

1, 4, 9, 16, 25, 36, 49, 64, 81, 10, 22, 36, 52, 70, 90, 112, 136, 162, 190, 40, 63, 88, 115, 144, 175, 208, 243, 280, 319, 90, 124, 160, 198, 238, 280, 324, 370, 418, 468, 160, 205, 252, 301, 352, 405, 460, 517, 576, 637, 250, 306, 364, 424, 486, 550, 616

Offset: 1

Felipe Bottega Diniz, Jun 16 2012

Differs from A057147 first at n=100.

a(n)/n = {1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 6, 7, 8, 9, 10, 11, 12, 13, 14, 7, 8, 9, 10, 11, 12, 13, 14, 15, 8, 9, 10, 11, 12, 13, 14, 15, ...}. - Alexander R. Povolotsky, Jun 19 2012

			a(123456) = (1+2)*12 + (3+4)*34 + (5+6)*56 = 890.
a(890) = (0+8)*08 + (9+0)*90 = 874.
a(15) = (1+5)*15 = 90.
a(7) = (0+7)*07 = 49.

Felipe Bottega Diniz and T. D. Noe, Table of n, a(n) for n = 1..1000 (first 100 terms from Felipe Bottega Diniz)
F. B. Diniz, About a New Family of Sequences, arXiv:1607.06082 [math.GM], 2016.

read("transforms"):
A213630 := proc(n)
    local dgs,a,i ;
    if n < 10 then
        n^2;
    else
        dgs := convert(n,base,10) ;
        a := 0 ;
        for i from 2 to nops(dgs) do
            a := a+(op(i-1,dgs)+op(i,dgs))*digcat2(op(i,dgs),op(i-1,dgs)) ;
        end do:
        a;
    end if;
end proc: # R. J. Mathar, Aug 10 2012

num[n_] := Module[{d = IntegerDigits[n], d2}, If[OddQ[Length[d]], d = Prepend[d, 0]]; d2 = Partition[d, 2]; Sum[(d2[[i, 1]] + d2[[i, 2]])*(10*d2[[i, 1]] + d2[[i, 2]]), {i, Length[d2]}]]; Table[num[n], {n, 100}] (* T. D. Noe, Jun 19 2012 *)

Let t_1t_2...t_n be a natural number.
a(t_1t_2...t_n) = (t_1 + t_2)*t_1t_2 + ... + (t_n-1 + t_n)*t_n-1t_n, if n even.
If n is odd, t_1 t_2...t_n <- 0t_1t_2...t_n and do a(0t_1 t_2...t_n).

A328683 Positive integers that are equal to 99...99 (repdigit with n digits 9) times the sum of their digits.

Original entry on oeis.org

81, 1782, 26973, 359964, 4499955, 53999946, 629999937, 7199999928, 80999999919, 899999999910, 9899999999901, 107999999999892, 1169999999999883, 12599999999999874, 134999999999999865, 1439999999999999856, 15299999999999999847, 161999999999999999838

Offset: 1

Bernard Schott, Feb 25 2020

The idea of this sequence comes from a problem during the annual Moscow Mathematical Olympiad (MMO) in 2001 (see reference).

			359964 = 36 * 9999 and the digital sum of 359964 = 36 , so 359964 = a(4).

Roman Fedorov, Alexei Belov, Alexander Kovaldzhi, Ivan Yashchenko, Moscow Mathematical Olympiads, 2000-2005, Level B, Problem 5, 2001, MSRI, 2011, p. 8 and 70/71.

Colin Barker, Table of n, a(n) for n = 1..995
Index to sequences related to Olympiads.
Index entries for linear recurrences with constant coefficients, signature (22,-141,220,-100).

Cf. A057147, A086580, A110807.

Maple
```
C:=seq(9*n*(10^n-1),n=1..20);
```

Table[9*n*(10^n - 1), {n, 1, 18}] (* Amiram Eldar, Feb 25 2020 *)
LinearRecurrence[{22,-141,220,-100},{81,1782,26973,359964},20] (* Harvey P. Dale, Feb 02 2025 *)

Vec(81*x*(1 - 10*x^2) / ((1 - x)^2*(1 - 10*x)^2) + O(x^20)) \\ Colin Barker, Feb 25 2020

a(n) = 9 * n * (10^n - 1).
From Colin Barker, Feb 25 2020: (Start)
G.f.: 81*x*(1 - 10*x^2) / ((1 - x)^2*(1 - 10*x)^2).
a(n) = 22*a(n-1) - 141*a(n-2) + 220*a(n-3) - 100*a(n-4) for n>4.
(End)
From Michel Marcus, Feb 25 2020: (Start)
a(n) = 9*A110807(n).
a(n) = n*A086580(n). (End)

A333814 Multiples of 12 whose sum of digits is 12.

Original entry on oeis.org

48, 84, 156, 192, 228, 264, 336, 372, 408, 444, 480, 516, 552, 624, 660, 732, 804, 840, 912, 1056, 1092, 1128, 1164, 1236, 1272, 1308, 1344, 1380, 1416, 1452, 1524, 1560, 1632, 1704, 1740, 1812, 1920, 2028, 2064, 2136, 2172, 2208, 2244, 2280, 2316, 2352, 2424

Offset: 1

Bernard Schott, Apr 06 2020

If m is a term, 10*m is also a term.

			732 = 12 * 61 and 7 + 3 + 2 = 12, hence 732 is a term.

Intersection of A235151 (sum of digits = 12) and A008594 (multiples of 12).
Multiples of k whose sum of digits = k: A011557 (k=1), A069537 (k=2), A052217 (k=3), A063997 (k=4), A069540 (k=5), A062768 (k=6), A063416 (k=7), A069543 (k=8), A052223 (k=9), A333834 (k=10), A283742 (k=11), this sequence (k=12), A283737 (k=13).
Cf. A008594 (multiples of 12), A235151 (sum of digits = 12).
Cf. A057147 (a(n) = n times sum of digits of n).

Select[12 * Range[200], Plus @@ IntegerDigits[#] == 12 &] (* Amiram Eldar, Apr 06 2020 *)

is(n)=sumdigits(n)==12 && n%4==0 \\ Charles R Greathouse IV, Apr 07 2020

a(n) ~ A235151(n). - Charles R Greathouse IV, Apr 07 2020

A333834 Multiples of 10 whose sum of digits is 10.

Original entry on oeis.org

190, 280, 370, 460, 550, 640, 730, 820, 910, 1090, 1180, 1270, 1360, 1450, 1540, 1630, 1720, 1810, 1900, 2080, 2170, 2260, 2350, 2440, 2530, 2620, 2710, 2800, 3070, 3160, 3250, 3340, 3430, 3520, 3610, 3700, 4060, 4150, 4240, 4330, 4420, 4510

Offset: 1

Bernard Schott, Apr 07 2020

If m is a term, 10*m is also a term.

Intersection of A052224 (sum of digits = 10) and A008592 (multiples of 10).

			2440 = 10 * 244 and 2 + 4 + 4 + 0 = 10, hence 2440 is a term.

Multiples of k whose sum of digits = k: A011557 (k=1), A069537 (k=2), A052217 (k=3), A063997 (k=4), A069540 (k=5), A062768 (k=6), A063416 (k=7), A069543 (k=8), A052223 (k=9), this sequence (k=10), A283742 (k=11), A333814 (k=12), A283737 (k=13).
Subsequence of A218292.
Cf. A008592 (multiples of 10), A052224 (sum of digits = 10).
Cf. A057147 (a(n) = n times sum of digits of n)

Select[10 * Range[500], Plus @@ IntegerDigits[#] == 10 &] (* Amiram Eldar, Apr 07 2020 *)

a(n) = 10*A052224(n). - Charles R Greathouse IV, Apr 07 2020

cp's OEIS Frontend

A229549 Numbers k such that k*(sum of digits of k) is a palindrome.

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A037478 Number of positive solutions to "numbers that are n times sum of their digits".

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A337816 Numbers that can be written as (m * sum of digits of m) for some m.

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A117570 Numbers of the form k * (sum of digits of k) listed sorted with multiplicity.

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A338976 Primes p such that p*A007953(p)+1 is prime.

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A089227 Numbers k such that 1 + k*ds(k) is prime, where ds(k) is the sum of digits of k.

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A213630 a(t_1 t_2...t_n) = (t_1 + t_2)*t_1 t_2 + ... + (t_n-1 + t_n)*t_n-1 t_n.

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A328683 Positive integers that are equal to 99...99 (repdigit with n digits 9) times the sum of their digits.

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A213630 a(t_1 t_2...t_n) = (t_1 + t_2)t_1 t_2 + ... + (t_n-1 + t_n)t_n-1 t_n.