A338038 -id:A338038 - cp's OEIS frontend

A338039 Numbers m such that A338038(m) = A338038(A004086(m)) where A004086(i) is i read backwards and A338038(i) is the sum of the primes and exponents in the prime factorization of i ignoring 1-exponents; palindromes and multiples of 10 are excluded.

18, 81, 198, 576, 675, 819, 891, 918, 1131, 1304, 1311, 1818, 1998, 2262, 2622, 3393, 3933, 4031, 4154, 4514, 4636, 6364, 8181, 8749, 8991, 9478, 12441, 14269, 14344, 14421, 15167, 15602, 16237, 18018, 18449, 18977, 19998, 20651, 23843, 24882, 26677, 26892, 27225

Offset: 1

Michel Marcus, Oct 08 2020

Palindromes (A002113) are excluded from the sequence because they obviously satisfy the condition.
Sequence is infinite since it includes 18, 1818, 181818, .... See link.
There are many cases of terms that are the repeated concatenation of integers like: 1818, 8181, 181818, ... , but also 131313131313131313131313131313 and more. See A338166.
If n is in the sequence and has d digits, and gcd(n, x) = gcd(A004086(n), x) where x = (10^((k+1)*d)-1)/(10^d-1), then the concatenation of k copies of n is also in the sequence. - Robert Israel, Oct 13 2020

			For m = 18 = 2*3^2, A338038(18) = 2 + (3+2) = 7 and for m = 81 = 3^4, A338038(81) = 7, so 18 and 81 are terms.

Michel Marcus, Table of n, a(n) for n = 1..2998
Chris Bispels, Muhammet Boran, Steven J. Miller, Eliel Sosis, and Daniel Tsai, v-Palindromes: An Analogy to the Palindromes, arXiv:2405.05267 [math.HO], 2024.
Muhammet Boran, Garam Choi, Steven J. Miller, Jesse Purice, and Daniel Tsai, A characterization of prime v-palindromes, arXiv:2307.00770 [math.NT], 2023.
Daniel Tsai, A recurring pattern in natural numbers of a certain property, arXiv:2010.03151 [math.NT], 2020.
Daniel Tsai, A recurring pattern in natural numbers of a certain property, Integers (2021) Vol. 21, Article #A32.
Daniel Tsai, v-palindromes: an analogy to the palindromes, arXiv:2111.10211 [math.HO], 2021.

Cf. A004086 (read n backwards), A002113, A029742 (non-palindromes), A338038, A338166.

rev:= proc(n) local L,i;
  L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
g:= proc(n) local t;
  add(t[1]+t[2],t=subs(1=0,ifactors(n)[2]))
end proc:
filter:= proc(n) local r;
  if n mod 10 = 0 then return false fi;
  r:= rev(n);
  r <> n and g(r)=g(n)
end proc:
select(filter, [$1..30000]); # Robert Israel, Oct 13 2020

s[1] = 0; s[n_] := Plus @@ First /@ (f = FactorInteger[n]) + Plus @@ Select[Last /@ f, # > 1 &]; Select[Range[30000], !Divisible[#, 10] && (r = IntegerReverse[#]) != # &&  s[#] == s[r] &] (* Amiram Eldar, Oct 08 2020 *)

f(n) = my(f=factor(n)); vecsum(f[,1]) + sum(k=1, #f~, if (f[k,2]!=1, f[k,2])); \\ A338038
isok(m) = my(r=fromdigits(Vecrev(digits(m)))); (m % 10) && (m != r) && (f(r) == f(m));

A141276 a(n) = A338038(A002808(n)).

Original entry on oeis.org

4, 5, 5, 5, 7, 7, 9, 8, 6, 7, 9, 10, 13, 8, 7, 15, 6, 11, 10, 7, 14, 19, 12, 9, 21, 16, 10, 12, 15, 10, 25, 9, 9, 9, 20, 17, 8, 16, 12, 22, 31, 12, 33, 12, 8, 18, 16, 21, 26, 14, 10, 39, 10, 23, 18, 18, 11, 7, 43, 14, 22, 45, 32, 16, 12, 20, 27, 34, 49, 24, 10, 11, 16, 11, 22, 18, 15, 55

Offset: 1

Juri-Stepan Gerasimov, Aug 08 2008

nonn

			Let k(n) = n-th composite (A002808(n)). Then:
a(4)=5 because k(4) = 9 = 3^2 and 5=3+2;
a(9)=6 because k(9) = 16 = 2^4 and 6=2+4;
a(10)=7 because k(10) = 18 = 2*3^2 and 7=2+3+2.

Cf. A002808, A338038.

A141276 := proc(n) local a, ifs, p, c ; if not isprime(n) then a := 0 ; ifs := ifactors(n)[2] ; for p in ifs do a := a + op(1,p) ; if op(2,p) > 1 then a := a+ op(2,p) ; fi; od: printf("%d,",a) ; end if; return ; end: for n from 4 to 200 do A141276(n); end do; # R. J. Mathar, Apr 28 2010

a[n_] := Plus @@ First /@ (f = FactorInteger[n]) + Plus @@ Select[Last /@ f, # > 1 &];a/@Select[Range[106], CompositeQ] (* James C. McMahon, Jul 19 2025 *)

Entries checked by R. J. Mathar, Apr 28 2010

Edited (with clearer definition) by N. J. A. Sloane, Jun 15 2021

A381202 a(n) is the sum of the elements of the set of bases and exponents (including exponents = 1) in the prime factorization of n.

Original entry on oeis.org

0, 3, 4, 2, 6, 6, 8, 5, 5, 8, 12, 6, 14, 10, 9, 6, 18, 6, 20, 8, 11, 14, 24, 6, 7, 16, 3, 10, 30, 11, 32, 7, 15, 20, 13, 5, 38, 22, 17, 11, 42, 13, 44, 14, 11, 26, 48, 10, 9, 8, 21, 16, 54, 6, 17, 13, 23, 32, 60, 11, 62, 34, 13, 8, 19, 17, 68, 20, 27, 15, 72, 5

Offset: 1

Paolo Xausa, Feb 16 2025

The prime factorization of 1 is the empty set, so a(1) = 0 by convention (empty sum).

			a(12) = 6 because 12 = 2^2*3^1, the set of these bases and exponents is {1, 2, 3} and 1 + 2 + 3 = 6.
a(31500) = 18 because 31500 = 2^2*3^2*5^3*7^1, the set of these bases and exponents is {1, 2, 3, 5, 7} and 1 + 2 + 3 + 5 + 7 = 18.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A008474, A338038, A381201, A381203, A381204, A381205.

A381202[n_] := If[n == 1, 0, Total[Union[Flatten[FactorInteger[n]]]]];
Array[A381202, 100]

a(n) = my(f=factor(n)); vecsum(setunion(Set(f[,1]), Set(f[,2]))); \\ Michel Marcus, Feb 18 2025

A338166 Terms of A338039 that are repeated concatenations of smaller integers.

Original entry on oeis.org

1818, 8181, 181818, 198198, 405405, 484848, 504504, 565656, 576576, 656565, 675675, 818181, 848484, 891891, 11311131, 13041304, 13111311, 18181818, 19981998, 22622262, 26222622, 33933393, 39333933, 40314031, 41544154, 45144514, 46364636, 63646364, 81818181, 87498749, 89918991, 94789478

Offset: 1

Michel Marcus, Oct 14 2020

Michel Marcus, Table of n, a(n) for n = 1..1050 (up to 15 digits).
Daniel Tsai, A recurring pattern in natural numbers of a certain property, arXiv:2010.03151 [math.NT], 2020.
Daniel Tsai, A recurring pattern in natural numbers of a certain property, Integers (2021) Vol. 21, Article #A32.

Cf. A338038, A338039.

Block[{f}, f[1] = 0; f[n_] := Plus @@ #[[All, 1]] + Plus @@ Select[#[[All, -1]], # > 1 &] &@ FactorInteger[n]; Select[Union@ Flatten@ Table[Union@ Flatten@ Map[Function[k, Map[FromDigits[Join @@ ConstantArray[IntegerDigits[#], n/k]] &, Range[10^(k - 1), 10^k - 1]]], Most@ Divisors[n]], {n, 3, 8}], And[Mod[#1, 10] != 0, #2 != #1, f[#1] == f[#2]] & @@ {#, IntegerReverse[#]} &] ] (* Michael De Vlieger, May 27 2021, after Amiram Eldar at A338039 *)

f(n) = my(f=factor(n)); vecsum(f[,1]) + sum(k=1, #f~, if (f[k,2]!=1, f[k,2])); \\ A338038
isok(m) = my(r=fromdigits(Vecrev(digits(m)))); if ((r != m) && (f(r) == f(m)), return(m));
listc(c) = {my(list = List()); fordiv(c, d, if ((d != 1) && (d != c), for(k=10^(d-1), 10^d, if (k % 10, my(sk = Str(k), skk = sk); for (j=1, c/d-1, sk = concat(sk, skk)); if (isok(eval(sk)), listput(list, eval(sk))););););); list;}
lista(nn) = {my(list = List()); forcomposite(c=1, nn, my(clist = Vec(listc(c))); for (k=1, #clist, listput(list, clist[k]));); vecsort(Vec(list),,8);}
lista(8) \\ to get terms up to 8 digits

A349674 a(n) is the least v-palindrome in base n.

Original entry on oeis.org

175, 1280, 6, 288, 10, 731, 14, 93, 18, 135, 22, 63, 26, 291, 109, 581, 34, 144, 38, 24, 51, 1145, 46, 273, 50, 260, 335, 63, 58, 360, 62, 141, 110, 513, 224, 1404, 74, 140, 294, 189, 82, 224, 86, 344, 105, 2410, 94, 417, 98, 176, 497, 56, 106, 76, 60, 189, 1385, 3952, 100

Offset: 2

Michel Marcus, Nov 24 2021

A v-palindrome in base n is a number k that is not palindromic in base n, but for which A338038(k) = A338038(reverse(k) in base n).

			a(10) = A338039(1) = 18.

Michel Marcus, Table of n, a(n) for n = 2..3000
Daniel Tsai, A recurring pattern in natural numbers of a certain property, arXiv:2010.03151 [math.NT], 2020.
Daniel Tsai, A recurring pattern in natural numbers of a certain property, Integers (2021) Vol. 21, Article #A32.
Daniel Tsai, v-palindromes: an analogy to the palindromes, arXiv:2111.10211 [math.HO], 2021. See Table 1 p. 9.

Cf. A338038, A338039.

s[1] = 0; s[n_] := Plus @@ First /@ (f = FactorInteger[n]) + Plus @@ Select[Last /@ f, # > 1 &]; a[b_] := Module[{k = b+1, r}, While[!(!Divisible[k, b] && k != (r = IntegerReverse[k,b]) && s[k] == s[IntegerReverse[k, b]]), k++]; k]; Array[a, 100, 2] (* Amiram Eldar, Nov 24 2021 *)

f(n) = my(f=factor(n)); vecsum(f[, 1]) + sum(k=1, #f~, if (f[k, 2]!=1, f[k, 2])); \\ A338038
isok(m, b) = my(r=fromdigits(Vecrev(digits(m, b)), b)); (m % b) && (m != r) && (f(r) == f(m));
a(n) = my(k=1); while (!isok(k, n), k++); k;

A338371 Integers for which there exists a self-repetition that is a term of A338166.

Original entry on oeis.org

13, 17, 18, 19, 26, 31, 37, 39, 48, 49, 56, 62, 65, 71, 73, 79, 81, 84, 91, 93, 94, 97, 103

Offset: 1

Michel Marcus, Oct 23 2020

			18 is a term since 1818 is a term of A338166.
48 is a term since 484848 is a term of A338166.
List of terms with their minimum number of repetitions : [13, 15], [17, 280], [18, 2], [19, 819], [26, 15], [31, 15], [37, 12], [39, 15], [48, 3], [49, 3243], [56, 3], [62, 15], [65, 3], [71, 280], [73, 12], [79, 624], [81, 2], [84, 3], [91, 819], [93, 15], [94, 3243], [97, 624], [103, 10234].

Chris Bispels, Muhammet Boran, Steven J. Miller, Eliel Sosis, and Daniel Tsai, v-Palindromes: An Analogy to the Palindromes, arXiv:2405.05267 [math.HO], 2024.
Daniel Tsai, A recurring pattern in natural numbers of a certain property, arXiv:2010.03151 [math.NT], 2020.
Daniel Tsai, A recurring pattern in natural numbers of a certain property, Integers (2021) Vol. 21, Article #A32.

Cf. A338038, A338039, A338166.

f(n) = my(f=factor(n)); vecsum(f[,1]) + sum(k=1, #f~, if (f[k,2]!=1, f[k,2])); \\ A338038
period(vp,n) = {my(p = 1, pten = 10^#Str(n)); for (i=1, #vp, if ((vp[i] != 2) && (vp[i] != 5), p = lcm(p, znorder(Mod(pten, vp[i]))); p = lcm(p, znorder(Mod(pten, vp[i]^2))););); p;}
isok(n) = {my(r = fromdigits(Vecrev(digits(n)))); my(vp = setunion(factor(n)[,1]~, factor(r)[,1]~)); my(nbmax = period(vp, n)); if (nbmax == 1, nbmax = 2); my(krep=1); my(pten = 10^#Str(n)); for (k=2, nbmax, krep = pten*krep+1; my(q=1); for (i=1, #vp, my(va = valuation(krep, vp[i])); q *= vp[i]^va;); if (f(n*q) == f(r*q), return(k);););}
ispal(n) = my(d=Vecrev(digits(n))); n == fromdigits(d);
lista(nn) = {for (n=1, nn, if ((n % 10) && !ispal(n), if (isok(n), print1(n, ", "));););}

cp's OEIS Frontend

A338039 Numbers m such that A338038(m) = A338038(A004086(m)) where A004086(i) is i read backwards and A338038(i) is the sum of the primes and exponents in the prime factorization of i ignoring 1-exponents; palindromes and multiples of 10 are excluded.

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A141276 a(n) = A338038(A002808(n)).

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A381202 a(n) is the sum of the elements of the set of bases and exponents (including exponents = 1) in the prime factorization of n.

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A338166 Terms of A338039 that are repeated concatenations of smaller integers.

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A349674 a(n) is the least v-palindrome in base n.

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A338371 Integers for which there exists a self-repetition that is a term of A338166.

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