A115738 -id:A115738 - cp's OEIS frontend

A115761 Numbers k such that the digits of k^2, reversed, include the digits of k as a substring.

1, 5, 6, 27, 74, 963, 5723, 6142, 6524, 9867, 53152, 65766, 69714, 351732, 409761, 497674, 624279, 664902, 697140, 781101, 2733747, 3879405, 4897317, 5031485, 6317056, 6971400, 8403702, 9324454, 16013465, 21646858, 34679962, 36909226, 38794051, 40781862, 41065012

Offset: 1

Giovanni Resta, Jan 30 2006

			963^2 = 927(369).

Giovanni Resta, Table of n, a(n) for n = 1..97 (terms < 4*10^12)

Contains A269588 as a subsequence.

Cf. A115762, A115763, A115764, A115738, A179803.

fQ[n_] := StringPosition[ IntegerString[n^2], StringReverse@ IntegerString@n] != {}; k = 1; lst = {}; While[k < 38794050, If[ fQ@k, AppendTo[lst, k]]; k++ ]; lst (* Robert G. Wilson v, Jul 28 2010 *)
Select[Range[37*10^6],SequenceCount[Reverse[IntegerDigits[#^2]], IntegerDigits[ #]]>0&] (* The program uses the SequenceCount function from Mathematica version 10 *) (* Harvey P. Dale, May 02 2016 *)

More terms from Robert G. Wilson v, Jul 28 2010

Terms a(33) and beyond from Giovanni Resta, May 04 2020

A115739 n is prime and digits of n^3 include digits of n as substring.

Original entry on oeis.org

5, 251, 499, 751, 1249, 4999, 8089, 18229, 31249, 49999, 281249, 4218751, 4999999, 9590417, 31622777, 68171507, 74218751, 574218751, 5740883989, 8829261293

Offset: 1

Giovanni Resta, Jan 30 2006

Subset of A029942.

			751^3 = 423564_751.

Cf. A029942, A115738.

A115763 Primes p such that the digits of p^2, reversed, include the digits of p as a substring.

Original entry on oeis.org

5, 87690769, 311314139, 44466725441, 38460053757197

Offset: 1

Giovanni Resta, Jan 30 2006

a(6) > 6*10^14. - Giovanni Resta, Oct 21 2018

			311314139^2 = 969164(931413113)21.

Cf. A115761, A115762, A115764, A115738.

a(5) from Giovanni Resta, Sep 11 2018

A239058 Numbers whose divisors all appear as a substring in their decimal expansion.

Original entry on oeis.org

1, 11, 13, 17, 19, 31, 41, 61, 71, 101, 103, 107, 109, 113, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 241, 251, 271, 281, 311, 313, 317, 331, 401, 419, 421, 431, 461, 491, 521, 541, 571, 601, 613, 617, 619, 631, 641, 661, 691, 701, 719, 751, 761, 811, 821, 881, 911, 919, 941, 971

Offset: 1

M. F. Hasler, Mar 09 2014

A subsequence of A092911 (all divisors can be formed using the digits of the number) which is a subsequence of A011531 (numbers having the digit 1).
Are 1 and 125 the only nonprime terms in this sequence?
No: 17692313, 4482669527413081, 21465097175420089, and 567533481816008761 are members. - Charles R Greathouse IV, Mar 09 2014
See A239060 for the nonprime terms of this sequence, which include in particular the squares of terms of A115738 (unless such a square would not have a digit 1).

			All primes having the digit 1 (A208270) are in this sequence, because {1, p} are the only divisors of a prime p.
The divisors of 125 are {1, 5, 25, 125}; it can be seen that all of them occur as a substring in 125, therefore 125 is in this sequence.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to decimal expansion of n

Cf. A092911, A011531, A121041, A121022-A121040, A018834.

is(n,d=vecextract(divisors(n),"^-1"))={ setminus(select(x->x<10,d),Set(digits(n)))&&return;!for(L=2,#Str(d[#d]),setminus(select(x->x
<10^L&&x>=10^(L-1),d),Set(concat(vector(L,o,digits(n\10^(L-o),10^L)))))&&return)}

overlap(long,short)=my(D=10^#digits(short)); while(long>=short, if(long%D==short,return(1));long\=10); 0
is(n)=my(d=divisors(n)); forstep(i=#d-1,1,-1, if(!overlap(n,d[i]), return(0))); 1 \\ Charles R Greathouse IV, Mar 09 2014

A239060 Nonprime numbers whose divisors all appear as a substring in the number's decimal expansion.

Original entry on oeis.org

1, 125, 17692313

Offset: 1

M. F. Hasler, Mar 09 2014

This is the subsequence of A239058 without the primes having a digit 1, A208270. It is thus a subsequence of A092911 (all divisors can be formed using the digits of the number) which is a subsequence of A011531 (numbers having the digit 1).
The term a(3)=17692313=A239058(870356), as well as the numbers 4482669527413081, 21465097175420089, and 567533481816008761 which are also members, were found by Charles R Greathouse IV, Mar 09 2014
The square of any term of A115738 is a member of this sequence. The above larger examples are of that form.
a(4) > 10^12. - Giovanni Resta, Sep 08 2018

			The divisors of 17692313 are {1, 23, 769231, 17692313}; it can be seen that all of them occur as a substring in 17692313, therefore 17692313 is in this sequence.

Cf. A092911, A011531, A121041, A121022-A121040, A018834.

PARI
```
is(n)=!isprime(n)&&is_A239058(n)
```

overlap(long,short)=my(D=10^#digits(short)); while(long>=short, if(long%D==short,return(1));long\=10); 0
is(n)=my(d=divisors(n)); #d!=2 && !forstep(i=#d-1,1,-1, if(!overlap(n,d[i]), return(0))) \\ Charles R Greathouse IV, Mar 09 2014

A318965 a(n) is the smallest number containing all its n prime factors in its decimal representation.

Original entry on oeis.org

2, 135, 735, 21372, 271362, 4773132, 113678565, 11317129824, 131175822960, 7113719552940, 255360234137190, 12411792985131540

Offset: 1

Giovanni Resta, Sep 06 2018

			a(2) = 135 = 3^3 * 5,
a(3) = 735 = 3 * 5 * 7^2,
a(4) = 21372 = 2^2 * 3 * 13 * 137,
a(5) = 271362 = 2 * 3 * 7^2 * 13 * 71,
a(6) = 4773132 = 2^2 * 3^2 * 7 * 13 * 31 * 47.
a(7) = 113678565 = 3 * 5 * 7 * 11 * 13 * 67 * 113.

Cf. A131929, A131930, A131931, A239060, A115738.

a[n_] := Block[{k=1, s, f}, While[True, k++; If[Length[f = FactorInteger[k]] == n, s = ToString@k; If[AllTrue[First /@ f, StringPosition[s, ToString@ #] != {} &], Break[]]]]; k]; Array[a, 5]

A329271 Numbers k such that k multiplied by the product of its divisors contains k as a substring.

Original entry on oeis.org

1, 5, 6, 10, 16, 24, 25, 30, 36, 40, 50, 51, 60, 70, 76, 90, 92, 100, 125, 176, 195, 240, 249, 250, 363, 375, 376, 430, 490, 500, 501, 510, 546, 556, 560, 568, 570, 600, 620, 624, 625, 648, 680, 730, 749, 750, 760, 810, 875, 909, 930, 972, 975, 976, 990, 999, 1000, 1001, 1010, 1636, 1680, 1930, 2354, 2400, 2490, 2500, 2510, 2512, 2943, 3000

Offset: 1

Scott R. Shannon, Nov 10 2019

Inspired by A328095. To avoid all primes being in the sequence the divisors of k includes k itself.

Contains 10^k, 5*10^k and 6*10^k for all k, 3*10^k, 4*10^k, 7*10^k and 9*10^k for all odd k. - Robert Israel, Nov 11 2019

			16 is in the sequence as the divisors of 16 are 1,2,4,8,16, and 16*(1*2*4*8*16) = 16*1024 = 16384, and '16384' contains '16' as a substring.
30 is in the sequence as the divisors of 30 are 1,2,3,5,6,10,15,30, and 30*(1*2*3*5*6*10*15*30) = 30*810000 = 24300000, and '24300000' contains '30' as a substring.

Scott R. Shannon, Table of n, a(n) for n = 1..1000. Note when searching for these numbers one needs to use arbitrary precision packages; the product for 24570000 has 1486 digits.

Cf. A007955, A328095, A326806, A328544.

The sequence of primes contained in their squares is A115738.

a:=[]; for k in [1..3000] do t:=IntegerToString(k*(&*Divisors(k))); s:=IntegerToString(k); if s in t then Append(~a,k); end if; end for; a; // Marius A. Burtea, Nov 10 2019

f[n_] := n^(1+DivisorSigma[0, n]/2); aQ[n_] := SequenceCount[IntegerDigits[f[n]], IntegerDigits[n]] > 0; Select[Range[3000], aQ] (* Amiram Eldar, Nov 10 2019 *)

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A115761 Numbers k such that the digits of k^2, reversed, include the digits of k as a substring.

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A115739 n is prime and digits of n^3 include digits of n as substring.

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A115763 Primes p such that the digits of p^2, reversed, include the digits of p as a substring.

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A239058 Numbers whose divisors all appear as a substring in their decimal expansion.

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A239060 Nonprime numbers whose divisors all appear as a substring in the number's decimal expansion.

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A318965 a(n) is the smallest number containing all its n prime factors in its decimal representation.

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Mathematica

A329271 Numbers k such that k multiplied by the product of its divisors contains k as a substring.

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