A083361 -id:A083361 - cp's OEIS frontend

A083359 Visible Factor Numbers, or VFNs: numbers n with the property that every prime factor of n can be found in the decimal expansion of n and every digit of n can be found in a prime factor.

735, 3792, 13377, 21372, 51375, 119911, 229912, 290912, 537975, 1341275, 1713192, 2333772, 2971137, 4773132, 7747292, 13115375, 13731373, 19853575, 22940075, 29090912, 29373375, 31373137, 35322592, 52979375, 71624133, 79241575

Offset: 1

Sven Simon, Apr 27 2003

Larger terms can be found with the factorization of 10^m+1. A prime p containing all the prime factors of 10^m+1 will give the VFN (pp), for example 13731373 = 73*137*1373 with 73*137 = 10001. Every prime 9090...9091 builds a VFN with the cofactor 2^5.
Sequence is probably infinite.
The prime p in the 10^m+1 example above must contain exactly m digits. Also, it can contain one of the prime factors wrapped around the end of p. For example, p=11909 contains 11 and 9091, the factors of 100001, with the 9091 wrapping around to the beginning of p. This forms a(44)=1190911909. - Deron Stewart, Feb 23 2019
The concatenation must be possible using the prime factors of the number, unlimited multiplicity of the distinct prime factors is not allowed. For example, 71153775 = 3*3*3*5*5*7*11*37*37 can be formed by 7||11||5||37||7||5 but the concatenation requires two 7's and there is only one 7 in the prime factorization, so it is not in the sequence. - Deron Stewart, Mar 01 2019

Lindon, Visible factor numbers, J. Rec. Math., 1 (1968), 217.

Giovanni Resta, Table of n, a(n) for n = 1..165 (first 64 terms except a(59) from Sven Simon, a(59) and a(64)-a(74) from Deron Stewart)

Cf. A083360, A083361.

A324257 Conceited Numbers: Composite numbers that are a concatenation of their distinct prime factors with multiplicity in some order allowing overlap.

Original entry on oeis.org

735, 3792, 13377, 21372, 51375, 67335, 119911, 229912, 290912, 537975, 1341275, 1713192, 2317312, 2333772, 2971137, 3719193, 4773132, 5117695, 7237755, 7747292, 11973192, 13115375, 13731373, 16749933, 19853575, 22940075, 29090912, 29373375

Offset: 1

Deron Stewart, Feb 19 2019

"Conceited Numbers" (they are full of themselves).
The decimal representation of these numbers can be formed typographically from their prime factors. Every distinct prime factor must appear at least once. Generalization of the sequence A083359.
Subsequences:
--No overlap: A324258
--Every prime factor appears in number (not just distinct prime factors): A324259
--No multiplicity: A324260
--Multiplicity only up to the exponent of the distinct prime factor: A083359
Other subsequences are formed by more than one constraint; e.g., A121342 is the intersection of A324258 and A324260, terms with no overlap and no multiplicity.

			67335 = 3*5*67^2 formed by 67||3|||3||5 (this term is not in A083359 because two 3's are required in the concatenation).
3719193 = 3*19*71*919 formed by 3||71||9(19)||3 where 19 and 919 overlap.

Giovanni Resta, Table of n, a(n) for n = 1..302 (first 103 terms from Deron Stewart)
Deron Stewart, C# method to evaluate candidate terms given the distinct prime factors.

Cf. A324258, A324259, A324260, A083359, A083360, A083361, A121342.

A083360 Subsequence of sequence A083359 in which factors do not overlap in the number.

Original entry on oeis.org

735, 3792, 13377, 290912, 537975, 1341275, 2333772, 7747292, 13115375, 19853575, 22940075, 29090912, 29373375, 52979375, 71624133, 79241575, 311997175, 319953792, 533334375, 1019127375, 1147983375, 1734009275

Offset: 1

Sven Simon, Apr 27 2003

173279232 deleted because it requires overlap of the 7 factor with either the 17 or the 79 factor. - Deron Stewart, Feb 21 2019

			71624133 = 7*7*162413*3*3.

Cf. A083359, A083361.

Incorrect term 173279232 deleted by Deron Stewart, Feb 21 2019

A121342 Composite numbers that are a concatenation of their distinct prime divisors in some order.

Original entry on oeis.org

735, 3792, 1341275, 13115375, 22940075, 29373375, 71624133, 311997175, 319953792, 1019127375, 1147983375, 1734009275, 5581625072, 7350032375, 17370159615, 33061224492, 103375535837, 171167303912, 319383665913, 533671737975, 2118067737975, 3111368374257

Offset: 1

Tanya Khovanova, Aug 28 2006

Larger terms of this sequence were calculated by Giovanni Resta and Farideh Firoozbakht. This sequence is a subsequence of A083360 (Subsequence of sequence A083359 in which factors do not overlap in the number), which is a subsequence of A083359 (Visible Factor Numbers, or VPNs: numbers n with the property that every prime factor of n can be found in the decimal expansion of n and every digit of n can be found in a prime factor. No additional 0's and 1's are allowed). Also, this sequence is a subsequence of A096595 (Numbers n with the property that n is an anagram of the digits of the distinct prime factors of n).

			For example: 735 = 3*5*7^2 and 3792 = 2^4*3*79.

Giovanni Resta, Table of n, a(n) for n = 1..30

Cf. A083359, A083360, A083361, A096595.

fQ[n_] := !PrimeQ@n && MemberQ[ FromDigits /@ (Flatten@# & /@ IntegerDigits[ Permutations[ First /@ FactorInteger@n]]), n]; Do[ If[fQ@n, Print@n], {n, 10^7/4}] (* Robert G. Wilson v, Sep 02 2006 *)

isok(n) = {if (isprime(n), return (0)); my(vp = factor(n)[,1], nb = #vp); for (i=0, nb!-1, my(vperm = numtoperm(nb, i), s = ""); for (i=1, #vperm, s = concat(s, vp[vperm[i]]);); if (eval(s) == n, return (1));); return (0);} \\ Michel Marcus, Feb 19 2019

a(14) from Emmanuel Vantieghem, Nov 30 2016
Missing term 5581625072=5581||62507||2 inserted by Deron Stewart, Feb 15 2019
a(16)-a(22) from Giovanni Resta, Mar 04 2019

A324259 Subsequence of A324257 (Conceited Numbers) where every prime factor is visible in the number, not just each distinct prime factor.

Original entry on oeis.org

21372, 119911, 229912, 2971137, 3719193, 13731373, 16749933, 31373137, 183171409, 221397372, 238903323, 241153517, 254724772, 271141332, 331191135, 1153115117, 1190911909, 1453312395, 2511176437, 2923699119, 2971193115, 3124516195

Offset: 1

Deron Stewart, Feb 19 2019

Includes all squarefree terms in A324257 by definition.

Generalization of A083361 without the restriction of limited multiplicity inherited from A083359.

			119911 = 11^2 * 991, formed by 11||(99(1)1), with 11 appearing twice in the number as required. (991 and 11 overlap.)

Deron Stewart, C# methods to evaluate candidate terms given the distinct prime factors.

Cf. A324257, A083361, A083359.

A096595 Numbers n with the property that n is an anagram of the digits of the distinct prime factors of n.

Original entry on oeis.org

735, 3792, 7236, 17482, 19075, 19276, 32104, 42175, 104392, 107329, 123678, 145273, 149782, 174082, 174298, 174982, 237951, 297463, 319675, 457192, 459728, 639175, 840175, 1093672, 1236874, 1259473, 1268374, 1283746, 1286374, 1374682

Offset: 1

Gil Broussard, Aug 13 2004

			Example: 104392, whose prime factors are 2*2*2*13049. The digits 2, 1, 3, 0, 4 and 9 appear exactly once within 104392.

Gil Broussard, Integers Anagrammed by Their Unique Primes.

Cf. A083359, A083360, A083361.

The digits of the unique prime factors of n appear exactly once within n in any order, accounting for all the digits of n

cp's OEIS Frontend

A083359 Visible Factor Numbers, or VFNs: numbers n with the property that every prime factor of n can be found in the decimal expansion of n and every digit of n can be found in a prime factor.

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A324257 Conceited Numbers: Composite numbers that are a concatenation of their distinct prime factors with multiplicity in some order allowing overlap.

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A083360 Subsequence of sequence A083359 in which factors do not overlap in the number.

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A121342 Composite numbers that are a concatenation of their distinct prime divisors in some order.

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A324259 Subsequence of A324257 (Conceited Numbers) where every prime factor is visible in the number, not just each distinct prime factor.

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A096595 Numbers n with the property that n is an anagram of the digits of the distinct prime factors of n.

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