A083359 -id:A083359 - cp's OEIS frontend

A083361 Subsequence of sequence A083359 in which every prime factor can be found in the number at least as often as it is factor of the number.

21372, 119911, 229912, 2971137, 13731373, 31373137, 183171409, 221397372, 241153517, 254724772, 271141332, 331191135, 1153115117, 1190911909, 1453312395, 2511176437, 2923699119, 2971193115, 3124516195, 3211433715

Offset: 1

Sven Simon, Apr 27 2003

			221397372 = 2*2*3*3*7*397*2213.

Deron Stewart, Table of n, a(n) for n = 1..27

Cf. A083359, A083360.

Offset corrected by Deron Stewart, Feb 21 2019

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

A324261 Subsequence of A083359 (Visual Factor Numbers) of the form (10^m+1)*p, where the decimal representation of prime p contains all the prime factors of 10^m+1.

Original entry on oeis.org

13731373, 31373137, 1190911909, 9091190911, 19090911909091, 7316763215964848081373167632159648480813, 111272689909091345969111272689909091345969, 111279090913268945969111279090913268945969, 112726894596919090913112726894596919090913

Offset: 1

Deron Stewart, Mar 13 2019

Subsequence of A020338 (Doublets: base-10 representation is the juxtaposition of two identical strings).
The prime factors of 10^m + 1 are also prime factors of 10^km + 1, where k is odd, so a(n) and a(km) have those prime factors in common.
Expanding on the comment in A083359 regarding finding large terms, we can generate large doublet terms in the following way:
-- for any composite number M = 10^m + 1 try to compose a prime number p of length m from the prime factors of M. Generally this will require the factors to overlap to reduce the length to m.
-- the factors can wrap around to the beginning of p. For example, if M has a factor of 137 then p can be of the form 7...13.
-- the term is formed by concatenating p with itself to form a(n) = p||p. The resulting number will consist entirely of the concatenation of its prime factors with allowed overlap as in A083359.

			With m = 4: 10^4 + 1 = 10001 = 73 * 137. We can form prime p = 1373 which concatenates with itself to give a(1) = 13731373 = 73 * 137 * 1373. We can also form the prime p = 3137 which gives a(2). The number 7313 also contains all the prime factors of 10001 but it is not prime.
With m = 33: 10^33 + 1 = 7*11*11*13*23*4093*8779*599144041*183411838171, there are 4932 m-digit numbers that contain all the factors, of which 227 of them are prime. Each of these primes generates a term in the sequence with 66 digits, the smallest of which is 112359914404134093877918341183817112359914404134093877918341183817. This is A324262(33).

Cf. A083359, A020338, A324262, A324257.

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.

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

A324260 Subsequence of A324257 (Conceited Numbers) where the distinct prime factors are concatenated without multiplicity.

Original entry on oeis.org

735, 3792, 1341275, 1713192, 2971137, 4773132, 13115375, 13731373, 22940075, 29373375, 31373137, 71624133, 121719472, 183171409, 221397372, 241153517, 311997175, 319953792, 331191135, 1019127375, 1147983375, 1190911909, 1453312395

Offset: 1

Deron Stewart, Feb 19 2019

Subsequence of A083359.

Overlap of the distinct primes is allowed per A324257. Terms without overlap form the subsequence A121342.

			4773132 = 2^2 * 3^2 * 7 * 13 * 31 * 47 formed by 47||7||[3(1][3)]||2. The 6 distinct prime factors are used once each, with 3, 13 and 31 overlapping.

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

Cf. A324257, A083359, A121342.

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.

A324262 a(n) is the smallest term of A324261 with 2n digits if it exists, otherwise 0.

Original entry on oeis.org

0, 0, 0, 0, 13731373, 1190911909, 0, 19090911909091, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7316763215964848081373167632159648480813, 111272689909091345969111272689909091345969, 10527889691056689261011052788969105668926101, 0, 0, 0, 0

Offset: 0

Deron Stewart, Mar 13 2019

There can be a large number of terms in A324261 with 2n digits. For example, there are 227 terms with length 66. Selecting only the smallest term for each length allows terms to be listed for larger values of n.
The terms of A324261 with length 2n are formed by concatenating two copies of prime p, where p has length n and the decimal representation of p contains all the prime factors of 10^n + 1 as described in A324261 and A083359.
Subsequence of A020338 (Doublets: base-10 representation is the juxtaposition of two identical strings).

			With m = 27 there are three prime p's: 114175966169705419295257913, 352579141759661697054192911 and 525791141759661697054192913. The smallest p concatenated with itself gives a(27) = 114175966169705419295257913114175966169705419295257913.
With m = 28 there are no solutions so a(28) = 0.

Deron Stewart, Table of n, a(n) for n = 0..61

Cf. A083359, A020338, A324261, A324257.

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

A319022 a(n) is the smallest number with n distinct prime factors whose decimal representation contains all its prime factors, taking multiplicity into account.

Original entry on oeis.org

2, 119911, 2510, 21372, 1943795, 73171842, 113678565, 13121675970, 297115923720, 73605381139290, 255360234137190, 43759729761726090

Offset: 1

Altug Alkan, Sep 08 2018

A version of A318965.
If a term t is divisible by a prime power p^k, then p must appear at least k times in the decimal representation of t. The copies of p are allowed to overlap; however, in the first 12 terms of the sequence, this does not occur, and only a(2), a(4), and a(9) are nonsquarefree.
a(n) >= A318965(n) by definition. If A318965(n) is squarefree, then a(n) = A318965(n). But a(n) = A318965(n) can also hold for a(n) that is nonsquarefree; e.g., a(4) = A318965(4).

			a(2) = 119911 = 11 * 11 * 991.
a(3) = 2510 = 2 * 5 * 251.
a(4) = 21372 = 2 * 2 * 3 * 13 * 137.
a(5) = 1943795 = 5 * 7 * 19 * 37 * 79.
a(6) = 73171842 = 2 * 3 * 17 * 31 * 73 * 317.
a(7) = 113678565 = 3 * 5 * 7 * 11 * 13 * 67 * 113.

Cf. A083359, A131930, A318965.

Terms computed by Giovanni Resta, Sep 08 2018

cp's OEIS Frontend

A083361 Subsequence of sequence A083359 in which every prime factor can be found in the number at least as often as it is factor of the number.

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

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A324261 Subsequence of A083359 (Visual Factor Numbers) of the form (10^m+1)*p, where the decimal representation of prime p contains all the prime factors of 10^m+1.

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

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A324260 Subsequence of A324257 (Conceited Numbers) where the distinct prime factors are concatenated without multiplicity.

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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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A324262 a(n) is the smallest term of A324261 with 2n digits if it exists, otherwise 0.

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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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A319022 a(n) is the smallest number with n distinct prime factors whose decimal representation contains all its prime factors, taking multiplicity into account.

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