A152242 -id:A152242 - cp's OEIS frontend

A166505 Numbers in A166504 which are not in A152242.

202, 203, 205, 207, 302, 303, 305, 502, 505, 507, 702, 703, 705, 707, 1102, 1105, 1107, 1302, 1305, 1702, 1703, 1705, 1707, 1902, 1903, 1905, 2002, 2005, 2007, 2013, 2019, 2022, 2023, 2025, 2031, 2032, 2033, 2035, 2037, 2041, 2043, 2047, 2052, 2055, 2057

Offset: 1

M. F. Hasler, Nov 02 2009

All terms have at least one zero digit and are composite.

Cf. A152242, A166506 (indices of these terms in A166504).

for(i=1,1e4, is_A166504(i) & !is_A152242(i) & print1(i", "))

a(n) = A166504(A166506(n))

Edited by Charles R Greathouse IV, Apr 23 2010

A166506 Indices of terms in A166504 which are not in A152242.

Original entry on oeis.org

70, 71, 72, 73, 115, 116, 117, 183, 185, 186, 250, 251, 252, 253, 365, 367, 368, 427, 429, 534, 535, 536, 537, 594, 595, 596, 640, 642, 643, 645, 647, 648, 649, 650, 653, 654, 655, 656, 657, 659, 660, 661, 662, 664, 665, 666, 667, 669, 671, 672, 673, 674

Offset: 1

M. F. Hasler, Nov 02 2009

			a(1)=70 since the first term in A166504 which is not in A152242 is A166504(70)=202.

Cf. A152242, A166504, A166505.

c=0; for(i=1,1e4, is_A166504(i) & c++ & !is_A152242(i) & print1(c", "))

A166504(a(n)) = A166505(n).

A165631 Numbers whose cube is a concatenation of primes, i.e., in A152242.

Original entry on oeis.org

3, 7, 9, 11, 13, 15, 17, 18, 27, 28, 29, 31, 33, 38, 39, 45, 47, 48, 49, 53, 55, 58, 59, 61, 63, 68, 71, 73, 75, 83, 85, 88, 91, 95, 98, 103, 108, 111, 113, 117, 121, 125, 127, 131, 133, 135, 137, 138, 148, 153, 157, 159, 161, 163, 167, 168, 173, 175, 177, 178, 179

Offset: 1

Zak Seidov and M. F. Hasler, Oct 16 2009

Cf. A038692, A019549, A105184, A166503.

for(n=1,999, is_A152242(n^3) & print1(n", "))

Edited by Charles R Greathouse IV, Apr 24 2010

A019549 Primes formed by concatenating other primes.

Original entry on oeis.org

23, 37, 53, 73, 113, 137, 173, 193, 197, 211, 223, 227, 229, 233, 241, 257, 271, 277, 283, 293, 311, 313, 317, 331, 337, 347, 353, 359, 367, 373, 379, 383, 389, 397, 433, 523, 541, 547, 557, 571, 577, 593, 613, 617, 673, 677, 719, 727, 733, 743, 757, 761, 773, 797, 977

Offset: 1

R. Muller

			113 is member as 11 and 3 are primes.
a(12)=227 = "2"+"2"+"7" is the first term not in A105184 (restricted to concatenation of two primes). [_M. F. Hasler_, Oct 15 2009]

M. F. Hasler, Table of n, a(n) for n = 1..17495
Sylvester Smith, A Set of Conjectures on Smarandache Sequences, Bulletin of Pure and Applied Sciences, (Bombay, India), Vol. 15 E (No. 1), 1996, pp. 101-107.

Cf. A105184, A152242.

is_A019549(n, recurse=0)={ isprime(n) == recurse & return(recurse); for(i=1, #Str(n)-1, isprime( n%10^i ) & is_A019549( n\10^i, 1) & n\10^(i-1)%10 & return(1)) } \\ M. F. Hasler, Oct 15 2009

from sympy import isprime
def c(n, m):
    if m > 0 and isprime(n): return True
    s = str(n)
    return any(s[i]!="0" and isprime(int(s[:i])) and c(int(s[i:]), m+1) for i in range(1, len(s)))
def ok(n): return isprime(n) and c(n, 0)
print([k for k in range(978) if ok(k)]) # Michael S. Branicky, Sep 01 2024

A038692 Square numbers that are concatenations of two or more prime numbers.

Original entry on oeis.org

25, 225, 289, 361, 529, 729, 2401, 2601, 2809, 4761, 5329, 5929, 7225, 7569, 11449, 11881, 13225, 15129, 19881, 21609, 22801, 23409, 24649, 25281, 26569, 27225, 29241, 29929, 31329, 32761, 34969, 36481, 39601, 47961, 52441, 53361, 54289, 55225, 57121, 58081

Offset: 1

Jeff Burch

Jinyuan Wang, Table of n, a(n) for n = 1..1000

Cf. A166503 (the square roots).

ric[d_] := PrimeQ@ FromDigits@ d || AnyTrue[ TakeDrop[d, #] & /@ Range[ Length[d] - 1], #[[2,1]] > 0 && PrimeQ@ FromDigits@ #[[1]] && ric@ #[[2]] &]; ok[n_] := If[ EvenQ[n] && Mod[n, 10] != 2, False, ric@ IntegerDigits@ n]; Select[ Range[300]^2, ok] (* Giovanni Resta, Mar 16 2020 *)

forstep(n=1, 300, 2, is_A152242(n^2) & print1(n^2, ", ")) \\ M. F. Hasler, Mar 19 2012

A038692(n) = A166503(n)^2. As a set, A038692 = A000290 intersection A152242 = A016754 intersection A152242. - M. F. Hasler, Mar 19 2012

Edited by N. J. A. Sloane, Oct 18 2009, incorporating corrections from M. F. Hasler and Zak Seidov.

A166504 Slime numbers: numbers which are the concatenation of primes, with "leading zeros" allowed.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 22, 23, 25, 27, 29, 31, 32, 33, 35, 37, 41, 43, 47, 52, 53, 55, 57, 59, 61, 67, 71, 72, 73, 75, 77, 79, 83, 89, 97, 101, 103, 107, 109, 112, 113, 115, 117, 127, 131, 132, 133, 135, 137, 139, 149, 151, 157, 163, 167, 172, 173, 175, 177, 179

Offset: 1

M. F. Hasler, Nov 02 2009

A number is in this sequence if and only if it is prime or of the form a(k)*10^m+a(n), where a(k), a(n) are in this sequence and 10^m >= a(n) (and from this follows that one among a(k), a(n) can be taken to be prime).
This contains A152242 as a subsequence, but also additional terms like e.g. 202 which can be split into two primes, 2 and 02 (= 2). Such a splitting, where some of the substrings contain leading zeros, is not allowed in A152242.
Terms not in A152242 are listed in A166505.

Henri Picciotto, Selected Integer Sequences

Cf. A152242 (no leading zeros allowed).

Cf. A085823 (super-slimes: all substrings are prime). - Henri Picciotto, Apr 01 2015

is_A166504(n)={ isprime(n) || ((bittest(n,0) || n%10==2) & for(i=1,#Str(n)-1, isprime(n%10^i) & is_A166504(n\10^i) & return(1)))}

is(n)=if(isprime(n),return(1));if(n<202,return(isprime(n%10)&&isprime(n\10)));my(k=n%10,v);if(k==5||k==2,return(if(n<6,1,n\=10;has(n/10^valuation(n,10)))));if(k%2==0,return(0));v=digits(n);for(i=1,#v,if(isprime(n%10^i)&&is(n\10^i),return(1)));0 \\ Charles R Greathouse IV, Apr 30 2013

Edited by Charles R Greathouse IV, Apr 23 2010

Name "Slime numbers", after Henri Picciotto, added by N. J. A. Sloane, Mar 25 2015

A066737 Composite numbers that are concatenations of primes.

Original entry on oeis.org

22, 25, 27, 32, 33, 35, 52, 55, 57, 72, 75, 77, 112, 115, 117, 132, 133, 135, 172, 175, 177, 192, 195, 213, 217, 219, 222, 225, 231, 232, 235, 237, 243, 247, 252, 253, 255, 259, 261, 267, 272, 273, 275, 279, 289, 292, 295, 297, 312, 315, 319, 322, 323, 325

Offset: 1

Joseph L. Pe, Jan 15 2002

			72 is the concatenation of primes 7 and 2. 132 is the concatenation of primes 13 and 2. 225 is the concatenation of 2, 2 and 5.

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

Cf. A121609.

ccat:= proc(m,n) 10^(1+ilog10(n))*m+n end proc:
C[1]:= {2,3,5,7}: P[1]:=C[1]:
for n from 2 to 3 do
  P[n]:= select(isprime, {seq(i,i=10^(n-1)+1..10^n-1,2)});
  C[n]:= P[n];
  for m from 1 to n-1 do
    C[n]:= C[n] union {seq(seq(ccat(p,q),p =P[m]),q=C[n-m])};
  od
od:
seq(op(sort(convert(remove(isprime,C[n]),list))),n=1..3); # Robert Israel, Jan 22 2020

for(n=1,999, !isprime(n) && is_A152242(n) && print1(n", ")) \\ M. F. Hasler, Oct 16 2009

A066737 = A152242 \ A000040 = A152242 intersect A002808. - M. F. Hasler, Oct 16 2009

More terms from Larry Reeves (larryr(AT)acm.org), Apr 03 2002

Missing terms added by M. F. Hasler, Oct 16 2009

A166503 Numbers k with property that k^2 is the concatenation of two or more prime numbers.

Original entry on oeis.org

5, 15, 17, 19, 23, 27, 49, 51, 53, 69, 73, 77, 85, 87, 107, 109, 115, 123, 141, 147, 151, 153, 157, 159, 163, 165, 171, 173, 177, 181, 187, 191, 199, 219, 229, 231, 233, 235, 239, 241, 243, 267, 269, 277, 279, 281, 289, 299, 319, 327, 331, 335, 337, 343, 357

Offset: 1

Zak Seidov, Oct 15 2009

Only odd numbers are eligible.

Jinyuan Wang, Table of n, a(n) for n = 1..10000

Cf. A019549, A038692, A105184, A129800.

for(n=1, 9999, is_A152242(n^2) & print1(n, ", ")) \\ M. F. Hasler, Oct 15 2009

a(n) = sqrt(A038692(n)).

Terms updated according to stricter definition of A152242 by M. F. Hasler, Oct 15 2009

A167459 Composite numbers in A166504, i.e., whose decimal expansion can be split up into prime numbers, with leading zeros allowed.

Original entry on oeis.org

22, 25, 27, 32, 33, 35, 52, 55, 57, 72, 75, 77, 112, 115, 117, 132, 133, 135, 172, 175, 177, 192, 195, 202, 203, 205, 207, 213, 217, 219, 222, 225, 231, 232, 235, 237, 243, 247, 252, 253, 255, 259, 261, 267, 272, 273, 275, 279, 289, 292, 295, 297, 302, 303

Offset: 1

M. F. Hasler, Nov 19 2009

In contrast to A066737 (which is a subsequence of this one), we allow for leading zeros in the "prime" substrings; the two sequences differ from n=24 on, with a(24)=202 which is not in A066737.

Sequence A166505 gives the difference, A167459 \ A066737 = A166504 \ A152242. Sequence A167458 gives the indices of the terms not in A066737.

Cf. A002808, A066737, A121609, A166504, A167505.

is_A167459(n) = !isprime(n) & is_A166504(n)

A167459 = A002808 n A166504, where "n" means intersection.

A167459 \ A066737 = A166504 \ A152242.

A381259 Numbers obtained by concatenating powers of 2, sorted into increasing order.

Original entry on oeis.org

1, 2, 4, 8, 11, 12, 14, 16, 18, 21, 22, 24, 28, 32, 41, 42, 44, 48, 64, 81, 82, 84, 88, 111, 112, 114, 116, 118, 121, 122, 124, 128, 132, 141, 142, 144, 148, 161, 162, 164, 168, 181, 182, 184, 188, 211, 212, 214, 216, 218, 221, 222, 224, 228, 232, 241, 242, 244, 248, 256, 264

Offset: 1

Stefano Spezia, Feb 18 2025

Take the list {2^i: i >= 0} and concatenate its terms (allowing multiple copies) in any order; then sort the result into increasing order.

The term a(32) = 128 is a power of 2 as well as the concatenation of several powers of 2. - Rémy Sigrist, Feb 20 2025

			11 is a term because it is the concatenation of 1 = 2^0 with itself;
12 is a term because it is the concatenation of 1 = 2^0 with 2 = 2^1;
32 is a term because it is equal to 2^5;
168 is a term because it is the concatenation of 16 = 2^4 with 8 = 2^3.
0 is not a term because it is not a power of 2.

Rémy Sigrist, Table of n, a(n) for n = 1..11921
Rémy Sigrist, PARI program

Supersequence of A028846.
Some subsequences: A000079, A045507, A178664.
Cf. A152242.

PARI
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A166505 Numbers in A166504 which are not in A152242.

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A166506 Indices of terms in A166504 which are not in A152242.

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A165631 Numbers whose cube is a concatenation of primes, i.e., in A152242.

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A019549 Primes formed by concatenating other primes.

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A038692 Square numbers that are concatenations of two or more prime numbers.

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Mathematica

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A166504 Slime numbers: numbers which are the concatenation of primes, with "leading zeros" allowed.

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A066737 Composite numbers that are concatenations of primes.

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A166503 Numbers k with property that k^2 is the concatenation of two or more prime numbers.

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