cp's OEIS Frontend

A298156 Composite numbers n of which the sum of prime divisors of n (with repetition) equals the concatenation of two integers k and k + 1.

%I A298156 #18 Feb 16 2025 08:33:53
%S A298156 35,42,50,60,64,72,76,81,86,93,136,145,153,159,164,174,253,273,289,
%T A298156 325,365,385,390,416,438,462,468,488,494,497,549,550,555,559,592,644,
%U A298156 658,660,664,666,686,703,704,710,737,747,792,836,852,884,885,891,920,940,944,946,980
%N A298156 Composite numbers n of which the sum of prime divisors of n (with repetition) equals the concatenation of two integers k and k + 1.
%C A298156 Composite numbers n, of which A001414(n) (sum of prime divisors of n with repetition, sopfr(n)) is in sequence A001704 (numbers m which are the concatenation of k and k+1).
%H A298156 Charles R Greathouse IV, <a href="/A298156/b298156.txt">Table of n, a(n) for n = 1..10000</a>
%H A298156 K. Alladi and P. Erdős, <a href="http://projecteuclid.org/euclid.pjm/1102811427">On an additive arithmetic function</a>, Pacific J. Math., Volume 71, Number 2 (1977), 275-294.
%H A298156 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SumofPrimeFactors.html">Sum of Prime Factors</a>
%F A298156 sopfr(n) = k||k+1 when n is not prime and k is a positive integer.
%e A298156 35 = 5*7, sopfr(35) = 5+7 = 12, 12 =k||k+1 when k = 1.
%t A298156 Select[Range[10^3], And[CompositeQ@ #, Subtract @@ Map[FromDigits, TakeDrop[#, Floor[Length[#]/2]]] == -1 &@ IntegerDigits@ Total[Flatten@ Map[ConstantArray[#1, #2] & @@ # &, FactorInteger[#]]]] &] (* _Michael De Vlieger_, Jan 14 2018 *)
%o A298156 (PARI) is(n)=my(f = factor(n), sopfr = sum(i = 1, #f~, f[i, 1] * f[i, 2]); d = digits(sopfr), v); if((#d) % 2 == 0, v = vector(#d / 2); v[#v] = -1; return(vector(#d / 2, j, d[j]) - vector(#d / 2, #d / 2 + j, d[j]) == v), return(d == concat(digits(10^(#d \ 2) - 1), digits(10^(#d \ 2))))) \\ _David A. Corneth_, Jan 13 2018
%o A298156 (PARI) sopfr(n,f=factor(n))=sum(i=1,#f~, f[i,1]*f[i,2])
%o A298156 has(n)=my(d=digits(n),k=#d); digits(fromdigits(d[1..k\2])+1) == d[k\2+1..k]
%o A298156 list(lim)=my(v=List()); forfactored(n=35,lim\1, if(n[2][,2]!=[1]~ && has(sopfr(0,n[2])), listput(v,n[1]))); Vec(v) \\ _Charles R Greathouse IV_, Jan 15 2018
%Y A298156 Cf. A001414 (sum of prime divisors of n with repetition, sopfr(n)).
%Y A298156 Cf. A001704 (numbers which are the concatenation of k and k+1).
%K A298156 base,nonn
%O A298156 1,1
%A A298156 _Daniel Blaine McBride_, Jan 13 2018